Voronoi weighted Steinhardt’s parameters

In order to improve the resolution of crystal structures Mickel et al [1] proposed weighting the contribution of each neighbor to the Steinhardt parameters by the ratio of the area of the Voronoi facet shared between the neighbor and host atom. The weighted parameters are given by,

\[q_{lm} (i) = \frac{1}{N(i)} \sum_{j=1}^{N(i)} \frac{A_{ij}}{A} Y_{lm}(\pmb{r}_{ij})\]

where \(A_{ij}\) is the area of the Voronoi facet between atoms \(i\) and \(j\) and \(A\) is the sum of the face areas of atom \(i\). In pyscal, the area weights are already assigned during the neighbor calculation phase when the Voronoi method is used to calculate neighbors in the find_neighbors(). The Voronoi weighted Steinhardt’s parameters can be calculated as follows,

sys.calculate_q([4, 6])
q = sys.get_qvals([4, 6])

The weighted Steinhardt’s parameters can also be averaged as described above. Once again, the keyword averaged=True can be used for this purpose.

sys.calculate_q([4, 6], averaged=True)
q = sys.get_qvals([4, 6], averaged=True)

It was also proposed that higher powers of the weight [2] \(\frac{A_{ij}^{\alpha}}{A(\alpha)}\) where \(\alpha = 2, 3\) can also be used, where \(A(\alpha) = \sum_{j=1}^{N(i)} A_{ij}^{\alpha}\) The value of this can be set using the keyword voroexp during the neighbor calculation phase.

sys.find_neighbors(method='voronoi', voroexp=2)

If the value of voroexp is set to 0, the neighbors would be found using Voronoi method, but the calculated Steinhardt’s parameters will not be weighted.

[1]Mickel, W, Kapfer, SC, Schroder-Turk, GE, Mecke, K, J Chem Phys 138, 2013.
[2]Haeberle, J, Sperl, M, Born, P Arxiv 2019.