  Heterogeneous interfaces are central to many energy-related applications in
the nanoscale. From the first-principles electronic structure perspective, one
of the outstanding problems is accurately and efficiently calculating how the
frontier quasiparticle levels of one component are aligned in energy with those
of another at the interface, i.e., the so-called interfacial band alignment or
level alignment. The alignment or the energy offset of these frontier levels is
associated with the charge-transfer barrier across the interface and therefore
dictates the interfacial dynamics. Although many-body perturbation theory
provides a formally rigorous framework for computing the interfacial electronic
structure, it is often associated with a high computational cost and is limited
by its perturbative nature. It is therefore of great interest to develop
practical alternatives, preferably based on density functional theory (DFT),
which is known for its balance between efficiency and accuracy. However,
conventional developments of density functionals largely focus on total
energies and thermodynamic properties, and the design of functionals aiming for
interfacial electronic structure is only emerging recently. This Review is
dedicated to a self-contained narrative of the interfacial electronic structure
problem and the efforts of the DFT community in tackling it. Since interfaces
are closely related to surfaces, we first discuss the key physics behind the
surface and interface electronic structure, namely the image potential and the
gap renormalization. This is followed by a review of early examinations of the
surface exchange-correlation hole and the exchange-correlation potential, which
are central quantities in DFT. Lastly, we survey two modern endeavors in
functional development that focus on the interfacial electronic structure,
namely the dielectric-dependent hybrids and local hybrids.
