  The simulation of transmission electron microscopy (TEM) images or
diffraction patterns is often required to interpret their contrast and extract
specimen features. This is especially true for high-resolution phase-contrast
imaging of materials, but electron scattering simulations based on atomistic
models are widely used in materials science and structural biology. Since
electron scattering is dominated by the nuclear cores, the scattering potential
is typically described by the widely applied independent atom model. This
approximation is fast and fairly accurate, especially for scanning TEM (STEM)
annular dark-field contrast, but it completely neglects valence bonding and its
effect on the transmitting electrons. However, an emerging trend in electron
microscopy is to use new instrumentation and methods to extract the maximum
amount of information from each electron. This is evident in the increasing
popularity of techniques such as 4D-STEM combined with ptychography in
materials science, and cryogenic microcrystal electron diffraction in
structural biology, where subtle differences in the scattering potential may be
both measurable and contain additional insights. Thus, there is increasing
interest in electron scattering simulations based on electrostatic potentials
obtained from first principles, mainly via density functional theory, which was
previously mainly required for holography. In this Review, we discuss the
motivation and basis for these developments, survey the pioneering work that
has been published thus far, and give our outlook for the future. We argue that
a physically better justified $\textit{ab initio}$ description of the
scattering potential is both useful and viable for an increasing number of
systems, and we expect such simulations to steadily gain in popularity and
importance.
