average-minimum-distance 1.3.5

Descriptors of crystals based on geometry (isometry invariants).

average-minimum-distance: geometry based crystal descriptors

• PyPI project: https://pypi.org/project/average-minimum-distance
• Documentation: https://average-minimum-distance.readthedocs.io
• Source code: https://github.com/dwiddo/average-minimum-distance
• References (bib references at the bottom of this page):
  • Average minimum distances of periodic point sets - foundational invariants for mapping periodic crystals. MATCH Communications in Mathematical and in Computer Chemistry, 87(3):529-559 (2022). https://doi.org/10.46793/match.87-3.529W
  • Resolving the data ambiguity for periodic crystals. Advances in Neural Information Processing Systems (Proceedings of NeurIPS 2022), to appear. https://arxiv.org/abs/2108.04798

What's amd?

The typical representation of a crystal as a motif and unit cell is ambiguous, because many choices of cell and motif define the same crystal. This package implements crystal descriptors designed to be isometry invariants, meaning they are always same for any two crystals which are geometrically equivalent, independent of the unit cell and motif. The descriptors can be compared to give a distance which is 0 for identical crystals, and close to 0 for similar crystals (a continuous metric).

The pointwise distance distribution (PDD) is a descriptor that records the environment of each atom in the unit cell by listing distances to neighbouring atoms. Two PDDs are compared using an optimal matching algorithm (Earth Mover's distance). Taking the average of a PDD gives a vector called the average minimum distance (AMD), which is significantly faster to compare but can still identify crystals with similar geometry. Both AMD and PDD take a parameter k, the number of neighbouring atoms considered for each atom in the unit cell.

Getting started

Use pip to install average-minimum-distance:

pip install average-minimum-distance

Then import average-minimum-distance with import amd.

amd.compare() compares crystals in cif files by AMD or PDD descriptors, e.g.

import amd

# compare all items in one cif by AMD, k=100
df = amd.compare('file.cif', by='AMD', k=100)
# compare all in file1 vs all in file2 by PDD, k=100
df = amd.compare('file1.cif', 'file2.cif', by='PDD', k=100)

The distance matrix is returned as a pandas DataFrame. amd.compare() can also accept a folder or list of cifs.

amd.compare() reads crystals, calculates their descriptors and compares them, but these steps can be done separately (e.g. to save the descriptors to a file, see below). amd.compare() accepts several optional parameters, see the documentation for a full list.

CSD Python API only: amd.compare() accepts one or more CSD refcodes or other file formats instead of cifs (by passing reader='ccdc').

Choosing a value of k

The parameter k is the number of neighbouring atoms considered for each atom in a unit cell. Two crystals with the same unit molecule will have a small AMD/PDD distance for small enough k (e.g. k = 5), and a larger k means the geometry must be similar up to a larger radius for the distance to be small. The default for amd.compare() is k = 100, but if this is significantly smaller than the number of atoms in the unit molecule, it may be better to choose a larger value e.g. k = 300. It is usually not useful to choose k too large (many times larger than the number of atoms in the unit molecule).

Reading crystals, calculating AMDs/PDDs

This code reads a cif file and computes the list of AMDs (k = 100):

import amd

reader = amd.CifReader('file.cif')
amds = [amd.AMD(crystal, 100) for crystal in reader]
# # To calculate the PDDs:
# pdds = [amd.PDD(crystal, 100) for crystal in reader]

CifReader accepts some optional parameters, e.g. for removing Hydrogen or handling disorder, see here for a full list.

CSD Python API only: CSD crystals can be read via the CSD Python API with amd.CSDReader, see the documentation for details. CifReader can accept file formats other than .cif by passing reader='ccdc'.

Comparing by AMD or PDD

To compare all crystals in one collection with each other, use amd.AMD_pdist() or amd.PDD_pdist(), which accept a list of AMDs/PDDs and return a condensed distance matrix like SciPy's pdist(). Here's a full example of reading crystals from a .cif, calculating the descriptors and comparing them:

import amd

# read and calculate AMDs and PDDs (k = 100)
crystals = list(amd.CifReader('path/to/file.cif'))
amds = [amd.AMD(crystal, 100) for crystal in reader]
pdds = [amd.PDD(crystal, 100) for crystal in reader]

amd_cdm = amd.AMD_pdist(amds) # compare AMDs pairwise
pdd_cdm = amd.PDD_pdist(pdds) # compare PDDs pairwise

# Use squareform for a symmetric 2D distance matrix
from scipy.distance.spatial import squareform
amd_dm = squareform(amd_cdm)

Note: AMDs can be quickly computed from PDDs with amd.PDD_to_AMD().

The default metric for comparison is chebyshev (L-infinity), though it can be changed to anything accepted by SciPy's pdist, e.g. euclidean.

To compare crystals in one set with those in another set, use amd.AMD_cdist or amd.PDD_cdist:

import amd

amds1 = [amd.AMD(c, 100) for c in amd.CifReader('set1.cif')]
amds2 = [amd.AMD(c, 100) for c in amd.CifReader('set2.cif')]
# dm[i][j] = AMD distance between amds1[i] & amds2[j]
dm = amd.AMD_cdist(amds)

Example: AMD-based dendrogram

This example compares some crystals in a cif by AMD (k = 100) and plots a single linkage dendrogram:

import amd
import matplotlib.pyplot as plt
from scipy.cluster import hierarchy

crystals = list(amd.CifReader('crystals.cif'))
names = [crystal.name for crystal in crystals]
amds = [amd.AMD(crystal, 100) for crystal in crystals]
cdm = amd.AMD_pdist(amds)
Z = hierarchy.linkage(cdm, 'single')
dn = hierarchy.dendrogram(Z, labels=names)
plt.show()

Cite us

Use the following bib references to cite AMD or PDD.

Average minimum distances of periodic point sets - foundational invariants for mapping periodic crystals. MATCH Communications in Mathematical and in Computer Chemistry, 87(3), 529-559 (2022). https://doi.org/10.46793/match.87-3.529W.

@article{widdowson2022average,
  title = {Average Minimum Distances of periodic point sets - foundational invariants for mapping periodic crystals},
  author = {Widdowson, Daniel and Mosca, Marco M and Pulido, Angeles and Kurlin, Vitaliy and Cooper, Andrew I},
  journal = {MATCH Communications in Mathematical and in Computer Chemistry},
  doi = {10.46793/match.87-3.529W},
  volume = {87},
  number = {3},
  pages = {529-559},
  year = {2022}
}

Resolving the data ambiguity for periodic crystals. Advances in Neural Information Processing Systems (NeurIPS 2022), v.35. https://openreview.net/forum?id=4wrB7Mo9_OQ.

@inproceedings{widdowson2022resolving,
  title = {Resolving the data ambiguity for periodic crystals},
  author = {Widdowson, Daniel and Kurlin, Vitaliy},
  booktitle = {Advances in Neural Information Processing Systems},
  year = {2022},
  url = {https://openreview.net/forum?id=4wrB7Mo9_OQ}
}