Glenn D. Hibbard and
John Çamkıran
Department of Materials Science and Engineering, University of Toronto, 184 College St, Toronto, ON M5S 3E4, Canada. E-mail: john.camkiran@utoronto.ca
First published on 11th September 2024
We call for a theory of the particle-scale structure of materials that is based on the general notion of information rather than its special case of symmetry. An inherent limitation to the symmetry-based understanding of structure is described. The rapid decay in interaction strength with interparticle distance is used to argue for the representability of a system locally through the neighbourhoods of its constituent particles. The extracopularity coefficient E is presented as a local quantifier of information and compared to point group order |G|, a local quantifier of symmetry. The former is found to have nearly double the resolution of the latter for a set of commonly encountered coordination geometries. A proof is given for the generality of extracopularity over point symmetry. Some practical challenges and future perspectives are discussed.
But while particle positions are what formally specify the state of a system, it is the distances between those particles that constitute the fundamental quantity underlying structure. This follows from the fact that particle positions can be recovered from interparticle distances up to translation, rotation, and reflection,10 which have no effect on the total energy of a system. Indeed, the information content of particle positions is simply the information content of interparticle distances plus some extraneous information regarding the system's overall position, orientation, and handedness. It is possible to express this relationship symbolically using Shannon's information entropy functional H as follows:8
H(P) = H(D) + H(T), | (1) |
Particle neighbourhoods tend to be approximately spherical (i.e. characterised by neighbours nearly equidistant from the central particle) as a consequence of the isotropy of space. By virtue of this approximate sphericality, the distance between any two neighbours of a given particle depends almost entirely on the angle between the vectors that indicate their position relative to the central particle; such vectors are commonly called bonds,11 and the angles between them, bond angles.12 Fig. 1 illustrates this dependence geometrically.
Just as with the position–distance relationship, the distance–angle relationship can be captured using the functional H. Observe that the triple consisting of bond angles Θ, bond length differences Δ, and the nearest-neighbour distance fully determines and is determined by interparticle distances D. These quantities are thereby equivalent in information,
H(D) = H(Θ, Δ, ![]() | (2) |
Twice applying the chain rule8 to the right-hand side, then invoking the independence between Θ and , we get the following general relationship:
H(D) = H(Θ) + H(Δ|![]() ![]() | (3) |
At the spherical neighbourhood limit, vanishing bond length differences render H(Δ) = 0. By monotonicity, H(Δ|,Θ) ≤ H(Δ), and nonnegativity, H(Δ|
,Θ) ≥ 0, we also have H(Δ|
,Θ) = 0.8 Thus, interparticle distances D are informationally equivalent to bond angles Θ, up to a contribution from the nearest-neighbour distance
,
H(D) = H(Θ) + H(![]() | (4) |
In systems with a simple crystalline ground state, H() tends to 0 with increasing density, leaving H(D) = H(Θ). This underscores the importance of bond angles to structure at the particle scale.
Consider, for instance, the icosahedral geometry, which is of special importance to systems like supercooled liquids14 and metallic glasses.15 Observe that the 66 bond pairs implied by its 12 bonds make only three different angles, namely ∼63.4°, ∼116.6°, and 180°. These pairs are redundant in the sense that only three different angles suffice to describe all 66 of them; they are thus losslessly compressible.8 Repeating this exercise for the three other geometries in Fig. 2, the number of different bond angles is found to be a quantity able to distinguish between the various ways of arranging 12 neighbours around a particle.
In a previous work,16 the occurrence of fewer different bond angles than combinatorially possible was termed extracopularity. This phenomenon was quantified by a coefficient E, defined as the conditional Hartley entropy17 of bond pairs given bond angles,
E = log2(n) − log2(m), n > 0, | (5) |
![]() | (6) |
As a tool, E has found early application in the studies of convex polyhedra,18 cellular materials,19 and radiation damage.20
Geometry | Point symmetry | Extracopularity | |||
---|---|---|---|---|---|
G | |G| | k | m | E | |
ICO | Ih | 120 | 12 | 3 | 4.46 |
FCC | Oh | 48 | 12 | 4 | 4.04 |
HCP | D3h | 12 | 12 | 6 | 3.46 |
BPP | D5h | 20 | 12 | 7 | 3.24 |
Where the advantage of the information-based approach shows is in the analysis of coordination geometries that are symmetrically identical. Table 2 presents E and |G| for cubic geometries, which are characterised by the point group Oh. Here, |G| fails at the most basic level, being unable to make any distinction between those coordination geometries much less rank them on an ordinal scale.21 By contrast, E succeeds in both respects. The ranking implied by E is further observed to agree with that of the familiar atomic packing fraction (ηFCC = 0.74, ηBCC = 0.68, ηSC = 0.52). In this way, E appears to be able to capture the insights of point symmetry and packing fraction in a single quantity.
Geometry | Point symmetry | Extracopularity | |||
---|---|---|---|---|---|
G | |G| | k | m | E | |
FCC | Oh | 48 | 12 | 4 | 4.04 |
BCC | Oh | 48 | 14 | 5 | 3.92 |
SC | Oh | 48 | 6 | 2 | 2.91 |
Table 3 expands the comparison to 22 of the most commonly encountered coordination geometries in the physical sciences. Both E and |G| are maximal for the icosahedral geometry but vary in their minima. For these 22 geometries, E is found to take 17 distinct values while |G| is found to take 9. With nearly double the resolution, E appears to be the more capable classifier. Moreover, in those 5 geometries with non-distinct E, differences in the bond angle distribution open up the possibility for an adjusted coefficient that can distinguish all 22 geometries. By contrast, the presence of only 13 distinct groups imposes a suboptimal upper bound on the resolution of symmetry-based approaches.
Abbreviation | Coordination geometry | Polyhedral classification | k | m | E | G | |G| |
---|---|---|---|---|---|---|---|
TBP | Trigonal bipyramidal | Deltahedral, bipyramidal | 5 | 3 | 1.737 | D3h | 12 |
SDS | Snub disphenoidal | Deltahedral | 8 | 6 | 2.222 | D2d | 8 |
CTP | Capped trigonal prismatic | Prismatic | 7 | 4 | 2.392 | C2v | 4 |
PBP | Pentagonal bipyramidal | Deltahedral, bipyramidal | 7 | 4 | 2.392 | D5h | 20 |
BTP | Bicapped trigonal prismatic | Prismatic | 8 | 5 | 2.485 | C2v | 4 |
TET | Regular tetrahedral | Platonic, deltahedral | 4 | 1 | 2.585 | Td | 24 |
HBP | Hexagonal bipyramidal | Bipyramidal | 8 | 4 | 2.807 | D6h | 24 |
CSA | Capped square antiprismatic | Antiprismatic | 9 | 5 | 2.848 | C4v | 8 |
CSP | Capped square prismatic | Prismatic | 9 | 5 | 2.848 | C4v | 8 |
TTP | Tricapped trigonal prismatic | Prismatic, deltahedral | 9 | 5 | 2.848 | D3h | 12 |
SC | Regular octahedral | Platonic, deltahedral, bipyramidal | 6 | 2 | 2.907 | Oh | 48 |
BSA | Bicapped square antiprismatic | Deltahedral, antiprismatic | 10 | 6 | 2.907 | D4d | 16 |
BSP | Bicapped square prismatic | Prismatic | 10 | 5 | 3.170 | D4 | 8 |
CPP | Capped pentagonal prismatic | Prismatic | 11 | 6 | 3.196 | C5v | 10 |
SA | Square antiprismatic | Antiprismatic | 8 | 3 | 3.222 | D4d | 16 |
HDR | Regular hexahedral | Platonic, prismatic | 8 | 3 | 3.222 | Oh | 48 |
BPP | Bicapped pentagonal prismatic | Prismatic | 12 | 7 | 3.237 | D5 | 20 |
HCP | Anticuboctahedral | Bicupolar | 12 | 6 | 3.459 | D3h | 12 |
BCC | Rhombic dodecahedral | Catalan | 14 | 6 | 3.923 | Oh | 48 |
FCC | Cuboctahedral | Bicupolar | 12 | 4 | 4.044 | Oh | 48 |
CPA | Capped pentagonal antiprismatic | Antiprismatic | 11 | 3 | 4.196 | C5v | 10 |
ICO | Regular icosahedral | Platonic, deltahedral, antiprismatic | 12 | 3 | 4.459 | Ih | 120 |
Theorem. Consider a geometry of k > 2 bonds, equal in length. Then, nontrivial point symmetry (|G| > 1) implies nontrivial extracopularity (E > 0), but not conversely.
Proof. We first prove the implication, then disprove its converse.
Part 1. The implication is proven directly.
Let |G| > 1. Then, there exists a map μ ∈ G such that {b,b′} and {β,β′} := {μ(b),μ(b′)} are two distinct bond pairs satisfying
![]() | (7) |
Since all four bonds are equal in length, the two pairs also make the same angle,
![]() | (8) |
But if two distinct bond pairs have the same angle, then the number of different bond angles is less than the total number of bond pairs, m < n. Recalling that E = log2(n) − log2(m), we have
E > 0. | (9) |
Part 2. The converse is disproven by counterexample.
Consider a geometry of k > 3 bonds for which there exists a unique pair of distinct bond pairs {b,b′} ≠ {b,b′′} such that
d(b,b′) = d(b,b′′). | (10) |
This has the following two consequences:
• There exists an angular equality. This implies that there are fewer bond angles than bond pairs, m < n. As a result,
E > 0. | (11) |
• There does not exist a nontrivial isometry μ ∈ G. Since point groups consist only of isometries, G must be trivial. Thus,
|G| = 1. | (12) |
This counterexample is illustrated in Fig. 4
![]() | ||
Fig. 4 Introducing a fourth bond pointing anywhere off the meridional circle M on this sphere trivialises point symmetry but not extracopularity. |
There are two noteworthy challenges to the informational study of structure through E in particular. Firstly, the number of different bond angles, m, can be difficult to ascertain in practice, with a variety of factors like thermal fluctuations ensuring that this is not as simple as counting the number of distinct angles.22 Secondly, the extracopularity coefficient does not in its current form account for variation in bond lengths; such variation could constitute an important feature of structure, especially in higher entropy systems like liquids and polydisperse packings.
In addition to addressing the above limitations, it will be necessary to link structure as it is observed locally by E to structure as it prevails globally to determine properties. Here, various statistical techniques present a promising avenue forward, in particular, simple descriptive statistics like the mean and variance, the joint distribution of neighbouring particle coefficients, and a coefficient correlation function similar in spirit to the radial distribution function.22
This journal is © The Royal Society of Chemistry 2024 |