Ideal symmetry is known to break down under almost any noise. One measure of asymmetry in a periodic crystal is the relative multiplicity Z ′ of geometrically non-equivalent units. However, Z ′ discontinuously changes under almost any displacement of atoms, which can arbitrarily scale up a primitive cell. This discontinuity was recently resolved by a hierarchy of invariant descriptors that continuously change under all small perturbations. We introduce a Continuous Invariant-based Asymmetry (CIA) to quantify (in physically meaningful ångstroms) the deviation of a periodic crystal from a higher-symmetry form. Our experiments on several crystal structure prediction datasets show that about a half of simulated crystals have high values of CIA, while all experimental structures in these datasets have CIA = 0. On another hand, many crystals with high values of Z ′ in the Cambridge Structural Database (CSD) turned out to be close to more symmetric forms with Z ′ ≤ 1 due to low values of CIAs.

With the advent of self-driving labs promising to synthesize large numbers of new materials, new automated tools are required for checking potential duplicates in existing structural databases before a material can be claimed as novel. To avoid duplication, we rigorously define the novelty metric of any periodic material as the smallest distance to its nearest neighbor among already known materials. Using ultra-fast structural invariants, all such nearest neighbors can be found within seconds on a typical computer even if a given crystal is disguised by changing a unit cell, perturbing atoms, or replacing chemical elements. This real-time novelty check is demonstrated by finding near-duplicates of the 43 materials produced by Berkeley's A-lab in the world's largest collections of inorganic structures, the Inorganic Crystal Structure Database and the Materials Project. To help future self-driving labs successfully identify novel materials, we propose navigation maps of the materials space where any new structure can be quickly located by its invariant descriptors similar to a geographic location on Earth.

The Cambridge Structural Database (CSD) played a key role in the recently established crystal isometry principle (CRISP). The CRISP says that any real periodic crystal is uniquely determined as a rigid structure by the geometry of its atomic centers without atomic types. Ignoring atomic types allows us to study all periodic crystals in a common space whose continuous nature is justified by the continuity of real-valued coordinates of atoms. Our previous work introduced structural descriptors pointwise distance distributions (PDD) that are invariant under isometry defined as a composition of translations, rotations, and reflections. The PDD invariants distinguished all nonduplicate periodic crystals in the CSD. This paper presents the first continuous maps of the CSD and its important subsets in invariant coordinates that have analytic formulas and physical interpretations. Any existing periodic crystal has a uniquely defined location on these geographic-style maps. Any newly discovered periodic crystals will appear on the same maps without disturbing the past materials.

Rigid structures such as cars or any other solid objects are often represented by finite clouds of unlabeled points. The most natural equivalence on these point clouds is rigid motion or isometry maintaining all inter-point distances. Rigid patterns of point clouds can be reliably compared only by complete isometry invariants that can also be called equivariant descriptors without false negatives (isometric clouds having different descriptions) and without false positives (non-isometric clouds with the same description). Noise and motion in data motivate a search for invariants that are continuous under perturbations of points in a suitable metric. We propose the first continuous and complete invariant of unlabeled clouds in any Euclidean space. For a fixed dimension, the new metric for this invariant is computable in a polynomial time in the number of points.

Conventional crystal representations by unit cells and motifs in a typical Crystallographic Information File (CIF) are unreliable for comparing real structures because any reduced cell can discontinuously change (even by volume) under almost any perturbation.

Crystallographic representations and machine learning predict inorganic synthesis conditions for arbitrary zeolites, as validated with literature-mined data.

Since crystal structures are determined in a rigid form, their most practical equivalence is rigid motion, which is a composition of translations and rotations.If we also allow mirror reflections, we get a general isometry maintaining all inter-point distances.Crystal structures can be distinguished up to isometry only by invariants that are preserved under all isometric transformations.Since atomic coordinates are not preserved even under translation, there are not isometry invariants.The parameters of Niggli's reduced is invariant but is discontinuous [1] under almost any tiny displacement of atoms, which can make a primitive cell larger.The physical density is a continuous invariant but is too weak to reliably distinguish many crystals.A strongest invariant is called complete and can be considered a materials genome or a DNA-style code that uniquely identifies any periodic crystal in practice.

Molecular set transformer is a deep learning architecture for scoring molecular pairs found in co-crystals, whilst tackling the class imbalance problem observed on datasets that include only successful synthetic attempts.

Mesoporous molecular crystals have potential applications in separation and catalysis, but they are rare and hard to design because many weak interactions compete during crystallization, and most molecules have an energetic preference for close packing. Here, we combine crystal structure prediction (CSP) with structural invariants to continuously qualify the similarity between predicted crystal structures for related molecules. This allows isomorphous substitution strategies, which can be unreliable for molecular crystals, to be augmented by a priori prediction, thus leveraging the power of both approaches. We used this combined approach to discover a rare example of a low-density (0.54 g cm-3) mesoporous hydrogen-bonded framework (HOF), 3D-CageHOF-1. This structure comprises an organic cage (Cage-3-NH2) that was predicted to form kinetically trapped, low-density polymorphs via CSP. Pointwise distance distribution structural invariants revealed five predicted forms of Cage-3-NH2 that are analogous to experimentally realized porous crystals of a chemically different but geometrically similar molecule, T2. More broadly, this approach overcomes the difficulties in comparing predicted molecular crystals with varying lattice parameters, thus allowing for the systematic comparison of energy-structure landscapes for chemically dissimilar molecules.

The fundamental model of any solid crystalline material (crystal) at the atomic scale is a periodic point set. The strongest natural equivalence of crystals is rigid motion or isometry that preserves all inter-atomic distances. Past comparisons of periodic structures often used manual thresholds, symmetry groups and reduced cells, which are discontinuous under perturbations or thermal vibrations of atoms. This work defines the infinite sequence of continuous isometry invariants (Average Minimum Distances) to progressively capture distances between neighbors. The asymptotic behaviour of the new invariants is theoretically proved in all dimensions for a wide class of sets including non-periodic. The proposed near linear time algorithm identified all different crystals in the world's largest Cambridge Structural Database within a few hours on a modest desktop. The ultra fast speed and proved continuity provide rigorous foundations to continuously parameterise the space of all periodic crystals as a high-dimensional extension of Mendeleev's table of elements.

The fundamental model of all solid crystalline materials is a periodic set of atomic centers considered up to rigid motion in Euclidean space. The major obstacle to materials discovery was highly ambiguous representations of periodic crystals that didn't allow fast and reliable comparisons and led to numerous (near-) duplicates in many databases of experimental and simulated crystals. This paper exemplarily resolves the ambiguity by invariants, which are descriptors without false negatives. The new Pointwise Distance Distributions (PDD) is a numerical matrix with a near-linear time complexity and an exactly computable metric. The strongest theoretical result is generic completeness (absence of false positives) for all finite and periodic sets of points in any dimension. The strength of PDD is shown by 200B+ pairwise comparisons of all periodic structures in the world's largest collection (Cambridge Structural Database) of existing materials over two days on a modest desktop.

6 The fundamental model of a periodic structure is a periodic set of points considered up to rigid 7 motion or isometry in Euclidean space. The recent work by Edelsbrunner et al defined the new 8 isometry invariants (density functions), which are continuous under perturbations of points and 9 complete for generic sets in dimension 3. This work introduces much faster invariants called higher 10 order Pointwise Distance Distributions (PDD). The new PDD invariants are simpler represented 11 by numerical matrices and are also continuous under perturbations important for applications. 12 Completeness of PDD invariants is proved for distance-generic sets in any dimension, which was also 13 confirmed by distinguishing all 229K known molecular organic structures from the world’s largest 14 Cambridge Structural Database. This huge experiment took only seven hours on a modest desktop 15 due to the proposed algorithm with a near linear or small polynomial complexity in terms of key 16 input sizes. Most importantly, the above completeness allows one to build a common map of all 17 periodic structures, which are continuously parameterized by PDD and explicitly reconstructible 18 from PDD. Appendices include first tree-based maps for several thousands of real structures. 19 2012 ACM Subject Classification Theory of computation → Computational geometry 20

A conventional representation of a periodic crystal by its primitive unit cell and motif is well-known to be ambiguous.Indeed, any crystal can be generated from infinitely many primitive unit cells and motifs containing differently located atoms.Niggli's reduced cell is unique but discontinuous under perturbations.Continuity of crystal representations is important for filtering out near duplicates in big datasets [1, Fig. 2d] of simulated crystals in Crystal Structure Prediction (CSP).Symmetry groups and many other descriptors discontinuously change under perturbations.So CSP landscapes are plotted only by two coordinates: the lattice energy and density.We describe a new geometric approach to generating a unique code (called a crystal isoset) of any periodic crystal, which continuously changes under perturbations of atoms [2-3].This isoset is a material genome or a DNA-type code that allows an inverse design of new periodic crystals.Using these complete isosets, one can compute invariants via density functions [4] and interatomic distances [5].For any crystal dataset irrespective of symmetries or chemical compositions, invariants of crystals can be joined in a minimum spanning tree via continuous distances that quantify crystal similarities.Our Python code of distance-based invariants produced a map of all 229K organic crystals in the Cambridge Structural Database overnight on a modest desktop [6 (appendix D), 7].Figure 1.A new invariant-based visualization is illustrated on the CSD Drug Subset of 12,576 structures colored by CSD ref codes.Left: A tree joins those structures that have close values of invariants extracted from isosets.Right: the interactive invariant-based map is zoomed to show families of chemically different aspirin and paracetamol, which are in close branches as in a pharmacy.

Periodic sets of points model all solid crystalline materials (crystals) by representing atoms as labeled points. Crystal structures are determined in a rigid form and are considered up to rigid motions or isometries. Modern tools of Crystal Structure Prediction output thousands of simulated structures, though only few of them can be really synthesized. The first obstacle is the presence of many near duplicate structures that can not be efficiently recognized on the fly by past tools. To continuously quantify a similarity between periodic sets, their isometry invariants should be continuous under perturbations when all discrete invariants such as symmetry groups can break down. This paper studies the isometry classification problem for periodic sets with the new continuity requirement and introduces the Average Minimum Distances, which form an infinite sequence of continuous isometry invariants. Their asymptotic behaviour for a wide class of sets is explicitly described in terms of a point packing coefficient. All results are illustrated by experiments on large datasets of crystals.

The fundamental model of a periodic structure is a periodic point set up to rigid motion or isometry. Our recent paper in SoCG 2021 defined isometry invariants (density functions), which are complete in general position and continuous under perturbations. This work introduces much faster isometry invariants (average minimum distances), which are also continuous and distinguish some sets that have identical density functions. We explicitly describe the asymptotic behaviour of the new invariants for a wide class of sets including non-periodic. The proposed near linear time algorithm processed a dataset of hundreds of thousands of real structures in a few hours on a modest desktop.