Molecular Orbital Tomography: Reconstruction of phase symmetry from circular dichrosim experiments

Imaging of molecular orbitals by angle resolved photoelectron spectroscopy (ARPES) has emerged as a particularly powerful tool for the investigation of molecular materials, and the photoelectron emission microscopy (PEEM) at NanoESCA beamline is an ideal set-up to measure a ARPES map within a single image. By mapping the angle dependent intensity patterns of photoelectrons this technique provides tomographic images of the density distribution of the respective orbitals in k-space and thus provides fascinating insight into the properties of molecules and interfaces. For molecular species it has been shown that the ARPES signals can be described using a plane wave approximation for the photoemission final state. This provides a relatively simple interpretation of ARPES data, since the angle dependent photoemission intensity becomes proportional to the square of the Fourier transformation of the real space molecular orbital.
It is an intrinsic problem of the measurement process, though, that the information about the phase of the respective wave function is generally lost during the experiment. This phase, however, is a key parameter for the full orbital information, and it is also mandatory for the reconstruction of the orbital in real space by backwards Fourier transformation. Since mathematical methods for phase retrieval have restricted significance, experimental approaches to recover the phase are highly desired.

Fig. 1a presents the parallel momentum dependent photoelectron intensity distribution recorded for the lowest unoccupied orbital (LUMO) of a brickwall layer of 3,4,9,10-perylenetetracarboxylic di-anhydride (PTCDA) on Ag(110). Note that the LUMO is filled due to charge transfer from the substrate. The image is the sum of two datasets with left- and right-handed polarized light at 27 eV photon energy, and thus equivalent to unpolarized light and was recorded using the photoelectron emission microscope (PEEM) at Elettra NanoESCA beamline, which considerably reduces the measurement time compared to conventional ARPES experiments. 

Figure 1.  (a) k//-dependent photoelectron intensity pattern of the PTCDA LUMO (monolayer on Ag(110), photon energy hν=27 eV, sum of left- and right-handed polarized light). (b) Difference between intensity patterns recorded with right- and left-circular polarized light (CDAD) for light incidence along the y direction and (c) for light incidence along the x direction. Dotted iso-intensity contours indicate the momentum-positions from (a). The PTCDA molecule is sketched on top together with the incidence geometry (green arrows).

Under certain conditions, deviations from the pure plane wave final state occur, which allows accessing the phase information which is not contained in the intensity pattern of Fig. 1a. By the use of circular polarized light interference effects of the different partial waves of the final state occur, which are related to the phase distribution within the respective orbital. This is demonstrated by Fig. 1b, which shows the circular dichroism in the angular distribution (CDAD) of the LUMO. If the symmetry of the molecule is known, the phase symmetry can be derived from CDAD experiments with different photon incidence along high symmetry directions of the molecule. While in Fig. 1b light incidence was in the plane of the short molecular axis and the sample normal (i.e. y-z-plane),  Fig. 1c was recorded with incidence in the x-z plane. For the C2v symmetric PTCDA the orbitals belong to one of the four different irreducible representations A1, A2, B1, B2, which have characteristic dichroism signals due to their handedness. This can be seen for the LUMO in Fig. 1 b and c, where the CDAD pattern changes when changing the direction of the incoming light. In Fig. 1b, a left–right asymmetry is obvious. In addition, the CDAD vanishes if the emitted photoelectrons are coplanar with the incident photon beam. If the light comes in the x-z-plane (Fig. 1c) the two prominent maxima reverse the sign, as do the two maxima on the right. The signals on the left, however, behave differently. Here the intensity on top is blue while the one on bottom is black. Thus, the LUMO has to be B1 or B2, i.e. two different mirror planes must exist. Since a nodal plane in the CDAD signal along kx=0 is absent in Fig. 1c but should occur for B1 symmetry, the LUMO must be B2 and the phase symmetry associated with the LUMO k// pattern in Fig. 1a can be reasoned (Fig. 2a).

Figure 2.  k//-dependent photoelectron intensity distributions (hν=55 eV) of the PTCDA LUMO with superimposed phase (positive: blue, negative: red) derived from the CDAD. Dashed lines indicate iso-intensity contours of the unpolarized signal. Inverse Fourier transformation leads to the real space images of the real part of the LUMO (b). The PTCDA molecular structure is superimposed. (c) Respective calculation for an isolated molecule.

Now, inverse Fourier transformation of the k-space orbital map of Fig. 1a to real space is possible. The result is the sectional view of the real part of the LUMO wave function in Fig. 2b. If compared to the corresponding calculation in Fig. 2c the agreement is striking. Although additional oscillations of the partial charge density in real space emerge at large distances from the molecule due to the finite k//-range, our results demonstrate the fascinating potential of the method in reconstructing full molecular wave functions in real space.


This research was conducted by the following research team:

  • M. Wießner, D. Hauschild, C. Sauer, A. Schöll, F. Reinert, University of Würzburg, Experimental Physics VII,Würzburg, Germany
  • V. Feyer, Peter Grünberg Institute (PGI-6), JARA-FIT, Research Center Jülich, Germany 

Contact person:
Achim Schöll:


 M. Wießner, D. Hauschild, C. Sauer, V. Feyer, A. Schöll, F. Reinert Complete determination of molecular orbitals by measurement of phase symmetry and electron density” Nature Communications 5, 4156 (2014), DOI: 10.1038/ncomms5156 

Last Updated on Wednesday, 12 November 2014 10:30