[QE-users] Numerical understanding of the pp.x-generated charge-density fields

[Wai-Ga Ho]

Hello,

I am currently trying to explore the importance of point-ion-electrostatics (i.e. Madelung energies) and its comparability to DFT results in different ionic solids. Using a more classical example, here I'll focus on NaCl in its B1 structure.


As a quick note, I am using PWSCF v.5.0.2 and Post Proc v.5.0.2 (pp.x). For my both the Na and Cl atoms, I use PAW pseudopotentials and PBE XC functionals; specifically I make use of (1) Na.pbe-spn-kjpaw_psl.1.0.0.UPF and (2) Cl.pbe-n-kjpaw_psl.1.0.0.UPF


First, doing several total energy calculations using PWSCF, I found a ground state structure whose cubic lattice parameter A lie in decent agreement with those produced experimentally (within .06 Angstroms). I then used this structure and computed its charge-density field using pp.x with the inputs plot_num=0 (for electron pseudo charge densities) and output_format=5 (xCrysden formatted output file)


Instead of using some charge visualization software like xCrysden, I was more interested in the numerical details concerning the charge distribution within the unit cell. Specifically, to compare to point-ion-electrostatics, I wanted to see how many electrons were localized to each Na or Cl position in the cell. Since the xCrysden formatted charge-density data lists the charge densities in column-major order (according to their documentation), I wrote a program to iterate through the charge-density data and assign these values onto a 3D grid corresponding to the discretized primitive unit cell of my NaCl structure. With this 3D grid, my program counts all of the charge-density contributions that are within a sphere which is centered at some user-specified position and has a user-specified radius. I took care to periodically repeat the user-specified position in order to account for charge contributions that lie outside of the unit cell. And I believe there are no implementation issues with my program.


My procedure for the analysis of the electron charge distribution data are as follows:

  1.  I enclosed both the Na and Cl positions in a sphere with a radius of half of the Na-Cl nearest neighbor separation distance.
  2.  Assuming each point in the grid occupies the same volume (the total volume of the unit cell divided by number of grid points) I summed over each charge contribution (charge density value times the volume of a grid point) within each sphere
  3.  I then counted the total charge within the entire unit cell


Doing so, I found that:

  1.  Spheres enclosing Na atoms contain ~9.2 electrons
  2.  Spheres enclosing Cl atoms contain ~6.6 electrons
  3.  The total number of electrons within the entire primitive unit cell was ~16.77 (slightly more than 9.2+6.6 since there are also interstitial regions not enclosed in any spheres)
  4.  These results were verified qualitatively by a 4D scatter plot of the entire grid where the first 3D correspond to positions while the 4th dimension indicated the charge-density magnitude and used a color scale


My questions now is the following:


Is this the correct way to interpret the charge-density data produced using pp.x? From the periodic table, isolated Na and Cl atoms have 11 and 17 electrons respectively. In an ionic solid, assuming full charge transfer, Na ions should then have 10 electrons while Cl ions should have 18. I understand that in the PAW formalism, pseudization does not preserve the charge. But from my numerical analysis and from the 4D scatter plot I produced, I found that more electrons were localized around the Na ions rather than the Cl. This seems like a strange result considering the fact that atomic Na has less electrons than atomic Cl and the fact that Na acts as a cation, further reducing the electrons it should have. Please let me know if there are any gaps in my understanding or problems with my procedure for these calculations.


Thanks in advance,

David Ho, Department of Physics at Florida State University


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