Similarity evaluation
On similarity index
The difference between displacement parameters of individual atoms can be compared using a metric called similarity index. This index was introduced by Whitten & Spackman as S12 = 100 (1 – R12), where R12 is a measure of the overlap between the probability density functions described by two ADPs, U1 and U2, in the Cartesian frame.
For two identical ADPs R12 = 1 and S12 = 0. The smaller the value of S12, the better the agreement between U1 and U2. When the similarity index is calculated for each pair of compared ADPs, an overall similarity index can be evaluated as the arithmetic mean of all obtained values. Here, hikari calculates S12 for all pairs of same-labelled atoms. Afterwards, the averaging is performed both for all atoms, as well as for all hydrogen atoms only, and the results are appended to the individual value list as avg(*) and avg(H), respectively.
The similarity index appears to be more sensitive to the directions of the principal axes of the ADP tensor, and less to the magnitude of the mean-square displacements – you can see the differences by accessing some additional materials available here. However, in order to negate the impact of ADP "size" altogether and compare the "shape" of individual displacements only, hikari can be told to normalise all cartesian matrices U before evaluating S12, which in the form above can be reached by toggling the "Normalize the thermal ellipsoid volume" option on.
Furthermore, thanks to the marvelous package uncertainties, hikari is also able to automatically estimate the uncertainty of calculated similarity indices. To this aim, uncertainties of all Uij are assumed to be uncorrelated and linearly propagated. Consequently, they should be treated rather qualitatively than quantitatively.
References
The description of the Similarity Index has been adapted from section 5. of the following article:
- Hoser. A. A. and Madsen A. Ø. Dynamic quantum crystallography: lattice-dynamical models refined against diffraction data. II. Applications to L-alanine, naphthalene and xylitol Acra Cryst. A 73, 2, 102–114 (2006)
If you are interested in similarity index and its applications, consult the following paper and other articles citing it:
- Whitten. A. E. and Spackman M. A. Anisotropic displacement parameters for H atoms using an ONIOM approach Acra Cryst. A 62, 5, 875–888 (2006)
If you have found this page useful or you have utilised any similarity evaluation tools in your work, please consider attributing the back-end library "hikari" by citing the following paper: