Methods¶
BaderKit is based on the Henkelman group's excellent Fortran code. The same methods available in their code are available here in addition to some variants that we find useful.
Summary¶
| Method | Speed | Accuracy | Single Basin Assignments |
|---|---|---|---|
| ongrid | Very Fast | Low | |
| neargrid | Medium | High | |
| hybrid-neargrid | Medium | High | |
| weight | Fast | Very High | |
| hybrid-weight | Fast | Very High |
Descriptions¶
This is the original algorithm proposed by Henkelman et. al. It is extremely fast, but prone to error. For example, it gives slightly different oxidation states for different orientations of a molecule or material. We recommend using it only if speed is a major concern.
For each point on the grid, the gradient is calculated for the 26 nearest neighbors, and the neighbor with the steepest gradient is selected as the next point in the path.
In the original code, this path is followed until a maximum is reached or a previous path is hit. In the former case, all of the points in the path are assigned to the maximum, and in the latter they are assigned to the same maximum as the colliding path.
In our implementation, the steepest gradient is calculated once at each point to establish a pointer to the best neighbor. The points are then assigned to maxima using a pointer doubling algorithm. This gives the same results as the original algorithm, while allowing us to parallelize the operation.
Reference
G. Henkelman, A. Arnaldsson, and H. Jónsson, A fast and robust algorithm for Bader decomposition of charge density, Comput. Mater. Sci. 36, 354-360 (2006)
This algorithm was developed by Henkelman et. al. several years after the ongrid method to fix orientation errors. It is generally less accurate than the weight method, but is useful in situations where it is desirable to have only one basin assignment per point on the grid.
A gradient vector is calculated at each point using the three nearest neighbors. A step is then made to the neighboring point that is closest to this gradient vector. To preserve information about the true (offgrid) gradient, a correction vector is stored that points from the new point to the original gradient vector.
At each step, the difference between the gradient and the ongrid step is added to this correction vector. If any component of the correction vector is ever closer to a neighboring point than the current one, a correction is made to keep the path closer to the true gradient.
After all of the points are assigned, a refinement must be made to the points on the edge, as the accumulation of the gradient is technically only correct for the first point in the path.
Note
Although the original paper and code suggests only one refinement of the edges is needed, we found that several are usually required. In our test case (a Ag structure) on a course grid, the Henkelman groups code assigns asymmetrical charges/volumes to symmetrical basins while our code reaches symmetry after several iterations.
By default we use iterative refinement, but this can be changed to the
original single refinement by setting refinement_method="single" in python or
--refinement-method single in the command-line.
Reference
W. Tang, E. Sanville, and G. Henkelman, A grid-based Bader analysis algorithm without lattice bias, J. Phys.: Condens. Matter 21, 084204 (2009)
This method reduces errors due to orientation by allowing each point to be partially assigned to multiple basins. This method tends to provide the most accurate charges, but may not be the best for workflows that rely on each point being assigned to a single basin.
A voronoi cell is generated at each point on the grid. A "flux" is calculated from this point to each neighbor sharing a voronoi facet using the difference in charge density modified by the distance to the neighbor and area of the shared voronoi facet. This flux is then normalized to create "weights" which indicate the fraction of the volume that flows to each neighbor.
Moving from highest to lowest, each point is assigned to basins by assigning the weight going to each neighbor to that neighbors own fractional assignments. The ordering from highest to lowest ensures that the higher neighbors have already received their assignment.
In the original implementation, the flux is calculated as the algorithm steps down the points in order. In our implementation, the flux is calculated in a separate first step, allowing for parallelization, then the flux is assigned as normal.
Note
The bader.basin_labels and bader.atom_labels properties
for this method are generated by assigning the points to the basin with
the highest weight, or the first basin index in the case of a tie.
Reference
M. Yu and D. R. Trinkle, Accurate and efficient algorithm for Bader charge integration, J. Chem. Phys. 134, 064111 (2011)
Because we found that iterative edge refinement is required for the neargrid method (see the Note), we've added a variation where the faster ongrid method is performed first, then refined using the neargrid method.
In the original weight method, the use of a voronoi cell restricts neighbors to exclusively those that share a voronoi facet. This is different from the other methods and can results in "local maxima" that have lower values than one of their 26 neighbors. This also tends to result in more basins being found.
This method performs the weight method, but merges basins that are not maxima relative to all 26 nearest neighbors into basins that are.
Note
The bader.basin_labels and bader.atom_labels properties
for this method are generated by assigning the points to the basin with
the highest weight, or the first basin index in the case of a tie.