Application of Woodward-Hoffmann ideas to solid-state polymorphic phase transitions with specific reference to polymerization of ${\mathrm{S}}_2$${\mathrm{N}}_2$ and the black phosphorus to $A7$ (arsenic) structural transformation

The analog of the Woodward-Hoffmann orbital correlation diagrams for solid-state polymorphic phase transitions are defined (within the tight-binding approximation) and the implications of the form of the correlation diagram $E(\overrightarrow{\mathrm{k}},q)$ for the reaction kinetics are discussed. It is proposed that if an occupied orbital of the parent structure is transformed into an unoccupied orbital of the new structure at a point in $\overrightarrow{\mathrm{k}}$ space, then the reaction cannot proceed with retention of translation symmetry in the intermediate, and so can only proceed via a "nucleation-and-growth"-type mechanism. On the other hand, if the $E(\overrightarrow{\mathrm{k}},q)$ surface shows a smooth correlation of occupied levels of the reactants and products the process may proceed "martensitically." Correlation diagrams $E(\overrightarrow{\mathrm{k}},q)$ are constructed with the use of group-theoretical analysis and extended-Hückel band-structure calculations), and discussed for the following transformations: sodium chloride from the rocksalt to cesium chloride structure, the bond breaking and reforming in a linear chain of diatomic molecules, an idealized model of the polymerization of ${\mathrm{S}}_2$ ${\mathrm{N}}_2$ to both the cis and trans polymers ${\left(\mathrm{SN}\right)}_x$, and the transformation of black phosphorus into the $A7$ arsenic-type structure.