Algebraic search for cooperative rotational rigid-unit modes

Abstract

Crystalline solids consisting of three-dimensional networks of interconnected polyhedra or other rigid units are ubiquitous amongst functional materials. In many cases, application-critical properties are sensitive to the rotations of individual rigid units. But the shared atoms that connect the rigid units together 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. Here, we present a purely algebraic approach to RUM identification wherein the constraints of interconnectedness are applied to a linear system of equations. The new approach appears to be fully general and quite straightforward to implement.

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

“…These theoretical and computational improvements to the algebraic RUM-search algorithm have now been employed in a new web-based software tool for RUM identification called ISOTILT, which is a new component of the ISOTROPY Software Suite (https://iso.byu.edu). A companion article on the ISOTILT software (Campbell et al, 2021) will contain more detail for each of the HTB, TTB and CAZO examples, and will show the reader how to reproduce the data in Tables 1-4 and to visualize the resulting RUMs.…”

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

“…This block separability property dramatically reduces the computational complexity of the algorithm. Campbell et al (2021) further improved the robustness and interpretability of the algorithm by constructing a smaller matrix with the same null space and by employing singular value decomposition rather than Gaussian row reduction. Symmetry-mode blocks are first arranged in ascending order according to their smallest singular values.…”

“…More recently, Campbell et al (2021) described a series of critical improvements to the original algorithm based on the observation that the split-passenger-atom-displacement pattern arising from a rotational symmetry mode belonging to a given irrep is also a symmetry mode of that same irrep, so that such patterns also enjoy special orthogonality properties. This means that the split-passenger-atom-displacement patterns arising from distinct irreps, or even from different components of the OPD of the same irrep, must be linearly independent.…”

“…Our present objective is to introduce the user-friendly and web-based ISOTILT software tool, which implements the linear-algebraic RUM-search algorithm and its improvements reported by Campbell et al (2018Campbell et al ( , 2021. ISOTILT is the newest member of the ISOTROPY Software Suite (https:// iso.byu.edu/iso/isotropy.php) and must be used in tandem with the ISODISTORT software (Campbell et al, 2006).…”

“…Section 4 walks through the key results obtained from several other examples from the literature, and Section 5 concludes with a discussion of limitations and opportunities. The examples presented in Section 4 are the same ones described by Campbell et al (2018Campbell et al ( , 2021: tetragonal tungsten bronze (TTB) as described by Whittle et al (2015) and Smirnov & Saint-Gre ´goire (2014), hexagonal tungsten bronze (HTB) as described by Whittle et al (2015), and Ca 3 Al 4 ZnO 10 (CAZO) as described by Kahlenberg, Albrecht et al (2019) and Kahlenberg, Hejny & Kru ¨ger (2019).…”