Octahedral tilting in Prussian blue analogues

Octahedral tilting is key to the structure and functionality of perovskites. Here we show how these distortions manifest in the related Prussian blue analogues (PBAs): cyanide versions of double perovskites with formula A$_x$M[M$^{\prime}$(CN)$_6$]$_{1-y}\Box _y\cdot n$H$_2$O (A = alkali metal, M and M$^{\prime}$ = transition metals, $\Box$ = vacancy/defect). Tilts are favoured by high values of $x$ if A = Na or K, whereas the transition metals play a less important role. External hydrostatic pressure can induce tilt transitions nearly irrespective of the stoichiometry, whereas thermal transitions are only reported for $x>1$. Interstitial water can alter the transitions induced by a different stimulus, but (de)hydration \textit{per se} does not lead to tilts. Implications for rational design of critical functionality -- including improper ferroelectricity and electrochemical performance -- are discussed. The results are important for a fundamental understanding of phase transitions as well as for the development of functional materials based on PBAs.