Theoretical and computational improvements to the algebraic method for discovering cooperative rigid-unit modes

A linear-algebraic algorithm for identifying rigid-unit modes in networks of interconnected rigid units has recently been demonstrated. This article presents a series of enhancements to the original algorithm, which greatly improve its conceptual simplicity, numerical robustness, computational efficiency and interpretability. The improvements include the efficient isolation of constraints, the observation of variable-block separability, the use of singular value decomposition and a quantitative measure of solution inexactness.

The ISOTILT software for discovering cooperative rigid-unit rotations in networks of interconnected rigid units

A user-friendly web-based software tool called `ISOTILT' is introduced for detecting cooperative rigid-unit modes (RUMs) in networks of interconnected rigid units (e.g. molecules, clusters or polyhedral units). This tool implements a recently described algorithm in which symmetry-mode patterns of pivot-atom rotation and displacement vectors are used to construct a linear system of equations whose null space consists entirely of RUMs. The symmetry modes are first separated into independent symmetry-mode blocks and the set of equations for each block is solved separately by singular value decomposition. ISOTILT is the newest member of the ISOTROPY Software Suite. Here, it is shown how to prepare structural and symmetry-mode information for use in ISOTILT, how to use each of ISOTILT's input fields and options, and how to use and interpret ISOTILT output.

Framework structure crystalline materials and Rigid Unit Modes (RUMs). Introducing the new concept of MLRUMs and skeletions Authors

This article reviews Framework Structures (FWSs), defined as crystalline materials built of rigid AXn polyhedra sharing vertices (like perovskites, tungsten bronzes, Dion-Jacobson, Ruddlesden-Popper, and Aurivillius phases, quartz, silicates, and others), and their pecularities resulting from this linkage. The situation of rigid units linked by common vertices may allow the units to accomplish concordant rotations without deformation, which gives rise to soft phonon modes called “Rigid Unit Modes” (RUMs). The condensation of a RUM can trigger structural phase transitions to a structure of lower symmetry, with tilted polyhedra, at the origin of spontaneous ferroic or multiferroic properties. We overview results precedently obtained on RUMs in perovskites, tetragonal tungsten bronzes, and quartz, and detail new results on “maximally localized RUMs” (MLRUMs), a fundamental new concept in the physics of RUMs. We introduce also the related new concept of “skeletions” that allows to generate all ferroelastic phases found in these systems, and generalizes the Glazer's tilt-system approach.

Tilts and shifts in molecular perovskites

A survey of the rigid unit modes in molecular perovskites is presented, showing how the prevalence of conventional tilts, unconventional tilts and columnar shifts vary across the different classes of molecular perovskites.

An algebraic approach to cooperative rotations in networks of interconnected rigid units

Crystalline solids consisting of three-dimensional networks of interconnected rigid units are ubiquitous amongst functional materials. In many cases, application-critical properties are sensitive to rigid-unit rotations at low temperature, high pressure or specific stoichiometry. The shared atoms that connect rigid units impose severe constraints on any rotational degrees of freedom, which must then be cooperative throughout the entire network. Successful efforts to identify cooperative-rotational rigid-unit modes (RUMs) in crystals have employed split-atom harmonic potentials, exhaustive testing of the rotational symmetry modes allowed by group representation theory, and even simple geometric considerations. This article presents a purely algebraic approach to RUM identification wherein the conditions of connectedness are used to construct a linear system of equations in the rotational symmetry-mode amplitudes.