E+: Software for Hierarchical Modeling of Electron Scattering from Complex Structures

In modern nanobeam transmission electron microscopy methods, such as 4D-STEM, a converged electron nanobeam is scanned across a sample. Its 2D scattering pattern is recorded at each sample position, mapping the local sample structure. One of the bottlenecks in electron scattering is the analysis of the scattering data obtained from complex atomic or molecular structures. On the basis of D+ software, we developed the software E+ for analyzing electron scattering data, enabling us to model the 2D scattering pattern from any complex structure in a single orientation or a fiber. In addition, the azimuthally integrated 1D scattering curve of isotropically oriented structures (as in solutions or powders) or any other distribution of orientations, can also be computed. E+ allows the docking of geometric and/or molecular atomic models into their assembly symmetry. The assembly symmetry contains the rotations and translations of repeating subunits within a large structure. This process can be repeated hierarchically, using a bottom-up approach, adding as many subunits as needed. This procedure can be used to model the scattering data from any complex supramolecular structure at any spatial resolution, down to atomic resolution. In addition, the contribution from the solvation layers of structures in solutions can be computed in a scalable manner for large complexes. Furthermore, the Python API of E+ can be used for advanced modeling of structure factor and pair distribution functions, taking into account various effects, including thermal fluctuations, polydispersity of any structural parameters, or the intermolecular interactions between subunits. We validate E+ against the abTEM software and show a few examples, demonstrating how E+ can be used to analyze 4D-STEM electron scattering data.