Background

In chemistry and materials science, we often find ourselves talking about the oxidation state of a given atom. We discuss the atom as if it has taken or given exactly one electron. However, in real molecules and materials, the charge density is a smooth continuous function, and the electrons do not "belong" to any given atom. Indeed, the concept of oxidation states is not founded in quantum mechanics, and is effectively a tool invented by chemists through the years. Still, it is an exceptionally useful tool that allows for qualitative understanding of a vast array of concepts from bonding in organic molecules, to doping in metal alloys, to the charging of lithium ion batteries.

There have been many methods for calculating oxidation states proposed through the years. One of the most popular was proposed by Bader in his Quantum Theory of Atoms in Molecules (QTAIM). This method relies solely on the charge density, which is relatively insensitive to the choise of basis set used to approximate the wavefunction of the system, and is observable through experiment. In the QTAIM method, the charge is separated into regions by a surface located at local minima throughout the charge density. More technically, this "zero-flux" surface is made up of points where the gradient normal to the surface is zero.

We describe the regions defined by these surfaces as basins, though they are also commonly called Bader regions. Each basin has exactly one local maximum, sometimes termed an attractor. Each attractor typically (though not always) correspond to an atom, and the charge and oxidation state of the atom can be determined by integrating the charge density within this region.

In practice, it is often difficult and computationally expense to thoroughly sample the zero-flux surface defining basins. To avoid this problem, Henkelman et. al. proposed a brilliantly simple alternative utilizing the common representation of the charge density as points on a regular grid. Each point on the grid is assigned to a basin by climbing the steepest gradient until a local maximum is reached. Repeat gradient calculations can be avoided by stopping the ascent when a previously assigned path is reached. The end result is a robust and efficient method for dividing the charge density into basins, without ever needing to calculate the exact location of the zero-flux surface.

Through the years, several methods for performing this steepest ascent have been developed. We have implemented the same methods that exist in the Henkelman group's excellent Fortran code, as well as several slightly altered versions of them. We highly select reading about each on our Methods page to determine which is the best for your use case.