Inside a solid, Wannier functions localize electronic states to individual atoms or bonds — the tight-binding basis that makes band structure intuitive. Outside the solid, scattering states describe electrons as plane waves modified by the surface potential. At the interface, neither description works alone: the localized basis doesn't extend into vacuum, and the plane-wave basis doesn't capture the bonding inside the crystal.
The paper constructs vacuum Wannier functions that bridge the two descriptions. Using a close-packing principle — the functions pack together to span the full Hilbert space without gaps — the authors build a basis that smoothly transitions from localized character inside the solid to scattering character outside. Born-series expansions in this basis produce scattering states that reduce to the expected forms deep inside the crystal and far into vacuum, while correctly handling the interface where both descriptions overlap.
The practical payoff: predictive photoemission calculations without semiempirical vacuum potentials. Photoemission probes the electronic structure by ejecting electrons from the solid into vacuum — the measurement is inherently an interface process. Previous calculations required fitted potentials in the vacuum region; the vacuum Wannier basis eliminates this fitting by deriving the vacuum behavior from the same first-principles calculation that determines the bulk electronic structure.
Tests on graphene and hexagonal boron nitride show corrections beyond first-order Born approximation, meaning the interface is not a perturbative problem — the coupling between inside and outside matters at multiple scattering orders. The basis that works is the one that doesn't choose sides.