  A series expansion of the interaction between a nucleus and its surrounding
electron distribution provides terms that are well-known in the study of
hyperfine interactions: the familiar quadrupole interaction and the less
familiar hexadecapole interaction. If the penetration of electrons into the
nucleus is taken into account, various corrections to these multipole
interactions appear. The best known one is a scalar correction related to the
isotope shift and the isomer shift. This paper discusses a related tensor
correction, which modifies the quadrupole interaction if electrons penetrate
the nucleus: the quadrupole shift. We describe the mathematical formalism and
provide first-principles calculations of the quadrupole shift for a large set
of solids. Fully relativistic calculations that explicitly take a finite
nucleus into account turn out to be mandatory. Our analysis shows that the
quadrupole shift becomes appreciably large for heavy elements. Implications for
experimental high-precision studies of quadrupole interactions and quadrupole
moment ratios are discussed. A literature review of other small quadrupole-like
effects is presented as well.
