# Difference between revisions of "Structure Similarity"

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

## 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.