An improved algorithm to locate critical points in a 3D scalar field as implemented in the program MORPHY

Abstract

A new algorithm for location of the critical points in general scalar fields is described. The new method has been developed as part of an on-going process to exploit the topologic analysis of general 3D scalar fields. Part of this process involves the use of topologic information to seed the critical point search algorithm. The continuing move away from topologic studies of just the electron density requires more general algorithms and the ability to easily "plug in" new functions, for example, the Laplacian of the electron density ( triangle down (2)rho), the Electron Localisation Function (ELF), the Localised Orbital Locator (LOL), the Lennard-Jones function (LJF), as well as any new functions that may be proposed in the future. Another important aspect of the current algorithm is the retention of all possible intermediate information, for example, the paths describing the connectivity of critical points, as well as an ability to restart searches, something that becomes increasingly important when analysing larger systems. This new algorithm represents a core part of a new local version of the MORPHY code. We distinguish nine universal types of gradient paths.

A grid-based Bader analysis algorithm without lattice bias

“…Other approaches suggested by Popelier include the use of the divergence theorem to replace the three-dimensional integration over the Bader volumes into a two-dimensional integral over the dividing surfaces [8], and the use of a tree search to treat complex bonding topologies between critical points [14,15].…”

Rapid topography mapping of scalar fields: Large molecular clusters

“…Locating and characterizing all CP's of a scalar field is an important part of the acclaimed AIM theory, proposed and developed by Bader and co-workers. [1][2][3] The MED CP's serve as starting points for tracing the gradient paths for a given system. There are several methods available for mapping the CP's of MED and MESP in literature.…”

“…Topography mapping summarizes the basic information of a scalar field and involves locating and characterizing its critical points (CP's), isosurfaces, contours, gradient paths, etc. [1][2][3][4] The CP of a three-dimensional scalar function f(x 1 , x 2 , x 3 ) is defined as a point P at which the partial derivatives vanish, viz.…”

“…The topography of the scalar field of MED has been pioneered and popularized over the past forty years by Bader and co-workers. [1][2][3] It has culminated into the celebrated atoms in molecules (AIM) theory. The AIM analysis of a molecule starts with the mapping of all the CP's followed by tracing the gradient paths, mapping out atomic basins, etc.…”

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

“…The evaluation of property densities at the CPs and their connectivity via special GPs is instrumental in recovering the chemical insight that QCT offers. An exhaustive and clear classification 7 of all possible GPs that topologically connect two CPs in an arbitrary scalar 3D function was published before. What concerns us here is the integration over the volume of a topological atom, which is called the (atomic) basin.…”