On the importance of the free energy for elasticity under pressure

It is shown that the theory of elasticity under hydrostatic pressure p at zero temperature is unified and simplified by the use of the Gibbs free energy G, rather than the energy E. The minima of G, but not of E, give the equilibrium structure; the second derivatives of G, but not of E, with respect to strains at the equilibrium structure give the zero-temperature elastic constants; the stability of a phase at p is then determined by the same Born stability conditions used at p = 0 when applied to the elastic constants from G. Examples are noted of mistakes due to use of E rather than G.