Resolving the data ambiguity for periodic crystals
Abstract
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 nearlinear 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.
Motivations for resolving the ambiguity challenge for periodic crystals
This paper resolves the long-standing challenge of ambiguous data representation for periodic structures that model all solid crystalline materials (periodic crystals).
Any real crystal is best modeled as a periodic set S ⊂ R n of points at all atomic centers, whose positions have a physical meaning and are determined via diffraction patterns.
Edges between points are excluded because they only abstractly represent inter-atomic bonds depending on thresholds for distances and angles [28].
The simplest example is a lattice Λ ⊂ R n consisting of all integer linear combinations of a basis whose vectors span a unit cell U .
Fig.
1 shows omly a few of inﬁnitely many possible choices of U . [Table]
Materials discovery still relies on trial-and-error because periodic crystals are traditionally represented by non-invariants (descriptors with false negatives) or discontinuous invariants such as symmetry groups that break down under tiny perturbations.
These conventional descriptions cannot identify fraudulent structures in experimental datasets that keep depositing numerous (near-) duplicates without reliable tools for justiﬁed comparisons [29].
The ambiguity challenge will be rigorously stated in Problem 1.1 as a classiﬁcation of periodic sets up to isometry preserving the crystal rigidity.
Fig.
1 illustrates the ﬁrst obstacle: the same lattice can be generated by inﬁnitely many different bases or unit cells, so distinguishing only lattices up to isometry is already non-trivial.
Any periodic point set S can be deﬁned as a sum Λ + M =
(cid:126)v + (cid:126)p : v ∈ Λ , p ∈ M } , where a motif M is a ﬁnite set of points in the basis of U .
Any lattice Λ is considered as a periodic set with a 1-point motif M = { p } .
A single point p can be arbitrarily chosen in a unit cell U , see the ﬁrst two pictures of Fig.
Basis vectors of U and atomic coordinates of motif points (atomic centers) in M constitute the data found in conventional Crystallographic Information File (CIF).
Fig.
2 (left) shows the ambiguity of the CIF pair (Λ , M ) even if a basis of U is ﬁxed.
The recent work by Widdowson et al. [66] rigorously stated the problem to continuosly classify periodic point sets up to isometry.
An isometry of Euclidean space R n is any map that maintains inter-point distances.
Any orientation-preserving isometry can be realized as a continuous rigid motion, for example any composition of translations and rotations in R 3 .
This equivalence is most natural for ﬁnite and periodic sets that represent real rigid structures
Crystals can be reliably distinguished up to isometry only by an isometry invariant I that takes the same value on all isometric sets. So I has no false negatives such that I ( S ) (cid:54) =
I ( Q ) for different representations of isometric crystals S ∼ = Q .
If a descriptor has false negatives, we can make no reliable conclusions because different inputs as in Fig.
1 and 2 (left) can reprsent isometric crystals.
Hence, non-invariants cannot be used to justiﬁably compare crystals [7].
It sufﬁces to classify crystals up to isometry f including reﬂections because, for any basis v 1 , . . .
, v n of R n , f preserves orientation if the determinant det( f ( v 1 ) , . . .
, f ( v n )) > 0 for the images of v 1 , . . . , v n in the same basis.
The traditional approach to identifying a periodic crystal is to use its conventional or reduced cell [39, section 9.3].
This reduced cell has been known to be discontinuous under perturbations [1] even for lattices when a motif M is a single point.
More formally, [25, section 1] and [66, Theorem 15] proved that a reduced cell cannot be continuously deﬁned for all lattices.
Discontinuity of many discrete invariants such as symmetry groups becomes clearer in Fig.
2 (right) showing that even real-valued invariants struggle to continuously quantify the similarity between nearly identical sets.
Under perturbations of points, a minimal cell U can double in size, while the density remains constant.
Problem 1.1. Find a function I on all periodic sets of unlabeled points in R n such that
(1.1a) invariance: if any periodic point sets S ∼ = Q are isometric in R n , then I ( S ) =
I ( Q ) , so the invariant I has no false negatives ; (1.1b) completeness: if I ( S ) = I ( Q ) for any periodic point sets S, Q , then S ∼ = Q are isometric, so the invariant I has no false positives ; (1.1c) metric: a distance d between values of I satisﬁes all axioms; 1) d ( I 1 , I 2 )
=
if and only if I
= I 2 , 2) symmetry d ( I 1 , I 2 ) =
d ( I 2 , I 1 ) , 3) triangle inequality d ( I 1 , I 2 ) + d ( I 2 , I 3 ) ≥
d ( I 1 , I 3 ) ; (1.1d) Lipschitz continuity: if Q is obtained from a periodic set S ⊂ R n by shifting each point of S by at most ε , then d ( I ( S ) , I ( Q ))
≤ Cε for a constant C and any periodic point sets S, Q ⊂ R n ; (1.1e) computability: the invariant I , the metric d and veriﬁcation of I ( S ) =
I ( Q ) should be computed in a near-linear time in the number of motif points of periodic sets for a ﬁxed dimension n ; (1.1f) inverse design: any periodic point set S ⊂ R n can be reconstructed from its invariant I ( S ) .
(cid:4)
Problem 1.1 is the ultimate data science challenge for all periodic crystals S whose non-invariant input (a cell basis and a motif) should be transformed into a complete invariant I ( S ) , which uniquely and unambiguously represents any S .
Such a complete invariant can be considered as a materials genome [22] or a DNA-style code that also allows an explicit reconstruction for any periodic crystal.
For example, computer vision tries to identify humans or other objects such as road signs by using pixel-based images as input.
Similar to other real objects, any periodic crystal can be given by (inﬁnitely) many inputs.
Problem 1.1 is an example of the data ambiguity challenge for almost any application that needs to uniquely identify a real object. Biology has partially solved this challenge by a DNA code, though a living creature cannot be easily created from such a code. Condition (1.1f) goes further and requires a full reconstruction of a periodic crystal S from its DNA-style code I ( S ) .
The proposed solution to Problem 1.1 is the isometry invariant I called the Pointwise Distance
Distribution PDD .
Theorems 3.2, 4.3, 5.1 , 4.4 prove that PDD satisﬁes all conditions of Problem 1.1, even (1.1b) at least for generic sets.
More exactly, Theorem 4.4 shows that any periodic point set
S = Λ + M ⊂ R n in a general position can be explicitly reconstructed from PDD and its lattice Λ .
The strength of PDD was experimentally checked for all 660K+ periodic crystals in the world’s largest collection CSD (Cambridge Structural Database).
Despite the CSD being curated to contain only real and distinct structures [29], the new invariants identiﬁed several pairs of duplicates.
All the underlying publications are now being investigated for data integrity by ﬁve journals, see section 6.
Problem 1.1 is stated in the hardest scenario when points are unordered and unlabeled because many real crystals have identical compositions.
For example, diamond and graphite (whose 2-dimensional layer is graphene) both consist of pure carbon but have vastly different physical properties.
Conditions (1.1cd) for a continuous metric are stronger than a complete classiﬁcation in (1.1ab): detecting an isometry gives a discontinuous metric d ( S, Q ) = 1 (or another positive number) for all non-isometric S (cid:54)∼ =
Q even if S, Q are near duplicates as in Fig.
2 (right).
Continuity under perturbations is practically important because atoms vibrate, and any real measurement of a crystal produces slightly different coordinates of its unit cell and motif.
Any simulation of periodic structures introduces ﬂoating point errors because of inevitable approximations by iterative optimization.
Thousands of near-duplicates are routinely produced, though only a few structures are synthesized.
Five real structures of 5679 predicted on a supercomputer over 12 weeks are a typical example [28].
This ‘embarrassment of over-prediction’ [51] wastes time and resources to run simulations and then analyze results, often by visual inspection, because of a lack of invariants that solve Problem 1.1.
Computability condition (1.1e) avoids the trivial function
I ( S ) =
S in Problem 1.1.
Inverse design in (1.1f) allows one to replace the traditional blind sampling (of ambiguous cells and motifs leading to (near-) duplicates via optimization) with a guided exploration of the crystal space parameterized by complete and reversible invariants.
Section 2 shows that the state-of-the-art tools remain stuck with conditions (1.1ab) while the new invariants satisfy the stronger practical requirements (1.1cdef).
A review of the related state-of-the-art on comparing periodic point sets
Computational Geometry methods studied isometry questions related to Problem 1.1 for ﬁnite sets of points.
The fastest algorithm to test an isometry between m -point sets in R n runs in time O ( m (cid:100) n/ 3 (cid:101) log m ) by [13], which can be improved to O ( m log m ) in R 4 [40].
Signiﬁcant results on matching bounded rigid shapes and registration of ﬁnite point sets were obtained in [55, 32, 30, 24].
These methods focus on true/false answers without continuous and easily computable metrics [57, 45]. The classical isometry invariant of a ﬁnite set in any metric space is the distribution of pairwise distances who completeness in general position was proved in 2004 [12].
The same paper described an inﬁnite family of singular counter-examples to the full completeness, see the non-isometric 4-point sets in Fig. 3 (left). Persistent homology is a more recent isometry invariant of ﬁnite point sets, which is continuous under perturbation but turned out to be weaker than previously anticipated [48, 26, 27, 60].
Mémoli’s seminal work on distributions of distances [44], also known as shape distributions [47, 9, 37, 34, 31] for bounded metric spaces is closest to the proposed Pointwise
Distance Distributions (PDD) for periodic point sets.
However, Problem 1.1 is not reducible to the ﬁnite sets by taking a cube or a ball of a ﬁxed (even very large) cut-off radius within a periodic point set.
Indeed, one can easily ﬁnd non-isometric subsets of the same periodic set as in Fig.
4 (left).
Computer Vision methods focused on alignment or registration of point clouds in dimensions 2, 3 [63, latest isometry invariant [17] deﬁned for a scalar ﬁeld R 3 → R 1 , which is more complicated than for a ﬁnite cloud. Completeness up to isometry was resolved by the invariants [41] with continuous metrics that have polymomial time in the number of points for a ﬁxed dimension. Crystallography and Materials Science developed many non-invariant descriptors by using quantities cell parameters and atomic coordinates in an ambiguous cell. Some approaches rely on ﬁnite subsets with a ﬁxed number of neighbors [18] or a ﬁxed cut-off radius [8, 7, 36].
A reduction to one subset can never guarantee a complete and continuous classiﬁcation because, under tiny perturbations, a minimal (by volume) cell can become larger than any subset of a ﬁxed size, see Fig.
2 (right).
Perturbations of atoms can change inter-atomic bonds depending on thresholds, which makes all graph-based descriptors [67, 20] discontinuous [6, 64] and unreliable for detecting near-duplicates. Another inconvenience of the tools such as SOAP (Smooth Overlapped Atomic Positions [7]) and ACSF (Atom-Centered Symmetry Functions [8]) is dependence on parameters that require manual choices. If these parameters change, then so do (possibly, discontinuously) the invariants, see Fig.
One of the oldest invariants of crystals is the X-ray diffraction pattern whose single crystal form is best for determining a 3D structure of an experimental crystal [21]. Since not all materials can be grown as single crystals, powder X-ray diffraction patterns (PXRD) are more common. All periodic structures with identical PXRDs are called homometric [53], see the periodic versions S (1) = { 0 , 1 , 3 , 4 } + 8 Z and Q (1) = { 0 , 3 , 4 , 5 } + 8 Z of the 4-point set T, K in Fig. The more general sets S ( r ) , Q ( r ) with the same Pair Distribution Function (PDF) will be distinguished by PDD in section 3.
The CCDC’s Mercury software compares periodic structures [18] by minimizing the Root Mean Square Deviation (RMSD) of atomic positions from up to a given number m (15 by default) of closest molecules in two structures. This comparison depends on many parameters (maximum number of matched molecules, thresholds for matched distances and angles), fails the triangle inequality in condition (1.1c), and is too slow for pairwise comparisons, see section 6. This RMSD tries to measure the maximum displacement of atoms only by using ﬁnite subsets, see the difﬁculties in Fig.
4 (left). Any point p in R n can be identiﬁed with the vector (cid:126)p from the origin 0 to p . The Euclidean distance between p, q ∈ R n is denoted by | p − q | , which is the length of the difference vector (cid:126)p − (cid:126)q .
For inﬁnite periodic crystals, the theoretically better alternative is the bottleneck distance d B ( S, Q ) equal to the maximum Euclidean distance needed to perturb every point p ∈ S to its unique match in Q .
Formally, d B ( S, Q ) =
inf g : S → Q sup p ∈ S | p − g ( p ) | is minimized over all bijections g and maximized over all points p ∈ S for a ﬁxed bijection. But d B is not a metric on all periodic sets because d B = + ∞ for nearly identical lattices, and d B is discontinuous under perturbations of lattice bases below. Example 2.1 (inﬁnite bottleneck) . We show that S =
Z and Q =
(1 + δ ) Z for any δ > 0 have d B ( S, Q ) = + ∞ . Assuming that d B ( S, Q ) is ﬁnite, consider an interval [ − N, N ] ⊂ R containing 2 N + 1 points of S . If there is a bijection g : S → Q such that | p − g ( p ) | ≤ d B for all points p ∈ S , the image of 2 N + 1 points S ∩ [ − N, N ] under this bijection g should be within the interval [ − N − d B , N + d B ] .
The last interval contains only 1 + 2( N + d B ) 1+ δ points, which is smaller than 1 + 2 N when N + d B 1+ δ < N or d B < δN .
We get a contradiction by choosing a large N > d B δ . [Equation] If we consider only periodic point sets S, Q ⊂ R n with the same density (or unit cells of the same size), the bottleneck distance d B ( S, Q ) becomes a well-deﬁned wobbling distance [16]. Example 2.2 (discontinuity of the wobbling distance) .
Slightly perturb the basis (1 , 0) , (0 , 1) of the integer lattice Z 2 to the basis vectors (1 , 0) , ( ε, 1) of the new lattice Λ .
We prove that d B (Λ , Z 2 ) ≥for
any ε > 0 .
Map R 2 by Z 2 -translations to the unit square [0 , 1] 2 with identiﬁed opposite sides (a torus). Then the whole square lattice Z 2 is mapped to the single point represented by the corners of the square [0 , 1] 2 . The perturbed lattice Λ maps to the {+sequence of points { kε (mod 1) } + ∞ k =
 ×{ 0 , 1 } in the horizontal edges.+}
If d B (Λ , Z 2 ) =
r < 13 , then all above points should be covered by the closed disks of the radius r centered at the corners of [0 , 1] 2 .
For 0 < ε < 13 − r , we can ﬁnd k ∈
Z so that kε is strictly between r, 1 − r , hence not covered by these disks, so d B (Λ , Z 2 ) ≥ 13 .
(cid:4)
The discontinuity under perturbations is the major weakness of many past invariants including
Voronoi diagrams, which should be matched via inﬁnitely many rotations [46], space groups and other group-theoretic invariants [35].
The key example in Fig.
2 (right) shows that a continuous distance between nearly identical sets should be close to 0, not exactly 0.
These sets have different symmetries and can be related only by pseudo-symmetries depending on manual thresholds [69].
Periodic Geometry develops invariants for Problem 1.1.
For any k ≥ 1 , the k -th density function ψ k ( t ) of a periodic set S = Λ+ M ⊂ R 3 is the total volume of the regions within the unit cell U of Λ covered by exactly k balls B ( p ; t ) with a radius t ≥ 0 and centres at motif points p ∈ M , divided by the cell volume Vol[ U ] .
The density function ψ k ( t ) was proved in [25] to be invariant under isometry, continuous under perturbations, complete for periodic sets satisfying certain conditions of general position in R 3 , and computable in a polynomial time in the motif size of S [59].
Section 5 in [25] gave the counter-example to completeness: the 1D sets S {+
+}
= X + Y +15 Z and Q 15 =
X − Y +15 Z for X = { 0 , 4 , 9 } and Y = { 0 , 1 , 3 } [38, section 4] have the same density functions for all k ≥ 1 [5, Example 10] but were distinguished by simpler invariants in [66, Example 5b].
The latest advance [4] reduces Problem 1.1 to an isoset of isometry classes of α -clusters around points in a motif at a certain radius α , which was motivated by the seminal work of Dolbilin [33, 11, 23]. The continuous metrics on isosets [3, section 5] have only an approximate algorithm.
Despite the progress in dimension[43] and lattices in dimension 2 [42, 14] and 3 [41, 15], Problem 1.1 remained open in general.
The Pointwise Distance Distribution of a ﬁnite or periodic point set
Distances to neighbors were considered in [66, Deﬁnition 5], though only their average was proved to be invariant.
Deﬁnition 3.1 introduces the matrix PDD whose ordered rows guarantee invariance in
Theorem 3.2 and weights of rows that guarantee continuity under perturbations in Theorem 4.3.
Deﬁnition 3.1 (Pointwise Distance Distribution PDD ) . For a motif of points M = { p 1 , . . .
, p m } in a unit cell U of a lattice Λ , let S ⊂ R n be a ﬁnite set coinciding with M or a periodic set S = Λ + M .
For an integer k ≥ 1 , consider the m × k matrix D ( S ; k ) , whose i -th row consists of the ordered distances d i 1 ≤ · · · ≤ d ik measured from p i to its ﬁrst k nearest neighbors in the full set S .
The rows of D ( S ; k ) are lexicographically ordered as follows.
A row ( d i 1 , . . .
, d ik ) is smaller than ( d j 1 , . . .
, d jk ) if a few ﬁrst distances coincide: d i 1 =
d j 1 , . . .
, d il = d jl for l ∈ { 1 , . . .
, k − 1 } and the next ( l + 1) -st
distances satisfy d i,l +1 <
d j,l +1 .
If w rows are identical to each other, any such group is collapsed to one row with the weight w/m .
For each row, put this weight in the ﬁrst column.
The ﬁnal m × ( k + 1)
-matrix is the Pointwise Distance Distribution
PDD
( S ; k ) .
(cid:4)
The matrix D ( T ; 3) in Table 1 has two pairs of identical rows, so the matrix PDD( T ; 3) consists of two rows of weight 12 below.
The matrix D ( K ; 3) in Table 1 has only one pair of identical rows, so
PDD( K ; 3) has three rows of weights 12 , 14 , 14 .
Hence PDD( T ; 3) (cid:54) =
PDD( K ; 3) .
Each point in T, K ⊂ R 2 from Figure 3 has ordered distances to three other points.
[Table]
[Equation]
[Table]
[Equation]
Theorem 3.2 (invariance of PDD , all theorems are proved in the extended version [65]) .
For any ﬁnite or periodic set S ⊂ R n , PDD( S ; k ) from Deﬁnition 3.1 is an isometry invariant of S , k ≥ 1 .
(cid:4)
[Table]
p
[Equation]
(2 + r ) − 0 | = 2 + r p 4 =| 4 − (2 + r ) | = 2 − r | 4 − r | = 4 − r | 4 − 0 | =
[Table]
points distance to neighbordistance to neighbordistance to neighborp 1 =| 0 − (2 + r ) |
= 2 + r | 0 − ( r + 4 − 8) | = 4 − r | 0 − 4 | =p 2 = 2 + r | (2 + r ) − 4 | = 2 − r | (2 + r ) − (4 + r ) | =| (2 + r ) − 0 | = 2 + r p 3 =| 4 − (4 + r ) | =
r | 4 − (2 + r ) | = 2 − r | 4 − 0 | =p 4 = 4 + r | (4 + r ) − 4 | = r | (4 + r ) − (2 + r ) | =| (4 + r ) − 8 | = 4 − r
For the 1D periodic sets S ( r ) = { 0 , r, 2 + r, 4 } + 8 Z and Q ( r ) = { 0 , 2 + r, 4 , 4 + r } + 8 Z with unit cell [0 , 8] in Fig.
3, Table 2 shows that S ( r ) , Q ( r ) are not isometric for any parameter 0 < r ≤ 1 .
PDD( S ( r ); 8) =   1 /r 2 + r  6 − r 8 − r 81 /r 4 − r 4 + r 8 − r 81 / 4 2 − r 2 + r 6 − r 6 + r 81 / 4 2 − r 4 − r  4 + r 6 + r 8  (cid:54) =
PDD( Q ( r ); 8) =   1 /r 2 − r  6 + r 8 − r 81 /r 4 − r 4 + r 8 − r 81 / 4 2 − r 2 + r 6 − r 6 + r 81 / 4 2 + r 4 − r  4 + r 6 − r 8  .
The averages of columns in PDD( S ; k ) form the vector AMD( S ; k ) of Average Minimum Distances [66, Deﬁnition 5].
Any lattice Λ ⊂ R n has a 1-point motif M = { p } , hence PDD( S ; k ) is a single row of increasing distances from p to all other points Λ − { p } .
Fig.
6 (right) shows a honeycomb set
S whose motif consists of two points that have the same distances to all their neighbors, hence two rows of D ( S ; k ) collapse to a single vector. So PDD( S ; k ) = AMD( S ; k ) in the cases above.
Example 3.3 (PDD is stronger than AMD) .
Since both sets S ( r ) , Q ( r ) in Fig.
3 (right) have period 8, the matrices PDD( S ( r ); k ) and PDD( Q ( r ); k ) have distance 8 in each row for columns 7 andas shown above.
All further distances are obtained from the ﬁrst eight by adding a multiple of period 8.
The vector AMD( S ( r ); k ) of column averages for any k ≥ 8 is determined by AMD( S ( r ); 8) = (1 , 2 . 5 , 3 . 5 , 4 . 5 , 5 . 5 , 7 , 8 , 8) .
Since the components of AMD( S ( r ); k ) do not depend on the parameter 0 < r < 1 , the sets S ( r ) are counter-examples to the completeness of AMD , now distinguished by PDD( S ( r ); k ) already for k = 1 .
Hence PDD( S ; k ) is strictly stronger than AMD( S ; k ) .
(cid:4)
For a periodic set S , the number k in PDD( S ; k ) can be considered as a degree of approximation (or a count of decimal places), not as a parameter that affects invariant values.
If we increase k , we extract more distant geometric data from S by adding more columns to PDD( S ; k ) and keeping all previous distances.
If some rows are identical in D ( S ; k − 1) and become different in D ( S ; k ) , we recompute weights but not distances.
The past tools [18, 7] strongly depend on extra parameters.
Now we compare PDD with the closest past invariant, the Pair Distribution Function (PDF).
For a periodic point set S ⊂ R n with a motif M , the exact ePDF( S ) consists of ordered distances from all points p ∈ M to all other points q ∈ S − { p } .
So the inﬁnite sequence ePDF( S ) is obtained from
PDD by combining all rows into one sequence and losing weights.
Additionally, we keep only one distance from each pair | p − q | = | q − p | .
For any ﬁxed 0 < r ≤ 1 , the sets S ( r ) , Q ( r ) have the same sequences starting with ePDF( S ( r ))
=
ePDF( Q ( r ))
=
{ r, 2 − r, 2 , 2 + r, 4 − r, 4 , 4 , 4 + r, . . . } .
This example shows that PDD for k = 1 is strictly stronger than ePDF as an isometry invariant.
For any lattice Λ ⊂ R n , the vector ePDF(Λ; k ) up to k distances concides with PDD( S ; k ) .
For the honeycomb set S in Fig.
6 (right), ePDF( S ; 2 k ) is obtained from PDD( S ; k ) by repeating every distance twice.
If a periodic set is perturbed and a unit cell doubles as in Fig.
2 (right), then every distance in ePDF is replaced by a couple of (near-) duplicate distances, so ePDF
( S ; k ) discontinuously changes by including twice as many short distances and losing longer distances.
This typical discontinuity was roughly repaired by replacing every single distance d with its Gaussian distribution exp( − ( x − d ) 2 / 2 σ ) with a parameter σ > 0 .
Then a normalized sum of such ‘blurred’ distances [62] becomes the smooth Pair Distribution Function PDF( S ; σ ) .
Since algorithms can compare only ﬁnite vectors, this PDF( S ; σ ) is then uniformly sampled, which creates dependence on σ and sampling.
So PDD provides an easy alternative to this counter-intuitive PDF pipeline { discrete sequence } → { smooth function } → { discrete sequence } , whose continuity wasn’t formally proved.
Continuity and generic completeness of Pointwise Distance Distributions
Continuity of PDD( S ; k ) under perturbations of S in the bottleneck distance d B will be measured by the Earth Mover’s Distance [54], which can be applied to any weighted distributions of different sizes.
Deﬁnition 4.1 is for any vector
I ( S ) =
([ w 1 ( S ) , R 1 ( S )] , . . . , [ w m ( S ) , R m ( S ) ( S )]) of pointwise invariants of a set S with weights w i ( S ) ∈ (0 , 1] satisfying the normalization m ( S ) (cid:80) i =1 w i ( S ) = 1 .
Later we consider only the case when the weighted vector [ w i , R i ] is the i -th row of PDD( S ; k ) .
Then m ( S ) is the number of rows in PDD( S ; k ) .
Each row R i ( S ) should have a size independent of S , for example a number k of neighbors in PDD( S ; k ) .
For any vectors R i =
( r i 1 , . . .
, r ik ) and R j = ( r j 1 , . . .
, r jk ) of a length k , we use the L ∞ -distance
| R i − R j | ∞ =
max l =1 ,...,k | r il − r jl | ∞ .
Deﬁnition 4.1 (EMD) . Let ﬁnite or periodic sets S, Q ⊂ R n have weighted vectors I ( S ) , I ( Q ) as discussed above.
A ﬂow from I ( S ) to I ( Q ) is an m ( S ) × m
( Q ) matrix whose element f ij ∈ [0 , 1] represents a partial ﬂow from R i ( S ) to R j ( Q ) .
The Earth Mover’s Distance is the minimum cost
EMD( I ( S ) , I ( Q )) =
m ( S ) (cid:80) i
= m ( Q ) (cid:80) j
=1 f ij |
R i ( S ) − R j ( Q ) | for f ij ∈ [0 , 1] subject to m ( Q ) (cid:80) j
=1 f ij ≤ w i ( S ) for i = 1 , . .
, m ( S ) , m ( S ) (cid:80) i =1 f ij ≤ w j ( Q ) for j = 1 , . .
, m ( Q ) , and m ( S ) (cid:80) i = m ( Q ) (cid:80) j =1 f ij = 1 .
(cid:4)
The ﬁrst condition m ( Q ) (cid:80) j =1 f ij ≤ w i ( S ) means that not more than the weight w i ( S ) of the component
R i ( S ) ‘ﬂows’ into all components R j ( Q ) via ‘ﬂows’ f ij , j = 1 , . . .
, m ( Q ) .
Similarly, the second condition m ( S ) (cid:80) i =1 f ij = w j ( Q ) means that all ‘ﬂows’ f ij from R i ( S ) for i = 1 , . . .
, m ( S ) ‘ﬂow’ into R j ( Q ) up to the maximum weight w j ( Q ) .
The last condition m ( S ) (cid:80) i = m ( Q ) (cid:80) j =1 f ij = 1 forces to ‘ﬂow’ all rows R i ( S ) to all rows R j ( Q ) .
The EMD satisﬁes all metric axioms [54, appendix], needs
O ( m 3 log m ) time for distributions of a maximum size m and is approximated in O ( m ) time [58, 56].
Theorem 4.2 (lower bound of EMD ) .
For ﬁnite or periodic point sets S, Q ⊂ R n , and k ≥ 1 , the distances satisfy EMD(PDD( S ; k ) , PDD( Q ; k )) ≥ ||
AMD( S ; k ) − AMD( Q ; k ))
|| ∞ .
(cid:4)
Theorem 4.3 uses the bottleneck distance d B ( S, Q ) =
inf g : S → Q sup p ∈ S | p − g ( p ) | and the packing radius r ( S ) , which is the minimum half-distance between any points of S .
Equivalently, r ( S ) is the maximum radius r to have disjoint open balls of radius r centered at all points of S .
Theorem 4.3 (continuity of PDD ) .
For any k ≥ 1 , if ﬁnite or periodic sets S, Q ⊂ R n satisfy d B ( S, Q ) < r ( S ) , then EMD(PDD( S ; k ) , PDD( Q ; k )) ≤ 2 d B ( S, Q ) .
(cid:4)
Continuity Theorem 4.3 means that any small perturbation of atomic positions in the bottleneck distance d B leads to a small change of the Pointwise Distance Distribution in the Earth Mover’s
Distance.
Theorem 4.3 extends the following fact for 2-point sets ( k = 1 ).
If we perturb two points by at most ε , the distance between them changes by at most 2 ε .
For any set S ⊂ R n of m points with distinct inter-point distances, completeness of PDD( S ; m − 1) follows from [66, Theorem 16].
Following the earlier work [25, section 5.1], the extended version [65] deﬁnes a distance-generic set that can approximate any periodic point set S = Λ + M ⊂ R n .
The number m of points in a unit cell U is an isometry invariant because any isometry maps U to another cell with the same number m of points.
Theorem 4.4 assumes that a lattice Λ is given and reconstructs a periodic point set S = Λ + M in any dimension n ≥ 2 . However, in low dimensions n = 2 , 3 , any lattice Λ can be reconstructed from its isometry invariants described in [19, 42, 41].
Theorem 4.4 (generic completeness of PDD ) . Let S = Λ+ M ⊂ R n be a distance-generic periodic set with m points in a motif M .
Let R (Λ) be the smallest radius such that all closed balls with centers p ∈ Λ cover R n .
Let 2 R (Λ) be smaller than all distances in the last column of PDD( S ; k ) for a big enough k .
The set S can be uniquely reconstructed up to isometry from Λ , m ,
(cid:4)
Polynomial time algorithms and experimental comparisons of PDD
The algorithm for PDD in Theorem 5.1 found several pairs of unexpected duplicates, which were missed by all past tools, through 200B+ pairwise comparisons of 660K+ real periodic crystals over a couple of days on AMD Ryzen 5 5600X (6-core) @4.60Ghz, 32GB DDR4 RAM @3600 Mhz.
The main input is a periodic point set S ⊂ R n given by a unit cell U and a motif of m points in U . The key parameters of PDD( S ; k ) are m and k .
The complexity in Theorem 5.1 is near-linear in both k and m for a ﬁxed dimension n .
The output PDD( S ; k ) is a matrix with at most m rows and exactly k + 1 columns.
The ﬁrst column contains the weights of rows, which sum to 1 and are proportional to the number of appearances of the row before collapsing, see the detailed code in [65].
Theorem 5.1 ( PDD complexity) . Let a periodic set S ⊂ R n have m points in a unit cell U .
For a ﬁxed dimension n , PDD( S ; k ) is computed in a near-linear time O ( km (5 ν ) n V n log( m ) log 2 ( k )) , where V n is the unit ball volume in R n , d and ν = d n √ Vol
[ U ] are the diameter and skewness of U .
(cid:4)
The near-linear time complexity in Theorem 5.1 goes far beyond the state-of-the-art. Sectionreviewed that all past tools are based on ambiguous non-invariant data or discontinuous invariants that miss (near-)duplicates, or are too slow for pairwise comparisons of millions of crystal structures.
The recent continuous invariants with theoretical (not exactly computable) metrics [25, section 6] and [2, section 8] require cubic algorithms, which turned out to be unrealistic for large crystal datasets.
The Cambridge Crystallographic Data Centre (CCDC) has curated the world’s largest collection of real solid crystalline materials since the 1960s, now having nearly 1.2M known structures.
New crystalline material is deposited in the CSD only after a peer-reviewed publication.
The CCDC should ideally check that a new structure is genuine and not a duplicate of an earlier one because their data is trusted by all pharmaceutical giants developing new drugs in a crystalline form.
However, the CSD is basically a huge list of Crystallographic Information Files representing crystals by unit cells and motifs of points in coordinates of a cell basis with limited search and comparison capabilities.
The papers [28, 68] reported ﬁve experimental T2 crystals (based on the same molecule T2) that were successfully synthesized after predicting 5679 crystals through 12-week simulations on a supercomputer.
All initial 2M+ randomly sampled crystals were iteratively optimized to the ‘most stable’ approximations of local energy minima.
This is a typical ‘embarrassment of over-prediction’ when many (near-)duplicates are found around the same local minimum but remain undetected.
One striking example is the pair of crystals 14 and 15 in Fig.
7 reported in [28], which have very different unit cells but are nearly identical.
When this pair was compared by another free software Platon [61], a bug was discovered, which required a few months to be ﬁxed.
Such bugs will keep emerging because the discontinuity of past invariants and metrics was not addressed as in Problem 1.1.
For example, a rough sampling of the density functions ψ k ( t ) from [25] of 5679 crystals for up to k = 8 took more than four days on a comparable machine.
This experiment detected the T2- δ crystal that was accidentally not deposited in the CSD because of visual confusion with another structure.
Conclusions and discussions of current limitations and duplicate detection
The popular packing similarity [18] algorithm COMPACK is available in the CCDC’s Mercury software.
The 4950 comparisons of the 100 lowest energy crystals close to T2- δ in density by packing similarity took 3 hours 53 min, 2.8 seconds.
Extrapolating this time for comparing any new structure with the whole CSD gives 38 days, hence more than 68 thousand years for pairwise comparisons of all 660K+ periodic crystals in the CSD.
A typical comparison by AMD and PDD runs in nanoseconds and milliseconds, respectively, so comparing 100 crystals pairwise takes less than one minute.
small numbers of molecules
[Table]
Most importantly, more than half of all comparisons in Table 3 matched only one molecule from two crystals.
Since all crystals consist of the same rigid molecule T2, this output means a complete failure: one common molecule, no other conclusions.
Since the CCDC deposits hundreds of new structures daily, the shortcut approach is to compare the chemical composition of atoms.
But even water (H 2 O) has at least 15 forms of ice crystals, while other compositions have many more polymorphic forms in the CSD.
This comparison by composition can miss duplicates where one atom is incorrect.
The unreliability of past comparisons fueled the growing crisis of fake crystallographic data [10] leading to the investigation of nearly 1000 structures in the CSD. The new invariants PDD detected ﬁve pairs of unexpected duplicates (‘needles in the haystack’) in the Cambridge Structural Database (CSD).
First, the simpler invariants AMD( S ; 100) were computed for all 660K+ periodic structures in the CSD, without disorder and with full geometric data.
More than 200 billion pairwise comparisons of AMD( S ; 100) vectors revealed 6371 pairs S, Q with | AMD( S ; 100) − AMD( Q ; 100) | ∞ ≤ 0 .
As an AMD is simpler and faster to compare, up to the order of 10 − 7 seconds per comparison, this step took around 8 hours.
This example and Theorem 4.2 show AMD is a good ﬁlter.
Second, computing the L ∞ -based EMD between the pairs above detected 182 pairs with EMD < 0 .
Most of these pairs were expected and were the same crystal, or different aliases for the same database entry.
The ﬁve pairs reported in [66, section 7] were unexpected because the underlying periodic sets of points at atomic centres were truly isometric (to the last decimal place) but one atom had different chemical elements in two crystals.
The crystals with the CSD codes HIFCAB and
JEPLIA are literally isometric but one Cadmium is replaced by Manganese at the same position.
All past tools taking into account atomic types see these geometric duplicates as different.
The CCDC agreed that such a coincidence is physically impossible because another atom should have slightly different distances to neighbors detected by PDD .
Hence at least one of the structures in each pair cannot be correct.
The ﬁve journals are investigating the data integrity of the underlying publications.
Detecting duplicates might have some negative impact on past publications that could be retracted
[-EMD = 
-]
In conclusion, Theorems 3.2, 4.3, 5.1 , 4.4 fulﬁlled almost all conditions of Problem 1.1, while all past tools remained discontinuous or too slow for billions of comparisons.
The only limitation is a hypothetical existence of singular sets S (cid:54)∼ = Q with PDD( S ; k ) =
PDD( Q ; k ) for all k ≥ 1 .
We found no such examples among all real materials in the CSD.
Even if such a singular pair of S, Q emerges, they can be distinguished by the slower and provably complete invariant isoset [4, 3].
Hence PDD and complete isoset can be considered as a DNA-style code or materials genome uniquely and continuously identifying any periodic crystal.
More importantly, the experiments conﬁrmed the
Crystal Isometry Principle [66, section 7]: the map { periodic crystals } → { periodic point sets } is injective (doesn’t lose information) modulo isometry.
Hence all existing and not yet undiscovered crystals live in the common Crystal Isometry Space (CRISP) of isometry classes of periodic point sets continuously parameterized by complete isometry invariants.
This CRISP can be considered a continuous universe containing all known crystals as visible stars in the universe, which requires mapping and exploration.
Since inter-atomic distances cannot be arbitrary, the next step is to describe subspaces of CRISP that represent realistic but still unknown materials.
This ﬁrst map of this space for 2D lattices appeared in [14].
We thank all reviewers for their valuable time and suggestions.