Reconstructing Molecular Wave Functions

Based on photoemission tomography, we have developed a technique to reconstruct the real-space probability amplitude density of molecular orbitals including their quantum mechanical phase.

The basis for a quantum-mechanical description of molecules is the many-electron wave function. At various levels of approximation, up to the exact configuration interaction wave function, one can write it in terms of single-electron wave functions: the orbitals. Strictly speaking, orbitals are not quantum mechanical observables. Nevertheless, they can be reconstructed in the framework of photoemission tomography.

For a full 3D reconstruction, we measure a series of distributions of the photoelectron parallel momentum, determined at multiple excitation energies in the range from 10 to 100 eV. This is used to create a four-dimensional dataset of photoelectron intensities as a function of (k_x, k_y, k_z). A vital requirement is the proper intensity normalisation of experimental data at different photon energies. To this end, we utilise the capabilities of the Metrological Light Source (MLS) of the Physikalisch-Technische Bundesanstalt (PTB), a unique beamline based on a metrological approach to accurate photon flux calibration and to filtering out parasitic higher harmonics of the synchrotron radiation.

In the framework of the one-step model of photoemission, and employing the plane-wave approximation for the final state of the photoelectrons, the experimentally measured 3D momentum space distribution of photoelectrons can be reconstructed into the 3D real-space probability amplitude density of the initial-state orbital [1]. The phase of the wave function that is required for this reconstruction is retrieved in an iterative algorithm based on the oversampling technique [2].

The best way to think about the as-measured orbitals is to interpret them as Dyson orbitals that quantify the change between the N electron wave function before and the N − 1 electron wave function after photoionisation. If Koopman's theorem holds, which should be the case for molecules with weakly correlated electrons, Dyson orbitals are closely related to canonical Hartree-Fock or Kohn-Sham orbitals.