Prussian blue analogues (PBAs) are a broad and important family of microporous inorganic solids, famous for their gas storage (1,2,3,4,5), metal-ion immobilisation (6,7), proton conduction (8,9), and stimuli-dependent magnetic (10, 11, 12), electronic (13) and optical (14) properties. The family also includes the widely-used double-metal cyanide (DMC) catalysts (15,16,17) and the topical hexacyanoferrate/hexacyanomanganate arXiv:1908.10596v1 [cond-mat.mtrl-sci] PBAs is the ability to transport mass reversibly, a process made possible by structural vacancies. Normally presumed random (21,22,23), vacancy arrangements are actually crucially important because they control the connectivity of the micropore network, and hence diffusivity and adsorption profiles (24,25). The long-standing obstacle to characterising PBA vacancy networks has always been the relative inaccessibility of single-crystal samples (26). Here we report the growth of single crystals of a range of PBAs. By measuring and interpreting their X-ray diffuse scattering patterns, we identify for the first time a striking diversity of non-random vacancy arrangements that is hidden from conventional crystallographic analysis of powder samples. Moreover, we show that this unexpected phase complexity can be understood in terms of a remarkably simple microscopic model based on local rules of electroneutrality and centrosymmetry. The hidden phase boundaries that emerge demarcate vacancy-network polymorphs with profoundly different micropore characteristics. Our results establish a clear foundation for correlated defect engineering in PBAs as a means of controlling storage capacity, anisotropy, and transport efficiency.The true crystal structures of PBAs-as of Prussian Blue itself-have long posed a difficult and important problem in solid-state chemistry because their ostensibly simple powder diffraction patterns [ Fig. 1(a)] belie a remarkable complexity at the atomic scale (27,28,29). The common parent structure is based on the cubic lattice and corresponds to the idealised composition M[M (CN) 6 ]. Atoms of type M and M (usually transition-metal cations) occupy alternate lattice vertices and are octahedrally coordinated by bridging cyanide ions (CN − ) at the lattice edges [ Fig. 1(b)]. There is a close conceptual parallel to the double perovskite structure (30); indeed the key considerations of covalency and octahedral coordination geometry that stabilise perovskites amongst oxide ceramics (31) also favour this same architecture for transition-metal cyanides, which accounts for the chemical diversity of PBAs (32). Charge balance

The central goal of crystal engineering is to control material function via rational design of structure. A particularly successful realisation of this paradigm is hybrid improper ferroelectricity in layered perovskite materials, where layering and cooperative octahedral tilts combine to break inversion symmetry. However, in the parent family of inorganic ABX3 perovskites, symmetry prevents hybrid coupling to polar distortions. Here, we use group-theoretical analysis to uncover a profound enhancement of the number of improper ferroelectric coupling schemes available to molecular perovskites. This enhancement arises because molecular substitution diversifies the range of distortions possible. Not only do our insights rationalise the emergence of polarisation in previously studied materials, but we identify the fundamental importance of molecular degrees of freedom that are straightforwardly controlled from a synthetic viewpoint. We envisage that the crystal design principles we develop here will enable targeted synthesis of a large family of new acentric functional materials.

We introduce columnar shifts-collective rigid-body translations-as a structural degree of freedom relevant to the phase behaviour of molecular perovskites ABX (X = molecular anion). Like the well-known octahedral tilts of conventional perovskites, shifts also preserve the octahedral coordination geometry of the B-site cation in molecular perovskites, and so are predisposed to influencing the low-energy dynamics and displacive phase transitions of these topical systems. We present a qualitative overview of the interplay between shift activation and crystal symmetry breaking, and introduce a generalised terminology to allow characterisation of simple shift distortions, drawing analogy to the "Glazer notation" for octahedral tilts. We apply our approach to the interpretation of a representative selection of azide and formate perovskite structures, and discuss the implications for functional exploitation of shift degrees of freedom in negative thermal expansion materials and hybrid ferroelectrics.