Whether one uses scanning probe microscopy to gather real-space images or one uses photoemission measurements to gather reciprocal-space images, only electron densities are experimentally accessible, all phase information being lost. (One exception is for simple molecules subjected to high-intensity femtosecond laser pulse excitation. In this case, one can gather intra-molecular interference information which permits phase recovery.)
In the January 14 issue of PNAS, researchers working at BESY II put forward an iterative refinement method for reconstructing slices through approximate single-electron molecular orbitals, based on ARPES momentum maps of molecular monolayers. Interestingly, the method can recover both the highest occupied molecular orbital (HOMO) and the lowest unoccupied molecular orbital (LUMO).
The method echoes the discovery by Sayre (1952) that Bragg diffraction under-samples diffracted intensity, and that over-sampling can lead to unique real space solutions.