Visualization and integration of quantum topological atoms by spatial discretization into finite elements

We present a novel algorithm to integrate property densities over the volume of a quantum topological atom. Atoms are grown outward, starting from a sphere centered on the nucleus, by means of a finite element meshing algorithm. Bond critical points and ring critical points require special treatment. The overall philosophy as well as intricate features of this meshing algorithm are given, followed by details of the quadrature over the finite elements. An effort has been made to design a streamlined and compact algorithm, focusing on the core of challenges arising in tracing the electron density's gradient vector field. The current algorithm also generates a new type of pictures that can be a Graphical User Interface. Excellent integration errors, L(Omega), are obtained, even for atoms with (narrow) tails or sharp corners.