[QE-users] Bader Charges for a Spin-Polarized System

[Alberto Otero de la Roza]

Hi Vic,

I haven't used pp.x for a while for spinpolarized systems so I'm not
sure about the state of the code in that regard but I can answer some
of your other questions. The plot_num=0 density (aka valence
pseudo-density) is built as the sum of the squares of the pseudo-KS
states, which have been smoothed near the nuclei. The valence
pseudo-density integrates to the number of valence electrons in your
calculation. Bader integration relies on having maxima at the nuclei
but the smoothing may cause some of them to be missing. Therefore, it
is a poor idea to use this density to partition the crystal space into
atomic regions.

In PAW, you can reconstruct the "correct" (i.e. not smoothed) valence
charge density. This is plot_num=17. If, on top of that, you add the
core density, then you have the all-electron charge density
(plot_num=21). These two densities are poorly represented by a uniform
grid and they do not integrate to an integer number of electrons. But
they are what you need to use to find the atomic basins. Once the
basins are found, you integrate the valence pseudo-density
(plot_num=0) inside them to find the atomic charges.

You can use plot_num=0 to find the basins. If you have enough valence
electrons and little charge transfer, you may very well have maxima on
top of all atoms. However, the results will be off. Also, regardless
of how you partition the system into atoms, the sum of all atomic
populations will trivially give the number of valence electrons
always, so that's not a good measure of quality.

I have some notes on the atomic integration topic here:

https://aoterodelaroza.github.io/critic2/examples/example_11_01_simple-integration/

using the critic2 program but I believe they should apply to
Henkelman's code as well.

As for why symmetry-equivalent atoms come out with different atomic
populations: because the uniform grid on which the density is written
doesn't have the same symmetry as the crystal. However, in the limit
of an infinitely fine grid the atomic populations should converge to
the same value. You may want to try the Yu and Trinkle method
implemented in critic2, which assigns fractional weights to grid
points and is in general a little more accurate than Henkelman's, but
I believe a finer grid is the ultimate solution.

Best,

Alberto

--
Dr. Alberto Otero de la Roza
Ramón y Cajal fellow,
Department of Physical and Analytical Chemistry, 
University of Oviedo