Scientists have always sought ways to explore the world on ever smaller length- and time-scales. Watching electrons moving in molecules whilst being excited by light is an ultimate chemist’s dream. This is because frontier orbitals are responsible for chemical reactivity and optical properties, and it would be very exciting to visualise them in space and time. In 2009, Puschnig et al.  spotted a remarkable relation fulfilled for organic molecules composed of light atoms. Namely, they demonstrated that angle-resolved photoelectron spectroscopy (ARPES) data are related to the squared modulus of the Fourier transform of molecular orbitals. The orbitals may be thereby reconstructed from experimental data, provided its phase distribution is known. This is the so-called phase problem, which consists in the reconstruction of an object distribution from the modulus of its Fourier transform. This mathematical problem is well known in optics and is solved computationally by means of phase retrieval algorithms . The method of computational reconstruction of molecular orbitals from photoemission data with phase retrieval algorithms is called orbital tomography. The requirements are the absence of scattering in the photoelectron final state, oversampling of experimental data data and uni-directional orientation of adsorbed molecules on the surface of a substrate.
In our work, published in Nature Communications, we initially aimed at reconstruction of frontier orbitals of cobalt-pyrphyrin and pyrphyrin molecules adsorbed on a silver substrate. Cobalt-pyrphyrin exhibits catalytic activity in the light-driven hydrogen evolution reaction and its electronic structure is thus of particular interest. We prepared a monolayer of molecules on a monocrystalline silver substrate and verified the coverage with low energy electron diffraction (LEED) and x-ray photoelectron spectroscopy. Upon acquisition of the ARPES data (Fig. 1a), followed by a few image processing steps, we successfully reconstructed real space images using our reconstruction procedures [3, 4]. Those images were expected to be the valence orbitals probed in the experiment and are shown in Figs. 1b and 1c. While the reconstructed images looked similar to the orbitals simulated using density functional theory (DFT) (Figs. 1d and 1e), we failed to establish a one-to-one correspondence with the momentum space data (Figs. 1a and 1f).
According to the DFT simulation, the molecules were likely to adsorb in several adsorption geometries. However, we found out that no single adsorption geometry alone could reproduce both reconstructed real space and experimental momentum data. By rigorous analysis of the information available from ARPES, LEED and DFT, we came to conclusion the molecules must be oriented in three adsorption domains, with the CN-groups located at 0° and ±39° with respect to the  high symmetry direction of the substrate. Thereby, we discovered a method to fully determine adsorption geometry as well as both energetics and spatial distributions of the valence electronic states.
It is important to note that, whilst solving the phase problem, one will always be able to reconstruct some amplitude and phase distributions in real space, provided the oversampling conditions are fulfilled. This means that the origin of the data used as an input for phase retrieval algorithms must always be clarified prior to any interpretation of reconstructed real space distributions. At the present moment, we could not find any reasonable interpretation for real space distributions reconstructed from ARPES data in the presence of multiple rotational domains. However, it might be possible to disentangle the photoelectron distributions from single rotational domains by means of independent component analysis . This statistical method decomposes a complex data set into its independent constituent parts. These parts have to be statistically independent, i.e. one of the pixel values of any one of the components must contain no information on the pixel values of the other components. In practice, this means that the experimental data must be recorded from multiple samples, in which molecular rotational domains are non-interacting and have different compositions (for example, in one sample, 20% of molecules in one domain and 80% in the other and, in another sample, 50% of molecules in one domain and 50% in the other). The use of statistical methods may enable computational reconstruction of molecular orbitals from experimental ARPES data even in the presence of multiple adsorption geometries. This is important for validation of the existing theories and is especially crucial in case of time-resolved studies when the spatial distribution and temporal evolution of orbitals may be difficult to predict theoretically.
These results were recently published in Nature Communications: Kliuiev, P., Zamborlini, G., Jugovac, M. et al. Combined orbital tomography study of multi-configurational molecular adsorbate systems. Nat Commun 10, 5255 (2019).
 P. Puschnig et al. Reconstruction of Molecular Orbital Densities from Photoemission Data, Science 326, 702 (2009).
 J. R. Fienup. Reconstruction of an object from the modulus of its Fourier transform, Opt. Lett. 3, 27 (1978).
 P. Kliuiev et al. Application if iterative phase-retrieval algorithms to ARPES orbital tomography data, New J. Phys. 18, 093041 (2016).
 P. Kliuiev et al. Algorithms and image formation in orbital tomography, Phys. Rev. B 98, 085426 (2018).
 A. Hyvärinen et al. Independent component analysis (Adaptive and Cognitive Dynamic Systems: Signal Processing, Learning, Communications and Control), Wiley (2001).