²There are several occasions when a consideration of imaginary frequencies may be useful. A lattice dynamics calculation can be used to test whether a purported equilibrium crystal structure really is at equilibrium. The presence of imaginary frequencies anywhere in the Brillouin Zone would indicate that the structure is unstable to a distortion along the corresponding eigenvector. Another occasion would be to investigate a transition state. By definition a transition state geometry is at a stationary saddle point of the energy hypersurface and must therefore exhibit precisely one imaginary frequency. A final example is the case of a system exhibiting a “soft-mode” phase transition. In such systems it can be useful to explore how the frequency approaches zero as some external variable such as pressure or lattice constant is varied, in order to precisely locate the phase transition.