  Density Functional Theory (DFT) sees prominent use in computational chemistry
and physics, however, problems due to the self-interaction error (SIE) pose
additional challenges to obtaining qualitatively correct results. An unphysical
energy an electron exerts on itself, the SIE impacts most practical DFT
calculations. We conduct an in-depth analysis of the one-electron SIE in which
we replicate delocalization effects for simple geometries. We present a simple
visualization of such effects, which may help in future qualitative analysis of
the one-electron SIE. By increasing the number of nuclei in a linear
arrangement, the SIE increases dramatically. We also show how molecular shape
impacts the SIE. Two and three dimensional shapes show an even greater SIE
stemming mainly from the exchange functional with some error compensation from
the one-electron error, which we previously defined [Phys. Chem. Chem. Phys.
22, 15805 (2020)]. Most tested geometries are affected by the functional error,
while some suffer from the density error. For the latter we establish a
potential connection with electrons being unequally delocalized by the DFT
methods. We also show how the the SIE increases if electrons occupy
higher-lying atomic orbitals; seemingly one-electron SIE free methods in a
ground are no longer SIE free in excited states, which is an important insight
for some popular, non-empirical DFAs. We conclude that the erratic behavior of
the SIE in even the simplest geometries shows that robust density functional
approximations are needed. Our test systems can be used as a future benchmark
or contribute towards DFT development.
