Continuous symmetry measures of density maps

Density maps are commonly used for data presentation, and their symmetry is a central characteristic. Quite often near symmetry can be detected but not the exact one. Here we present a method for estimating the symmetry content of density maps. We thus extend the application of the continuous symmetry measure (CSM) methodology to many-points problems. We demonstrate the applicability of the method on electrondensity probability maps and use, as a model, a series of copper clusters: ¹Cu₃⁺, ¹Cu ₄ (planar and tetrahedral), ²Cu₇, and ¹Cu₆. These were distorted in various ways, and the electron density maps of the distorted clusters were analyzed with respect to the content of the elementary symmetry point groups, namely reflection, inversion rotation, and improper rotation. In general, it has been found that symmetry distortions of nuclei arrangement are amplified in the electronic arrangement.