  Although real materials are finite in size, electronic structure theory is
built on the assumption of infinitely large solid, which led to a longstanding
controversy: where is the vacuum level? Here, we introduce an analytic
real-space potential-unfolding approach to uncover the vacuum level in
infinitely large solid. First-principles calculations show that, in the absence
of a physical surface, the bulk band structure, often measured with respect to
an average bulk potential, is offset by a hereto unknown and
orientation-dependent bulk quadrupole with respect to the vacuum level. By
identifying intrinsic contributions of a bulk solid to its surface and
interface properties, our theory eliminates the ambiguities surrounding the
physical origin of the band alignment between matters.
