  The lattice vibrations (phonon modes) of crystals underpin a large number of
material properties. The harmonic phonon spectrum of a solid is the simplest
description of its structural dynamics and can be straightforwardly derived
from the Hellman-Feynman forces obtained in a ground-state electronic structure
calculation. The presence of imaginary harmonic modes in the spectrum indicates
that a structure is a local maximum on the athermal structural potential-energy
surface and can yield important insight into the fundamental nature and
physical properties of a material. In this review article, we discuss the
physical significance of imaginary harmonic modes and distinguish between cases
where imaginary modes are indicative of such phenomena, and those where they
reflect technical problems in the calculations. We outline basic approaches for
exploring and renormalising imaginary modes, and demonstrate their utility
through a set of three case studies in the materials sciences.
