Difference between revisions of "Structure Similarity"

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The fingerprints from sites in a given structure are subsequently statistically processed to yield the minimum, maximum, mean, and standard deviation of each coordination percentage. The resultant ordered vector defines a structure fingerprint, '''v'''<sup>struct</sup>:
 
The fingerprints from sites in a given structure are subsequently statistically processed to yield the minimum, maximum, mean, and standard deviation of each coordination percentage. The resultant ordered vector defines a structure fingerprint, '''v'''<sup>struct</sup>:
  
<math>\mathbf{v}^\mathrm{struct} = [min(w(\mathrm{CN}=1)), max(w(\mathrm{CN}=1)), mean(w(\mathrm{CN}=1)), std(w(\mathrm{CN}=1)), min(w(\mathrm{CN}=2)), \dots, min(w(\mathrm{CN}=12)), max(w(\mathrm{CN}=12)), mean(w(\mathrm{CN}=12)), std(w(\mathrm{CN}=12))]^\mathrm{T}</math>
+
<math>\mathbf{v}^\mathrm{struct} = [</math>
 +
 
 +
<math>\min(w(\mathrm{CN}=1)), \max(w(\mathrm{CN}=1)), \mathrm{mean}(w(\mathrm{CN}=1)), \mathrm{std}(w(\mathrm{CN}=1)), \dots,</math>
 +
 
 +
<math>\min(w(\mathrm{CN}=12)), \max(w(\mathrm{CN}=12)), \mathrm{mean}(w(\mathrm{CN}=12)), \mathrm{std}(w(\mathrm{CN}=12))</math>
 +
 
 +
<math>]^\mathrm{T}</math>
  
 
== Structure Distance ==
 
== Structure Distance ==

Revision as of 23:22, 7 March 2018

The similarity between two structures i and j is assessed on the basis of local coordination information from all sites in the two structures. [1] [2]


Site Fingerprints

The similarity calculation begins with computing a crystal site fingerprint, vsite, for each site in the two structures. The fingerprint is a 12-dimensional vector in which an element at position k (e.g., 4) provides the percentage of how much the given site should be considered k-fold coordinated (i.e., w(CN=4)):

\mathbf{v}^\mathrm{site} = [w(\mathrm{CN}=1), w(\mathrm{CN}=2), w(\mathrm{CN}=3), \dots, w(\mathrm{CN}=12)]^\mathrm{T}

So, we are testing the coordination percentages up to 12-fold coordination.

Structure Fingerprints

The fingerprints from sites in a given structure are subsequently statistically processed to yield the minimum, maximum, mean, and standard deviation of each coordination percentage. The resultant ordered vector defines a structure fingerprint, vstruct:

\mathbf{v}^\mathrm{struct} = [

\min(w(\mathrm{CN}=1)), \max(w(\mathrm{CN}=1)), \mathrm{mean}(w(\mathrm{CN}=1)), \mathrm{std}(w(\mathrm{CN}=1)), \dots,

\min(w(\mathrm{CN}=12)), \max(w(\mathrm{CN}=12)), \mathrm{mean}(w(\mathrm{CN}=12)), \mathrm{std}(w(\mathrm{CN}=12))

]^\mathrm{T}

Structure Distance

Finally, structure similarity is determined by the distance, d, between two structure fingerprints vistruct and vjstruct:

d = || \mathbf{v}_{i}^\mathrm{struct} - \mathbf{v}_{j}^\mathrm{struct} ||

A small distance value indicates high similarity between two structures, whereas a large distance (around 1) suggests that the structures are very dissimilar. Note that the structure fingerprint vectors are normalized before calculating the distance measure.

References

  1. N. E. R. Zimmermann, A. Jain, in preparation (2018)
  2. N. E. R. Zimmermann, M. K. Horton, A. Jain, M. Haranczyk, Front. Mater., 4, 34, (2017)

Authors

Nils Zimmermann, Donny Winston