cshm

Computes the Continuous Shape Measures(CShM) between a structure and one or more idealized reference shapes. Cosmochlore includes 92 built-in reference polyhedra, and user-defined ones can be supplied via YAML.

cosmochlore cshm <NAME> [OPTIONS]

Options

Flag	Value	Description
-n, --nc		Treat the problem shape as non-centered, that is, the structure does not include an explicit central atom (only ligand/vertex coordinates are given).
-c, --center	<POSITION>	1-based position of the atom in the .xyz table that is the centre of the structure. Defaults to the first atom (position 1) when omitted. Cannot be combined with --nc.
-s, --sh	<SHAPES>...	Restrict the comparison to specific built-in reference shapes by index, for the detected vertex count. If omitted, all applicable built-in shapes are used. Separate the indices with whitespace.
-r, --ref	<USER_SHAPES>...	Paths to the YAML files that contain user-defined shapes to include in the CShM calculation. Separate the files with whitespace.
-t, --table		Write a name_cshm_table.csv file containing the output of the calculation.
-i, --ideal		Write a name_ideal.xyz file containing the reconstructed idealised structures for all the selected reference shapes.

Oh…, I lost my crab here. 🦀 Thanks for finding it!

Optimal transformation finding

For any given shapes, it can be shown that the optimal translation and scaling are obtained by centring both shapes on their geometric centres and normalizing their size. The optimal rotation $R$ is then the one that minimizes ${\sum }_i^N|R{q}_i-p_i{|}^2$, which is solved via singular-value decomposition of the covariance matrix $H=P^{T}Q$.

Vertex pairing

Naively finding the best point-pair matches between $Q$ and $P$, grows with $N!$. The combinatorial problem is tackled by:

• Fixed centre: for centred structures, the central atom is paired with the reference shape’s centre, reducing the dimensionality of the problem to $(N-1)!$
• Branch and bound: after each partial assignment of point pairs, a best-scenario $S$ value is calculated. Permutations that cannot beat the best $S$ found so far are discarded early.

Computational cost

Even by taking these measures, the calculation becomes impractical at $N>13$ and completely unaccessible for $N>24$. There is a plan to include heuristic matching for large shapes in cshm in the future.

Examples

Distorted octahedral iron(II) complex, with a user-defined shape

Given the following FeHS.xyz file:

FeHS.xyz

7
High-spin iron(ii) complex
Fe6 4.93013 10.3906 12.9236
N004 6.21021 10.6927 14.7824
N006 4.99632 8.50231 13.9489
N008 6.81046 10.0777 11.7691
N00B 5.46476 12.2907 12.1187
N00C 3.30099 11.3236 13.8608
N00D 3.74774 9.47633 11.4653

Running:

cosmochlore cshm FeHS.xyz --ref ebcT-6.yaml -t -i

Will output:

Input file: FeHS.xyz
============================================================
Symbol   Shape                           Symmetry   CShM
------------------------------------------------------------
HP-6     Hexagon                         D6h        33.210
PPY-6    Pentagonal pyramid              C5v        23.050
OC-6     Octahedron                      Oh         2.089
TPR-6    Trigonal prism                  D3h        11.079
JPPY-6   Johnson pentagonal pyramid J2   C5v        27.050
ebcT-6   Edge-bicapped tetrahedron       D2d        14.277
------------------------------------------------------------
Writing output table to FeHS_cshm_table.csv...
Writing idealised polyhedra coordinates to table to FeHS_ideal.xyz...
Program finished in 6.8154ms

The lowest CShM is against the octahedron (OC-6, 2.089), meaning the iron centre has a moderately distorted octahedral coordination environment.

The output of the calculation is saved in the file FeHS_cshm_table.csv by passing the -t/--table flag.

FeHS_cshm_table.csv

Symbol,Name,Symmetry,CShM
HP-6,Hexagon,D6h,33.210
PPY-6,Pentagonal pyramid,C5v,23.050
OC-6,Octahedron,Oh,2.089
TPR-6,Trigonal prism,D3h,11.079
JPPY-6,Johnson pentagonal pyramid J2,C5v,27.050
ebcT-6,Edge-bicapped tetrahedron,D2d,14.277

The FeHS_ideal.xyz file was produced by passing the -i/--ideal flag. The coordinates of the ideal octahedron placed in the correct position of the structure can be extracted from it.

Note that ebcT-6 is a non-standard reference shape included by passing the --ref flag. Its YAML definition is on the user-defined shapes page.

Non-centred structure: a pyridine ring

A pyridine ring has no central atom, so the -n/--nc flag is needed. All six atoms are then treated as vertices.

Co110_pyridine.xyz

6
Co110 part 0
N14 2.178776 6.987902 12.271005
C19 0.680230 6.204588 14.464450
C20 1.483930 5.816450 12.233931
C25 0.735126 5.411294 13.325930
C30 1.384024 7.394160 14.494005
C17 2.124076 7.759082 13.373102

cosmochlore cshm Co110_pyridine.xyz -n

Will output:

Input file: Co110_pyridine.xyz
============================================================
Symbol   Shape                           Symmetry   CShM
------------------------------------------------------------
HP-6     Hexagon                         D6h        0.014
PPY-6    Pentagonal pyramid              C5v        27.994
OC-6     Octahedron                      Oh         33.244
TPR-6    Trigonal prism                  D3h        33.535
JPPY-6   Johnson pentagonal pyramid J2   C5v        25.135
------------------------------------------------------------
Program finished in 906.3µs

The ring is an almost perfect hexagon (HP-6, 0.014). CShM only compares positions, so the nitrogen atom is treated like any other vertex.

Restricting the comparison to selected shapes

For a five-coordinate cadmium centre, only the trigonal bipyramid and the square pyramid are of interest. Their indices for five vertices are 2 (TBPY-5) and 3 (SPY-5):

CdONCl3.xyz

6
Five-coordinate Cd1 (CdONCl3) of Acta Cryst. E81, 393-400 (CCDC 2441688)
Cd1     6.18403    12.69910     7.20280
O1      5.03750    12.82268     5.15546
N3      7.81940    12.17629     5.69285
Cl1     4.30070    11.74221     8.44098
Cl2     6.38607    15.10826     7.98044
Cl3     8.04110    12.06770     9.04477

cosmochlore cshm CdONCl3.xyz -s 2 3

Will output:

Input file: CdONCl3.xyz
=======================================================
Symbol   Shape                      Symmetry   CShM
-------------------------------------------------------
TBPY-5   Trigonal bipyramid         D3h        2.871
SPY-5    Spherical square pyramid   C4v        2.039
-------------------------------------------------------
Program finished in 470.9µs

The cadmium centre is closer to a square pyramid than to a trigonal bipyramid, but it is distorted from both.