Disentangling the multiorbital contributions of excitons by photoemission exciton tomography

Excitons are realizations of a correlated many-particle wave function, specifically consisting of electrons and holes in an entangled state. Excitons occur widely in semiconductors and are dominant excitations in semiconducting organic and low-dimensional quantum materials. To efficiently harness the strong optical response and high tuneability of excitons in optoelectronics and in energy-transformation processes, access to the full wavefunction of the entangled state is critical, but has so far not been feasible. Here, we show how time-resolved photoemission momentum microscopy can be used to gain access to the entangled wavefunction and to unravel the exciton’s multiorbital electron and hole contributions. For the prototypical organic semiconductor buckminsterfullerene (C60), we exemplify the capabilities of exciton tomography and achieve unprecedented access to key properties of the entangled exciton state including localization, charge-transfer character, and ultrafast exciton formation and relaxation dynamics.

Excitons, quasiparticles consisting of bound electron-hole pairs, are at the heart of the optoelectronic response of all organic semiconductors, and exciton formation and relaxation processes are largely responsible for energy conversion and light harvesting applications in these materials.At the atomic level, excitons are described by a two-particle correlated quantum-mechanical wavefunction that includes both the excited electron and the remaining hole.This wavefunction covers the complete shape of the exciton wave and thus provides access to a number of critical exciton properties such as the orbital character, the degree of (de)localization, the degree of charge separation, and whether this involves charge transfer between molecules.Consequently, in order to fully understand exciton dynamics and to exploit them in, e.g., an organic solar cell, an accurate and complete measurement of the exciton wavefunction would be ideal.Exemplary in this situation is the ongoing work to understand the optoelectronic response of C 60 , a prototypical organic semiconductor that is commonly used in organic solar cells 1-3 .Here, a topic of research has been the optical absorption feature that occurs at 2.8 eV for multilayer and other aggregated structures of C 60 4 .Interestingly, time-and angle-resolved photoelectron spectroscopy and optical absorption spectroscopy studies have indirectly found that this optical transition corresponds to the formation of charge-transfer excitons with significant electron-hole separation [5][6][7][8][9] .Although these hints are supported by time-dependent density functional theory calculations that show the importance of delocalized excitations in C 60 clusters [10][11][12] , quantitative measurements of the exciton localization and charge separation have so far not been possible.Thus, the C 60 case highlights the need for a more direct experimental access to the wavefunctions of the electron-hole pair excitations.
From an experimental point of view, our method of choice to access exciton wavefunctions is time-resolved photoemission orbital tomography (tr-POT, see Methods for the experimental realization used in this work) [13][14][15][16] .In POT, the comparison with density functional theory calculations (DFT) provides a direct connection between photoemission data and the orbitals of the electrons. 13The extension to the time-domain promises valuable access also to the spatial information of excited electrons.However, at least for organic semiconductors, it has not been explicitly considered that photoemission of excitons requires the break-up of the two-particle electron-hole pair and that only the photoemitted electron, but as correlated particles (shaded blue-red areas) with a blue sphere for the hole and a red sphere for the electron.We probe the excitons in C 60 with extreme ultraviolet (EUV) pulses (purple pulse in (a), hν = 26.5 eV) that break up the electron-hole pairs and photoemit the corresponding electrons (red spheres), of which we detect the kinetic energies and the momentum emission patterns (yellow-green-colored disks in (a)).The optical excitation of excitons in C 60 is known to lead to the formation of a decay sequence of singlet exciton states with varying charge-transfer character 5,6 (see (b), S i : i th singlet excited state, S 0 : ground state).We are able to measure these exciton dynamics and the corresponding orbital tomography momentum patterns by adjusting the temporal delay between the optical excitation and the EUV probe pulses.
not the hole, is directly detected.In fact, it is not clear to what extent (tr-)POT can be reasonably used for the interpretation and analysis of such strongly interacting correlated quasiparticles.Here we address this open question and show how tr-POT can probe the exciton wavefunction in the example system of a C 60 multilayer.
We employ our recently developed setup for photoelectron momentum microscopy [17][18][19] and use ultrashort laser pulses to optically excite bright excitons in C 60 thin films that were deposited on Cu(111) (measurement temperature T ≈ 80 K; see Methods and Figure 1a).In the time-resolved photoemission experiment, we detect the energy and momentum emission pattern of the photoemitted electrons, which were initially part of the bound electron-hole pairs, i.e. the excitons.Following the time-evolution of the photoelectron spectrum, we can observe how the optically excited states relax to energetically lower-lying dark exciton states with different localization and charge-transfer character 5,6 (Figure 1b and data in Extended Fig. 6).In addition to the energy relaxation, we collect tr-POT data, and investigate in how far these patterns can be used to access real-space properties of the exciton wavefunctions.
Specifically, we will address two questions: which orbitals contribute to the formation of the excitons and how this key information is imprinted in the energy-and momentum-resolved photoemission spectra.

A. The exciton spectrum of buckminsterfullerene C 60
To lay the foundation for our study, we first discuss the theoretical electronic properties of the C 60 film.On top of a hybrid-functional DFT ground state calculation, we obtain the exciton spectrum by employing the many-body framework of GW and Bethe-Salpeter-Equation (GW +BSE) calculations (see Methods for full details).As shown in Fig. 2a, we model the C 60 low-temperature phase by the two symmetry-inequivalent C 60 dimers 1-2 and 1-4, respectively 20 , which are properly embedded to account for polarization effects in the film.The calculated single-particle energy levels are shown in Fig. 2b, where we group the electron removal and electron addition energies into four bands, denoted as HOMO-1, HOMO, LUMO, and LUMO+1 according to the orbitals of the parent orbitals of an gasphase C 60 molecule.These manifolds consist of 18, 10, 6, and 6 energy levels per dimer, respectively, originating from the g g +h g , h u , t 1u , and t 1g irreducible representations of the gas phase C 60 orbitals 21 .We emphasize that the calculated GW ionization levels of HOMO and HOMO-1 of 6.7 eV and 8.1 eV are in excellent agreement with experimental data for this C 60 film (see SI and Ref. 22).
Building upon the GW single-particle energies, we solve the Bethe-Salpeter equation and compute the energies Ω m of all correlated electron-hole pairs (excitons).The resulting absorption spectrum (bottom panel of Fig. 2c) agrees well with literature 4 .In addition, we obtain the weights X (m) vc on the specific electron-hole pairs that coherently contribute to the m th exciton state, from which the exciton wavefunctions are constructed in the Tamm-Dancoff approximation as follows: This means that each exciton ψ m with energy Ω m consists of a weighted coherent sum of multiple electron-hole-transitions φ v (r h )χ c (r e ) each containing one electron orbital χ c and one hole orbital φ v .
To gain more insight into the character of the excitons ψ m , we qualitatively classify them according to the most dominant orbital contributions that are involved in the transitions.This is visualized in the four sub-panels above the absorption spectrum in Fig. 2c.For a given exciton energy Ω m , the black bars in each sub-panel show the partial contribution |X (m) v,c | 2 of characteristic electron-hole transitions φ v (r h )χ c (r e ) to a given exciton ψ m .Looking at individual sub-panels, we see first that characteristic electron-hole transitions can belong to different excitons ψ m that have very different exciton energies Ω m .For example, the blue panel in Fig. 2c shows the contributions of HOMO→LUMO (abbreviated H→L) transitions as a function of exciton energy Ω m , and we see that these transitions contribute to excitons that are spread in energy over a scale of more than 1 eV (from Ω m ≈ 1.7 eV to 3 eV).This spread of H→L contributions (and also H−n→L+m contributions) is caused by the fact that there are already many orbital energies per dimer (cf.Fig. 2b) which combine to form excitons with different degrees of localization and delocalization of the electrons and holes on one or more molecules.
We now focus on four exciton bands of the C 60 film, denoted as S 1 -S 4 , which are centered around Ω S 1 , Ω S 2 , Ω S 3 and Ω S 4 at 1.9, 2.1, 2.8 and 3.6 eV, respectively (cf.Ref. 6).It is important to emphasize that each exciton band S 1 -S 4 arises from many individual excitons ψ m with similar exciton energies Ω m within the exciton band.Looking again at the subpanels, we see that the S 1 and S 2 exciton bands are made up of excitons ψ m that are almost exclusively composed of transitions from H→L.On the other hand, the S 3 shows in addition to H→L also significant contributions from H→L+1 transitions (pink-dashed panel).The S 4 exciton band can be characterized as arising from H→L+1 (pink-dashed panel) and H−1→L (orange-dash-dotted panel) as well as transitions from the HOMO to several higher lying orbitals denoted as H→L+n (yellow-dotted panel).We emphasize that although orbitals from several different GW energies contribute, e.g., to an exciton in the S 4 band, the exciton energy Ω m of each exciton ψ m has a single well-defined value.

B. Photoemission signatures of multiorbital contributions
In the following, we investigate whether these theoretically predicted multiorbital characteristics of the excitons can also be probed experimentally.As will be shown below, time-resolved photoemission spectroscopy can indeed provide access not only to the dark exciton landscape [23][24][25][26][27] , but also to the distinct orbital contributions of exciton states.A key step in extracting this information from the experimental data lies in a thorough compar-ison with simulations that specifically consider the pump-probe photoemission process, a topic that has recently attracted increased attention [28][29][30][31] .Here, we rely on the formalism of Kern et al. 32 , which is based on a common Fermi's golden rule approach to photoemission 33 .
Assuming the exciton of Eq. (1) as the initial state and applying the plane-wave final state approximation of POT, the photoemission intensity of the exciton ψ m is formulated as Here A is the vector potential of the incident light field, F the Fourier transform, k the photoelectron momentum, hν the probe photon energy, ε v the v th ionization potential, Ω m the exciton energy, and E kin the energy of the photoemitted electron.Note that ε v directly indicates the final-state energy of the left-behind hole.In the context of our present study, delving into Eq.( 2) leads to two striking consequences, which we discuss in the following.
First, we illustrate the consequences of the multiorbital character of the exciton states on the photoelectron spectrum, and sketch in Fig. 2d the typical single-particle energy level diagrams for the HOMO and LUMO states and then indicate the contributing orbitals to the two-particle exciton state by blue holes and red electrons in these states, respectively.
For the S 1 exciton band (left panel), we already found that the main orbital contributions to the band are of H→L character (Fig. 2d, left, and cf.In the case of the S 3 exciton band, we find that in contrast to the S 1 and S 2 excitons not only H→L, but also H→L+1 transitions contribute (Fig. 2d, middle panel, and cf.Finally, for the S 4 exciton band at Ω S 4 = 3.6 eV, we find three major contributions (Fig. 2d, right panel), where not only the electrons but also the holes are distributed over two energetically different levels, namely the HOMO (cf.pink-dashed and yellow-dotted panels in Fig. 2c,d) and the HOMO-1 (cf.orange-dash-dotted panels in Fig. 2c,d).Thus, there are two different final states available for the hole, each with a different binding energy.Consequently, the photoemission spectrum of S 4 is expected to exhibit a doublepeak structure with intensity appearing ≈ 3.6 eV above the HOMO kinetic energy E H , and ≈ 3.6 eV above the HOMO-1 kinetic energy E H−1 , as illustrated in the right-most panel of Fig. 2d.Relating this specifically to the single-particle picture of our GW calculations, the two peaks are predicted to have a separation of ε H−1 -ε H = 8.1 -6.7 = 1.4 eV.
In addition, Eq. ( 2) now also provides the theoretical framework for interpreting momentumresolved tr-POT data from excitons.Ground state POT can be easily understood in terms of the Fourier transform F of single-particle orbitals.A naive extension to excitons might imply an incoherent, weighted sum of all LUMO orbitals χ c contributing to the exciton wavefunction.However, as Eq. ( 2) shows, such a simple picture proves insufficient.Instead, the momentum pattern of the exciton wavefunction is related to a coherent superposition of the electron orbitals χ c weighted by the electron-hole coupling coefficients vc .The implications of this finding are sketched in the k x -k y plots in Fig. 2d and are most obvious for the S 3 band.Here, the exciton is composed of transitions with a common hole position, i.e., H→L and H→L+1, leading to a coherent superposition of all 12 electron orbitals from the LUMO and LUMO+1 in the momentum distribution.In summary, multiple hole contributions can be identified in a multi-peak structure in the photoemission spectrum, and multiple electron contributions will result in a coherent sum of the electron orbitals that can be identified in the corresponding energy-momentum patterns from tr-POT data.
These very strong predictions about multi-peaked photoemission spectra due to multiorbital excitons can be directly verified in an experiment on C 60 by comparing spectra for resonant excitation of either the S 3 or the S 4 excitons (cf.Fig. 2).The corresponding experimental data are shown in Fig. 3a and 3c, respectively.Starting from the excitation of the S 3 exciton band with hν = 2.9 eV photon energy (which is sufficiently resonant to excite the manifold of exciton states that make up the S 3 band around Ω S 3 = 2.8 eV), we can clearly identify the direct excitation (at 0 fs delay) of the exciton S 3 feature at an energy of E ≈ 2.8 eV above the kinetic energy E H of the HOMO level.Shortly after the excitation, additional photoemission intensity builds up at E − E H ≈ 2.0 eV and ≈ 1.7 eV, which is known to be caused by relaxation to the S 2 and S 1 dark exciton states 6 and is in good agreement with the theoretically predicted energies of E − E H ≈ 2.1 eV and ≈1.9 eV.
Changing now the pump photon energy to hν = 3.6 eV for direct excitation of the S 4 exciton band (Fig. 3c), two distinct peaks at ≈ 3.6 eV above the HOMO and the HOMO-1 are expected from theory.While photoemission intensity at E − E H ≈ 3.6 eV above the HOMO level is readily visible in Fig. 3c, the second feature at 3.6 eV above the HOMO-1 is expected at E − E H ≈ 2.2 eV above the HOMO level (corresponding to E − E H−1 ≈ 3.6 eV) and thus almost degenerate with the aforementioned S 2 dark exciton band at about E − E H ≈ 2.0 eV, which appears after the optical excitation due to relaxation processes.Therefore, we need to pinpoint this second H−1→L contribution to the S 4 exciton at the earliest time of the excitation.Indeed, a closer look around 0 fs delay shows additional photoemission intensity at about E − E H ≈ 2.2 eV.Using difference maps (Fig. 3b) and direct comparisons of energy-distribution-curves at selected time-steps (Fig. 3d), we clearly find a double-peak structure corresponding to the energy difference of ≈1.4 eV of the HOMO and HOMO-1 levels.Thereby, we have shown that photoelectron spectroscopy, in contrast to other techniques (e.g., absorption spectroscopy), is indeed able to disentangle different orbital contributions of the excitons.In this way, we have validated the theoretically predicted multi-peak structure of the multiorbital exciton state that is implied by Eq. ( 2).We also see that the photoelectron energies in the spectrum turn out to be sensitive probes of the corresponding hole contributions of the correlated exciton states.
We note that the signature of the S 3 excitons, even if not directly excited with the light pulse in this measurement, is still visible and moreover with significantly higher intensity than the multiorbital signals of the resonantly excited S 4 exciton band.This observation strongly suggests that there is a very fast relaxation from the S 4 exciton to the S 3 exciton, with relaxation times well below 50 fs (see Extended Fig. 6).

D. Time-resolved photoemission orbital tomography of exciton wavefunctions
Based on the excellent agreement between the experiment and the GW +BSE theoretical results, we are now ready to investigate to what extent the momentum patterns from tr-POT data of excitons in organic semiconductors contain information about the real-space spatial distribution of the exciton wavefunction.In the experiment, we once again excite the S 3 exciton band in the C 60 film with hν = 2.9 eV pump energy, and we now use femtosecond tr-POT to collect the momentum fingerprints of the directly excited S 3 excitons around 0 fs and the subsequently built-up dark S 2 and S 1 excitons that appear in the exciton relaxation cascade in the C 60 film (see Fig. 4a-c, where the momentum map of the lowest energy S 1 exciton band is plotted in (a), the S 2 in (b) and the highest energy S 3 exciton band in (c); see Extended Fig. 6 for time-resolved traces of the exciton formation and relaxation dynamics).We note that the collection of these data required integration times of up to 70 hours, and that a measurement of the comparatively low-intensity S 4 feature when excited with hν = 3.6 eV has not yet proved feasible.For the interpretation of the collected POT momentum maps from the S 1 , S 2 , and S 3 excitons, we also calculate the expected momentum fingerprints for the wavefunctions obtained from the GW +BSE calculation for both dimers, each rotated to all occurring orientations in the crystal.Finally, for the theoretical momentum maps, we sum up the photoelectron intensities of each electron-hole transition in an energy range of 200 meV centered on the exciton band.The results are shown in Fig. 4d-f below the experimental data for direct comparison.
First, we observe that the experimental momentum maps of the S 1 and S 2 states are largely similar (Fig. 4a,b), showing six lobes centered at k ≈ 1.2 Å-1 .These features, as well as the energy splitting between S 1 and S 2 (cf.Fig. 2c), are accurately reproduced by the GW +BSE prediction (Fig. 4d,e).Furthermore, also the GW +BSE calculation shows very similar momentum maps for S 1 and S 2 , suggesting a similar spatial structure of the excitons.This is in contrast to a naive application of static POT to the unoccupied orbitals of the DFT ground state of C 60 , which does show a similar momentum map for the LUMO, but cannot explain a kinetic energy difference in the photoemission signal, nor give any indication of differences in the corresponding exciton wavefunctions.With this agreement between experiment and theory, we now extract the spatial properties of the GW +BSE exciton wavefunctions.To visualize the degree of charge-transfer of these two-particle exciton wavefunctions ψ m (r h , r e ), we integrate the electron probability density over all possible hole positions r h , considering only hole positions at one of the C 60 molecules in the dimer.This effectively fixes the hole contribution to a particular C 60 molecule (blue circles in Fig. 4g-i indicate the boundary of considered hole positions around one molecule, hole distribution not shown), and provides a probability density for the electronic part of the exciton wavefunction in the dimer, which we visualize by a yellow isosurface (see Fig. 4g-i).Obviously, in the case of S 1 and S 2 (Fig. 4g,h), when the hole position is restricted to one molecule of the dimer, the electronic part of the exciton wavefunction is localized at the same molecule of the dimer.Our calculations thus suggest that the S 1 and S 2 excitons are of Frenkel-like nature.Their energy difference originates from different excitation symmetries possible for the H→L transition (namely t 1g , t 2g , and g g for the S 1 and h g for the S 2 ). 12 In contrast to the S 1 and S 2 excitons, the momentum map of the S 3 band shows a much more star-shaped POT fingerprint in both theory and experiment (Fig. 4c,f).This is to be expected, since the electronic part of the S 3 excitons contains not only contributions from the LUMO orbital, but also contributions from the LUMO+1 orbital.Note, however, that we find the experimentally observed star-shaped pattern to be only partially reproduced by the GW +BSE calculation.An indication towards the cause of this discrepancy is found by considering the electron-hole separation of the excitons making up the S 3 band.Here, we find that the positions of the electron and the hole contributions are strongly anticorrelated (Fig. 4i), with the electron confined to the neighboring molecule of the dimer.In fact, the mean electron-hole separation is as large as 7.6 Å, which is close to the core-to-core distance of the C 60 molecules.Although these theoretical results confirm the previouslyreported charge-transfer nature of the S 3 excitons 5,6 , they also reflect the limitations of the C 60 dimer approach.Indeed, the dimer represents the minimal model to account for an intermolecular exciton delocalization effect, but it cannot fully account for dispersion effects 22 (cf.Extended Fig. 5), which are required for a quantitative comparison with experimental data.Besides the discrepancy in the S 3 momentum map, this could also be an explanation why the S 2 in the present work is of Frenkel-like nature, but could have charge-transfer character according to previous studies 5,6 .However, future developments will certainly allow scaling up of the cluster size in the calculation, so that exciton wavefunctions with larger electron-hole separation can be accurately described.Most importantly, we find that the present dimer GW +BSE calculations are clearly suited to elucidate the multiorbital character of the excitons, which is an indispensable prerequisite for the correct interpretation of tr-POT data of excitons in organic semiconductors.

III. CONCLUSION
In conclusion, we have shown how the energy-and momentum-resolved photoemission spectrum of excitons in an organic semiconductor depends on the multiorbital nature of these excitons.By extending POT to fully-interacting exciton states calculated in the framework of the Bethe-Salpeter equation, we found that the energy of the photoemitted electron of the exciton quasiparticle is determined by the position of the hole and the exciton energy in combination with the probe photon energy.This leads to the prediction of multiple peaks in the photoelectron spectrum, which we verify experimentally, and allows disentangling the differ-ent orbital contributions, the wavefunction localization, and the charge-transfer character.
Similarly, the momentum fingerprint provides access to the electron states that make up the exciton.Most importantly, we introduce time-resolved photoemission orbital tomography as a key technique for the study of exciton wavefunctions in organic semiconductors.We apply full multidimensional time-and angle-resolved photoelectron spectroscopy (tr-ARPES) to a multilayer C 60 crystal evaporated onto Cu(111), where the film thickness was such that no photoemission signature of the underlying Cu(111) could be observed in our experiment.We verified the sample quality by performing momentum microscopy of the occupied HOMO and HOMO-1 states simultaneously to the measurement of the excited states (see Extended Fig. 5).Femtosecond exciton dynamics were induced using ≈100 fs, hν = 2.9 eV or ≈100 fs, hν = 3.6 eV laser pulses derived from the frequency-doubled output of a optical parametric amplifier.The exciton dynamics were probed using our custom photoemission momentum microscope with a 500 kHz ultrafast 26.5 eV extreme ultraviolet (EUV) light source 18 that enables us to map the photoelectron momentum distribution over the full photoemission horizon in a kinetic energy range exceeding 6 eV and an overall time resolution of ≈100 fs (the EUV pulse length is about 20 fs).The pump fluence was set to 90 (10) µJ/cm 2 and 20(5) µJ/cm 2 for the hν = 2.9 eV and the hν = 3.6 eV measurement, respectively.To prevent the free rotation of C 60 molecules, we cooled the sample down to ≈80 K 20 .In addition to the resulting long-range periodic ordering of the C 60 crystal, cooling was also observed to prevent light-induced polymerization.

Space-charge effects
In the momentum microscopy experiment, a balance has to be found between sufficiently low pump and probe light intensities to avoid space-charge effects, but also having sufficient intensity for the optical excitation (pump) and reasonably short integration times (probe).
Most of the time, for our settings, small space-charge effects are present in the data, but do not lead to strong distortions in the band-structure data and can be easily corrected.
Therefore, the first step in the data analysis was to subtract a space-charge-induced delayand momentum-dependent kinetic energy shift.For this purpose, the central kinetic energy of the HOMO was determined for normalization.To avoid the influence of the C 60 crystal C. Calculation of the C 60 exciton spectrum The ab initio calculation of the exciton spectrum of the C 60 film was performed in two steps, using a GW +BSE approach.For the static electronic structure, we perform calculations for two unique C 60 dimers, which have been extracted from the known structure of the molecular film 36,37 (see Fig. 2c, dimers 1-2 and 1-4 respectively).Starting from Kohn-Sham orbitals and energies of a ground state DFT calculation (6-311G*/PBE0+D3) [38][39][40][41] using ORCA 5.0.1 42,43 , we employ the Fiesta code 44 to self-consistently correct the molecular energy levels by quasi-particle self-energy calculations with the GW approximation.To account for polarization effects beyond the molecular dimer, we embed the dimer cluster in a discrete polarizable model using the MESCal program [45][46][47] .We found that mimicking 2 layers of the surrounding C 60 film in such a way resulted in the convergence of the band gap within 0.1 eV with a removal energy from the highest valence level of 6.65 eV.The close agreement of this quasi-particle energy with the experimentally determined work function of 6.5 eV gives us additional confidence in the choice of our embedding environment.The In line with our classification of the GW energy levels, one can group the composition of the excitons into four categories according to the contributing quasi-particle energy levels.
As visualized in Fig. 2c, this shows that S 1 and S 2 are almost completely described by HOMO to LUMO transitions.On the other hand, S 3 is predicted to have a small contribution of HOMO-1 character, however this contribution is too small to be reliably measured in our experiment.Finally, for S 4 we observe a clear and almost equal mixture of HOMO-1 and HOMO contributions, which is also confirmed by our measurements.Contributions from lower lying valence bands are negligibly small in the studied energy window.

D. Calculation of the exciton momentum maps
Based on our Kohn-Sham orbitals and BSE excitation coefficients, we calculate theoretical momentum maps for each exciton according to Eq. 2 following the derivation of Kern et al. 32 .Note that for better readability, Eqs.(1) and ( 2) are given within the TDA, however, can in general be extended to include also de-excitation terms. 32In the present case, we found the de-excitation contributions to be marginal (below 1%) without affecting the appearance of the momentum maps and our interpretation.The energy conservation term comprises the BSE excitation energies (Ω m ), the GW quasi-particle energies for electron removal (i.e.ionization potential ε v ), and the probe energy of hν = 26.5 eV in accordance with our experimental setup.Furthermore, we include an inner potential to correct for the photoemission intensity variation of 3D molecules along the moment vector component perpendicular to the surface.Here, we choose a value of 12.5 eV, which has already been shown to match with experimental C 60 data 22,49 .(Note that while considering the inner potential of the film is essential to describe the ARPES fingerprint, a variation of the inner potential between 12 and 14 eV indicated no influence on the interpretation of our results.)As we exploited the plane-wave approximation, the calculated photoemission intensity is modulated

FIG. 1 .
FIG. 1. Schematic overview of time-resolved photoemission orbital tomography of exciton states in C 60 .a, b, a femtosecond optical pulse (blue pulse and blue arrow in (a) and (b), respectively) excites optically bright excitons in a C 60 film.The exciton electron-hole pairs are sketched in (a)

FIG. 2 .
FIG.2.Ab-initio calculation of the electronic structure and exciton spectrum of C 60 dimers in a crystalline multilayer sample.a, the unit cell for a monolayer of C 60 , for which GW +BSE calculations for the dimers 1-2 and 1-4 were performed.b, electron addition/removal single-particle energies as retrieved from the self-consistent GW calculation.These energies directly provide ε v in Eq. (2).c, results of the full GW +BSE calculation, showing as a function of the exciton energy Ω from bottom to top: the calculated optical absorption, the exciton band assignment S 1 -S 4 , and the relative contributions to the exciton wavefunctions of different electron-hole pair excitations φ v (r h )χ c (r e ).Full details on the calculations are given in the Methods section.d, sketch of the composition of the exciton wavefunction of the S 1 -S 4 bands and their expected photoemission signatures based on Eq. (2).In order to visualize the contributing orbitals, blue holes and red electrons are assigned to the single-particle states as shown in (b).
photoelectron depends on the ionization energy of the involved HOMO hole state ε v = ε H and the correlated electron-hole pair energy Ω ≈ Ω S 1 .Therefore, we expect to measure a single photoelectron peak, as shown in the lower part of the left panel of Fig.2d.In the case of the S 2 exciton the situation is similar, since the main orbital contributions are also of H→L character.However, since the S 2 exciton band has a different energy Ω S 2 , the photoelectron peak is located at a different kinetic energy with respect to the S 1 peak.

Fig
Fig 2c, blue and pink-dashed panels, respectively).However, we still expect a single peak in the photoemission, because the same hole states are involved for both transitions (i.e., same ε v = ε H in the sum in Eq. 2), and all orbital contributions have the same exciton

FIG. 3 .
FIG. 3. a-d, comparison of the time-resolved photoelectron spectra of multilayer C 60 for hν = 2.9 eV excitation and hν = 3.6 eV excitation ((a) and (c), respectively), both normalized and shifted in time to match the intensity of the S 3 signals (see Methods for full details on the data analysis).As can be seen in the difference (b), for hν = 3.6 eV pump we observe an enhancement of the photoemission yield around E − E H ≈ 3.6 eV as well as around E − E H ≈ 2.2 eV.We attribute this signal to the S 4 exciton band, which has hole contributions stemming from both the HOMO and the HOMO-1.To further quantify the signal of the S 4 exciton, (d) shows energy distribution curves for both measurements at early delays, showing the enhancement in the hν = 3.6 eV measurement.

FIG. 4 .
FIG.4.a-f, comparison of the (a-c) experimental momentum maps acquired for the three exciton bands observed in C 60 with the (d-f ) predicted momentum maps retrieved from GW +BSE.Note that the center of the experimental maps could not be analyzed due to a space-charge-induced background signal in this region (gray area, see Methods).g-i, isosurfaces of the integrated electron probability density (yellow) within the 1-2 dimer for fixed hole positions on the bottom-left molecule (blue circle) of the dimer for the (g) S 1 , the (h) S 2 , and the (i) S 3 exciton bands.
IV. ACKNOWLEDGEMENTSThis work was funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) -432680300/SFB 1456, project B01 and 217133147/SFB 1073, projects B07 and B10.G.S.M.J. acknowledges financial support by the Alexander von Humboldt Foundation.A.W., C.S.K., and P.P acknowledge support from the Austrian Science Fund (FWF) project I 4145.The computational results presented were achieved using the Vienna Scientific Cluster (VSC) and the local high-performance resources of the University of Graz.R.H., M.A., and B.S. acknowledge financial support by the DFG -268565370/TRR 173, projects B05 and A02.B.S. acknowledges further support by the Dynamics and Topology Center funded by the State of Rhineland-Palatinate. V. AUTHOR CONTRIBUTIONS D.St., M.R., S.S., M.A., B.S., P.P., G.S.M.J. and S.M. conceived the research.W.B., D.Sch.and J.P.B. carried out the time-resolved momentum microscopy experiments.W.B. analyzed the data.W.B. and R.H. prepared the samples.A.W., C.S.K., G.D.A, X.B. and P.P. performed the calculations and analyzed the theoretical results.All authors discussed the results.G.S.M.J. and S.M. were responsible for the overall project direction and wrote the manuscript with contributions from all co-authors.
calculated quasi-particle energy levels are shown in Fig.2a.Here the finite width of the black bars actually arises from multiple energy levels forming bands on the energy axis.We characterize them according to symmetry 21 as HOMO-1, HOMO, LUMO, and LUMO+1 bands, each consisting of 18, 10, 6 and 6 energy levels per dimer, respectively.Note that we combine HOMO-1 is made up by states from two different irreducible representations of the isolated gas phase C 60 molecule which are practically forming a single band.Building upon the GW energies, we compute neutral electron-hole excitations by solving the Bethe-Salpeter equation beyond the Tamm-Dancoff approximation (TDA).This yields the excitation energies Ω m and the electron-hole coupling coefficients X (m) vc for a series of excitons labelled with m.We first analyze the resulting optical absorption spectrum which is shown in Fig.2cas black solid line.It reveals a prominent absorption band around hν = 3.6 eV that is well-known from gas-phase spectroscopy48 .Secondly, the dimer calculation reveals a strong optical absorption at hν = 2.8 eV as well as a weakly dipole-allowed transition at hν = 2 eV.Both of these transitions are known to only appear in aggregated phases of C 604 , and cannot be understood by considering only a single C 60 molecule.We refer to the exciton bands around Ω = 1.9, 2.1, 2.8 and 3.6 eV as S 1 , S 2 , S 3 and S 4 , respectively.

22 FIG. 6 .
photoemission horizon of the theoretical momentum maps.Finally, to arrive at the theoretical momentum maps shown in Fig.4, we sum up the photoelectron intensities of each contributing electron-hole transition in an kinetic energy range of 200 meV centered on the respective exciton band.