For a long time, physicists and chemists have dreamed of experimentally imaging molecular orbitals, which contain information on the spatial distribution of the electrons at a certain energy. To date, experimental methods for this purpose include femtosecond laser spectroscopy and scanning probe microscopy, which have gained widespread interest but also have some limitations. For example, high-order harmonic generation process based on ultrashort laser pulses is limited to the study of simple gas-phase molecules, and methods based on scanning probe microscopy require extremely low temperature to prevent molecular diffusion. In comparison, photoemission orbital tomography (POT) is based on angle-resolved photoemission spectroscopy (ARPES) at room temperature with less restricted experimental conditions. As a combined experimental and theoretical technique, POT focuses on establishing a direct link between the photoelectron angular distribution and the initial-state molecular orbital structure. By using a relatively simple plane-wave approximation, one can perform accurate analysis of the angle-resolved photoemission spectra for π-conjugated molecules on surface to study their respective properties.

Progress Since POT deals with the photoemission experimental data, the correct description should be Dyson orbitals. Because of the fact that Dyson orbitals closely resemble one-electron Kohn-Sham orbitals calculated from density functional theory (DFT), the simple one-electron picture is often used. By approximating the final state of photoelectrons by a plane wave, it can be shown that the photocurrent arising from one particular initial state, appearing as an intensity map (so-called k-map) in momentum space, is proportional to the Fourier transform (FT) of the initial state wave function (Fig. 1). This tomographic relationship can be refined by taking two additional aspects into consideration: the first is to treat the final state with a damped plane wave in the z direction (Fig. 2), and the second is to include a modulating term of polarization-dependent geometry factor (Fig. 3). Experimentally, it is favorable to obtain a large detectable momentum-space range for the applications of POT. Suitable ARPES apparatuses include photoemission electron microscope, NanoESCA, toroidal electron analyzer, and time-of-flight momentum microscope.

The accuracy and precision of the POT method have been proved by accumulating results in the past decade and several applications are introduced. Agreement between experimental ARPES results and calculated k-maps can be well utilized to determine the molecular orientation, if all orientation domains are properly considered. On the other hand, one could also infer the molecular orientations by comparing the measured k-maps with the theoretical ones based on the single molecule (Fig. 4). POT, being capable of mapping the angular distribution of the wave function for each peak in conventional photoemission spectra, is an ideal technique to unambiguously identify the energy levels of molecular orbitals (Fig. 5). If multiple molecular orbitals coexist in a small energy window, POT can provide an orbital-by-orbital characterization via deconvolution and create benchmark for ab initio electronic structure theory (Fig. 6). Efforts have also been made to regain the lost phase information of orbital in the FT process (Fig. 7) and even reconstruct the orbital in real space (Fig. 8). Quantitative studies using POT have covered topics such as intra- and intermolecular band dispersion (Fig. 9), delocalized π-states and localized σ-states, nonplanar molecules, electron-phonon coupling (Fig. 10), aromaticity, and the list continues.

Conclusions and Prospects The overall experimental simplicity of POT and the convenience to approach molecular orbital and its Fourier transform under the plane-wave final state approximation are key reasons for the wide application of the technique. Albeit not applicable to some data from photoelectron diffraction and circular dichroism experiments, this approximation can provide a new perspective to study physical processes such as orbital hybridization by analyzing the discrepancy between theory and experiment. Future research on POT will improve the theoretical basis of the technique by implementing the time-dependent density functional theory (TDDFT), providing more accurate predictions of the photoelectrons’ momentum distribution. In combination with ultrafast lasers, POT has great potential to be extended into the time domain. For example, real-time observation of molecular orbitals in momentum space during excitation processes (Fig. 11) or chemical reactions will greatly improve the understanding of ultrafast dynamics. With the help of advanced momentum microscopy, such as spin-resolved ARPES and NanoARPES, it is likely to further enhance the characterization capabilities of POT in momentum and real spaces and expand its application in materials science, physical chemistry, and nanophotonics.

Development of nanotechnology in the last few decades have witnessed the miniaturization of semiconductor devices and new challenges such as building a quantum computer using novel physical phenomena. On the fundamental level, however, many aspects in the understanding of physical properties at interfaces between materials are still in the exploration stage, and precise experimental and theoretical descriptions of complex surface structures are always difficult but crucial for practical applications. In the subfield of organic electronics that studies the solid-molecule interfaces, such as in organic light-emitting diode (OLED) displays or organic solar cells, the frontier orbitals of molecules are prime determinants of chemical, optical and electronic properties in the devices.