  Twisted double bilayer graphene (tDBLG) is a moir\'e material that has
recently generated significant interest because of the observation of
correlated phases near the magic angle. We carry out atomistic Hartree theory
calculations to study the role of electron-electron interactions in the normal
state. In contrast to twisted bilayer graphene (tBLG), we find that such
interactions do not result in significant doping-dependent deformations of the
electronic band structure. However, interactions play an important role for the
electronic structure in the presence of a perpendicular electric field as they
screen the external field. Finally, we analyze the contribution of the Hartree
potential to the crystal field, i.e. the on-site energy difference between the
inner and outer layers. We find that the on-site energy obtained from Hartree
theory has the same sign, but a smaller magnitude compared to previous studies
in which the on-site energy was determined by fitting tight-binding results to
ab initio density-functional theory (DFT) band structures. To understand this
quantitative difference, we analyze the ab initio Kohn-Sham potential obtained
from DFT and find that a subtle interplay of electron-electron and electron-ion
interactions determines the magnitude of the on-site potential.
