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Page 2 of 9 Wang et al. J. Mater. Inf. 2026, 6, 1

INTRODUCTION
Predicting the thermodynamic stability remains one of the most fundamental challenges in materials science.
Stability is not only a prerequisite for synthesizability but also a critical filter before exploring target
properties such as conductivity, magnetism, or catalytic activity . As generative models, including diffusion
[1-3]
models and large language models, become increasingly central to materials discovery, the ability to generate
stable candidates has emerged as a core design objective . However, these approaches often lack simple,
[4-6]
interpretable, and transferable criteria that can guide generation toward chemically and geometrically
feasible regions of the compositional space.

This gap highlights the renewed relevance of tolerance factors (T f), which provide a geometric measure of
structural stability for a given composition. Initially formulated for perovskite materials , the concept has
[7,8]
since been generalized to other families, such as spinels , garnets [10,11] , and quaternary compounds [12,13] , where
[9]
packing efficiency and ionic size compatibility govern the formation and stability of crystal structures. T f has
the appeal of physical interpretability and computational efficiency, making it ideal for rapid screening.
However, despite its simplicity and relevance, T f has not been fully integrated into modern AI-driven
materials pipelines. The challenge lies in how to embed such universal structural priors into generative or
predictive models in a flexible way, leading to realizable materials. For example, generative models guided by
geometric constraints or reinforced by structural stability feedback from density functional theory (DFT) or
ab initio molecular dynamics (AIMD) simulations can potentially learn to discover novel but stable
geometries [14,15] , extending the scope of materials far beyond canonical design spaces [16,17] .

Despite the historical significance, T f has received limited attention in the broader landscape of AI-driven
materials discovery. To date, its application has been largely confined to a few prototypical systems, many
other materials exhibiting similar geometric correlations remain underexplored. Here, we aim to fill this gap
by summarizing the T f frameworks used in representative systems, and demonstrating how T f can be
integrated into AI-based materials pipelines. Finally, we articulate key challenges that need to be addressed to
generalize T f across broader domains, and propose potential solutions. Together, these strategies suggest a
promising route toward discovering stable materials more efficiently, with T f serving as lightweight yet
informative guide in the design space.

GEOMETRIC TOLERANCE FACTORS FOR REPRESENTATIVE MATERIALS SYSTEMS
The concept of the T f originated from geometric considerations of ionic packing in perovskite structures .
[7,8]
The classical Goldschmidt T f is defined as

where r A, r B, and r X are the ionic radii of the A-site, B-site, and X-site, respectively. This formulation is
derived from the idealized cubic ABX 3 perovskite, assuming hard-sphere models and corner-sharing BX 6
octahedra. A T f close to unity indicates a stable cubic or slightly distorted structure, while values outside this
range often imply instabilities or transformations to non-perovskite phases. Over time, similar radius-based
formulations have been generalized to other compound families, by adapting the geometric motif and
coordination environments specific to each lattice type.

In this study, we curated geometric T f expressions across eight major structural families [Figure 1], including
simple perovskite (ABX 3) [7,8] , double perovskite (A 2BB’X 6) , spinel (AB 2X 4) , garnet (A 3B 2C 3X 12) [10,11] ,
[9]
[18]
quaternary family (A 2B 4C 2nX 7+n) , huntite [AC 3(BO 3) 4] , pyrochlore (A 2B 2O 7) , and photovoltaic
[20]
[21]
[19]
chalcogenide family (A 2BCX 4) [12,13] , where A, B, B’, and C are cations (e.g., Ca , Pb ), and X denotes anions
2+
2+
(e.g., N , O , F ). The structural T f is generally derived by considering the geometric fit between constituent
3-
2-
-