Electron-density critical points analysis and catastrophe theory to forecast structure instability in periodic solids

The critical points analysis of electron density, i.e. ρ(x), from ab initio calculations is used in combination with the catastrophe theory to show a correlation between ρ(x) topology and the appearance of instability that may lead to transformations of crystal structures, as a function of pressure/temperature. In particular, this study focuses on the evolution of coalescing non-degenerate critical points, i.e. such that ∇ρ(x_c) = 0 and λ₁, λ₂, λ₃ ≠ 0 [λ being the eigenvalues of the Hessian of ρ(x) at x_c], towards degenerate critical points, i.e. ∇ρ(x_c) = 0 and at least one λ equal to zero. The catastrophe theory formalism provides a mathematical tool to model ρ(x) in the neighbourhood of x_c and allows one to rationalize the occurrence of instability in terms of electron-density topology and Gibbs energy. The phase/state transitions that TiO₂ (rutile structure), MgO (periclase structure) and Al₂O₃ (corundum structure) undergo because of pressure and/or temperature are here discussed. An agreement of 3–5% is observed between the theoretical model and experimental pressure/temperature of transformation.