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Second tutorial on the Projector Augmented-Wave (PAW) technique

Generation of PAW atomic datasets

This tutorial aims at showing how to create your own atomic datasets for the Projector Augmented-Wave (PAW) method.

You will learn how to generate these atomic datasets and how to control their softness and transferability. You already should know how to use ABINIT in the PAW case (see the tutorial PAW1 ).

This tutorial should take about 2h00.

[TUTORIAL_README]

1. The PAW atomic dataset - introduction

The PAW method is based on the definition of spherical augmentation regions of radius r_c around the atoms of the system in which a basis of atomic partial-waves \phi_i, of pseudized partial-waves \tphi_i, and of projectors \tprj_i (dual to \tphi_i) have to be defined. This set of partial-waves and projectors functions (and some additional atomic data) are stored in a so-called PAW dataset. A PAW dataset has to be generated for each atomic species in order to reproduce atomic behavior as accurate as possible while requiring minimal CPU and memory resources in executing ABINIT for the crystal simulations. These two constraints are obviously conflicting.

The PAW dataset generation is the purpose of this tutorial. It is done according the following procedure (all parameters that define a PAW dataset are in bold):

Choose and define the concerned chemical species (name and atomic number).
Solve the atomic all-electrons problem in a given atomic configuration. The atomic problem is solved within the DFT formalism, using an exchange-correlation functional and either a Schrodinger (default) or scalar-relativistic approximation. This spherical problem is solved on a radial grid. The atomic problem is solved for a given electronic configuration that can be an ionized/excited one.
Choose a set of electrons that will be considered as frozen around the nucleus (core electrons). The others electrons are valence ones and will be used in the PAW basis. The core density is then deduced from the core electrons wave functions. A smooth core density equal to the core density outside a given r_{core} matching radius is computed.
Choose the size of the PAW basis (number of partial-waves and projectors). Then choose the partial-waves included in the basis. The later can be atomic eigen-functions related to valence electrons (bound states) and/or additional atomic functions, solution of the wave equation for a given l quantum number at arbitrary reference energies (unbound states).
Generate pseudo partial-waves (smooth partial-waves build with a pseudization scheme and equal to partial-waves outside a given r_c matching radius) and associated projector functions. Pseudo partial-waves are solutions of the PAW Hamiltonian deduced from the atomic Hamiltonian by pseudizing the effective potential (a local pseudopotential is built and equal to effective potential outside a $r_{Vloc} matching radius). Projectors and partial-waves are then orthogonalized with a chosen orthogonalization scheme.
Build a compensation charge density used later in order to retrieve the total charge of the atom. This compensation charge density is located inside the PAW spheres and based on an analytical shape function (which analytic form and localization radius r_{shape} can be chosen).

The user can choose between two PAW dataset generators to produce atomic files directly readable by ABINIT. The first one is the PAW generator ATOMPAW (originally by N. Holzwarth) and the second one is the Ultra-Soft USPP generator (originally written by D. Vanderbilt). In this tutorial, we concentrate only on ATOMPAW.

It is highly recommended to refer to the following papers to understand correctly the generation of PAW atomic datasets:

"Projector augmented-wave method" - cite:Bloechl1994
"A projector Augmented Wave (PAW) code for electronic structure" - cite:Holzwarth2001
"From ultrasoft pseudopotentials to the projector augmented-wave method" - cite:Kresse1999
"Electronic structure packages: two implementations of the Projector Augmented-Wave (PAW) formalism" - cite:Torrent2010
"Notes for revised form of atompaw code" (by N. Holzwarth) - PDF

2. Use of the generation code

Before continuing, you might consider to work in a different subdirectory as for the other tutorials. Why not Work_paw2?

cd $ABI_TESTS/tutorial/Input
mkdir Work_paw2
cd Work_paw2

You have now to install the ATOMPAW code. In your internet browser, enter the following URL:

 https://users.wfu.edu/natalie/papers/pwpaw/

Then, download the last version of the tar.gz file, unzip and untar it. Enter the atompaw-4.x.y.z and execute:

    mkdir build
    cd build
    ../configure
    make

if all goes well, you get the ATOMPAW executable at atompaw-4.x.y.z/build/src/atompaw.
If not, Go into the directory doc, open the file atompaw-usersguide.pdf, go p.3 and follow the instructions.

!!! Note On MacOS, you can use homebrew package manager and install ATOMPAW by typing:

        brew install atompaw/repo/atompaw

!!! Note In the following, we name atompaw the ATOMPAW executable.

How to use ATOMPAW?

The following process will be applied to Nickel in the next paragraph:

Edit an input file in a text editor (content of input explained here)
Run: atompaw < inputfile

Partial waves \phi_i, PS partial-waves \tphi_i and projectors \tprj_i are given in wfn.i files. Logarithmic derivatives from atomic Hamiltonian and PAW Hamiltonian resolutions are given in logderiv.l files. A summary of the atomic all-electron computation and PAW dataset properties can be found in the Atom_name file (Atom_name is the first parameter of the input file).

Resulting PAW dataset is contained in:

Atom_name.XCfunc.xml file
Normalized XML file according to the PAW- XML specifications (recommended).
Atom_name.XCfunc-paw.abinit file
Proprietary legacy format for ABINIT

3. First (and basic) PAW dataset for Nickel

Our test case will be nickel; electronic configuration: [1s^2 2s^2 2p^6 3s^2 3p^6 3d^8 4s^2 4p^0].

In a first stage, copy a simple input file for ATOMPAW in your working directory (find it in $ABI_HOME/doc/tutorial/paw2/paw2_assets/Ni.atompaw.input1). Edit this file.

{% dialog tutorial/paw2_assets/Ni.atompaw.input1 %}

This file has been built in the following way:

All-electron calculation parameters:
- 1st line: define the material.

    Ni 28

* 2nd line: choose the <span style="color:green">exchange-correlation functional</span> (LDA-PW or GGA-PBE)
  and select a scalar-relativistic <span style="color:green">wave equation</span>
  (nonrelativistic or scalarrelativistic)
  and a (2000 points) logarithmic <span style="color:green">grid</span>.

    GGA-PBE scalarrelativistic loggrid 2000

Electronic configuration: How many electronic states do we need to include in the computation? Besides the fully and partially occupied states, it is recommended to add all states that could be reached by electrons in the solid. Here, for Nickel, the 4p state is concerned. So we decide to add it in the computation.
- 3rd line: define the electronic configuration. A line with the maximum n quantum number for each electronic shell; here 4 4 3 means 4s, 4p, 3d.

    4 4 3 0 0 0

Following lines : definition of occupation numbers. For each partially occupied shell enter the occupation number. An excited configuration may be useful if the PAW dataset is intended for use in a context where the material is charged (such as oxides). Although, in our experience, the results are not highly dependent on the chosen electronic configuration. We choose here the [3d^8 4s^2 4p^0] configuration. Only 3d and 4p shells are partially occupied (3 2 8 and 4 1 0 lines). A 0 0 0 ends the occupation section.

    3 2 8
    4 1 0
    0 0 0

Selection of core and valence electrons. In a first approach, select only electrons from outer shells as valence. But, if particular thermodynamical conditions are to be simulated, it is generally needed to include "semi-core states" in the set of valence electrons. Semi-core states are generally needed with transition metal and rare-earth materials. Note that all wave functions designated as valence electrons will be used in the partial-wave basis. Core shells are designated by a c and valence shells by a v. All s states first, then p states and finally d states. Here:

means:

    1s core
    2s core
    3s core
    4s valence
    2p core
    3p core
    4p valence
    3d valence

Partial-waves basis generation:
- A line with l_{max} the maximum l for the partial-waves basis. Here l_{max}=2.

A line with the r_{PAW} radius.
Select it to be slightly less than half the inter-atomic distance in the solid (as a first choice). Here r_{PAW}=2.3\ a.u.

2.3

Next lines: add additional partial-waves \phi_i if needed. Choose to have 2 partial-waves per angular momentum in the basis (this choice is not necessarily optimal but this is the most common one; if r_{PAW} is small enough, 1 partial-wave per l may suffice). As a first guess, put all reference energies for additional partial-waves to 0 Rydberg. For each angular momentum, first add "y" to add an additional partial-wave. Then, next line, put the value in Rydberg units. Repeat this for each new partial-wave and finally put "n". Note : For each angular momentum, valence states already are included in the partial-waves basis. Here 4s, 4p and 3d states already are in the basis. In the present file:

    y
    0.5
    n

means that an additional s- partial-wave at E_{ref}=0.5 Ry as been added,

    y
    1.
    n

means that an additional p- partial-wave at E_{ref}=1. Ry has been added,

    y
    1.
    n

means that an additional d- partial-wave at E_{ref}=1. Ry as been added. Finally, partial-waves basis contains two s-, two p- and two d- partial-waves. * Next line: definition of the generation scheme for pseudo partial waves \tphi_i, and of projectors \tprj_i. We begin here with a simple scheme (i.e. "Bloechl" scheme, proposed by P. Blochl cite:Bloechl1994). This will probably be changed later to make the PAW dataset more efficient.

    bloechl

  * Next line: <span style="color:green">generation scheme for local pseudopotential
    $V_{loc}$</span>. In order to get PS partial-waves, the atomic potential has to be "pseudized"
    using an arbitrary pseudization scheme.
    *We choose here a "Troullier-Martins" using a wave equation at $l_{loc}=3$ and $E_{loc}=0.$ Ry.
    As a first draft, it is always recommended to put $l_{loc}=1+l_{max}$.*

    3 0. troulliermartins

  * Next two lines: `XMLOUT` makes `ATOMPAW` generate a PAW dataset in XML format;
    The next line contains options for this ABINIT file. "default" set all parameters to their default value.

    XMLOUT
    default

  * The `END` keyword ends the file.

END

At this stage, run ATOMPAW. For this purpose, simply enter:

atompaw < Ni.atompaw.input1

Lot of files are produced. We will examine some of them. A summary of the PAW dataset generation process has been written in a file named Ni.
Open it. It should look like:

 Completed calculations for Ni
 Exchange-correlation type: GGA, Perdew-Burke-Ernzerhof
  Radial integration grid is logarithmic 
r0 =   2.2810899E-04 h =   6.3870518E-03   n =      2000 rmax =   8.0000000E+01
 Scalar relativistic calculation
   AEatom converged in          32  iterations
	 for nz =  28.00
	 delta  =    9.5504321957145377E-017
  All Electron Orbital energies:         
  n  l     occupancy       energy
 1  0      2.0000000E+00 -6.0358607E+02
 2  0      2.0000000E+00 -7.2163318E+01
 3  0      2.0000000E+00 -8.1627107E+00
 4  0      2.0000000E+00 -4.1475541E-01
 2  1      6.0000000E+00 -6.2083048E+01
 3  1      6.0000000E+00 -5.2469208E+00
 4  1      0.0000000E+00 -9.0035739E-02
 3  2      8.0000000E+00 -6.5223644E-01

  Total energy
	 Total                    :    -3041.0743834110435     
 Completed calculations for Ni
 Exchange-correlation type: GGA, Perdew-Burke-Ernzerhof
  Radial integration grid is logarithmic 
r0 =   2.2810899E-04 h =   6.3870518E-03   n =      2000 rmax =   8.0000000E+01
 Scalar relativistic calculation
   SCatom converged in           1  iterations
	 for nz =  28.00
	 delta  =    8.7786021384577619E-017
   Valence Electron Orbital energies:         
  n  l     occupancy       energy
 4  0      2.0000000E+00 -4.1475541E-01
 4  1      0.0000000E+00 -9.0035739E-02
 3  2      8.0000000E+00 -6.5223644E-01

  Total energy
	 Total                    :    -3041.0743834029249     
	 Valence                  :    -185.18230020196870     
  paw parameters: 
	   lmax =            2
		 rc =    2.3096984974114871     
		irc =         1445
 Vloc: Norm-conserving Troullier-Martins with l= 3;e=   0.0000E+00
 Projector type: Bloechl + Gram-Schmidt ortho.
 Sinc^2 compensation charge shape zeroed at rc

Number of basis functions     6
 No.   n    l      Energy         Cp coeff         Occ
	1    4    0 -4.1475541E-01 -9.5091487E+00  2.0000000E+00
	2  999    0  5.0000000E-01  3.2926948E+00  0.0000000E+00
	3    4    1 -9.0035739E-02 -8.9594191E+00  0.0000000E+00
	4  999    1  1.0000000E+00  1.0610645E+01  0.0000000E+00
	5    3    2 -6.5223644E-01  9.1576184E+00  8.0000000E+00
	6  999    2  0.0000000E+00  1.3369076E+01  0.0000000E+00
 Completed diagonalization of ovlp with info =        0

 Eigenvalues of overlap operator (in the basis of projectors):
	1        7.27257365E-03
	2        2.25491432E-02
	3        1.25237568E+00
	4        1.87485118E+00
	5        1.05720648E+01
	6        2.00807906E+01

  Summary of PAW energies
		Total valence energy       -185.18230536120760     
		  Smooth energy             11.667559552612433     
		  One center               -196.84986491382003     
		  Smooth kinetic            15.154868503980399     
		  Vloc energy              -2.8094614733964467     
		  Smooth exch-corr         -3.3767012052640886     
		  One-center xc            -123.07769380742522

The generated PAW dataset (contained in Ni.GGA-PBE.xml file) is a first draft. Several parameters have to be adjusted, in order to get accurate results and efficient DFT calculations.

!!! Note The Ni.GGA-PBE.xml file is directly usable by ABINIT.

4. Checking the sensitivity to some parameters

4.a. The radial grid

Let's try to select 700 points in the logarithmic grid and check if any noticeable difference in the results appears. You just have to replace 2000 by 700 in the second line of Ni.atompaw.input1 file.
Then run:

atompaw < Ni.atompaw.input1

again and look at the Ni file:

   Summary of PAW energies
		 Total valence energy       -185.18230027710337     
		   Smooth energy             11.634042318422193     
		   One center               -196.81634259552555     
		   Smooth kinetic            15.117782033814152     
		   Vloc energy              -2.8024659321861067     
		   Smooth exch-corr         -3.3712015132649102     
		   One-center xc            -123.08319475096027

As you see, results obtained with this new grid are very close to previous ones, expecially the valence energy. We can keep the 700 points grid.

!!! Note We could try to decrease the size of the grid. Small grids give PAW dataset with small size (in kB) and run faster in ABINIT, but accuracy can be affected.

!!! Note The final r_{PAW} value (rc = ... in Ni file) changes with the grid; just because r_{PAW} is adjusted in order to belong exactly to the radial grid. By looking in ATOMPAW user's guide, you can choose to keep it constant.

4.b. The relativistic approximation of the wave equation

The scalar-relativistic option should give better results than non-relativistic one, but it sometimes produces difficulties for the convergence of the atomic problem (either at the all-electron resolution step or at the PAW Hamiltonian solution step). If convergence cannot be reached, try a non-relativistic calculation (not recommended for high Z materials).

!!! Note For the following, note that you always should check the Ni file, especially the values of valence energy. You can find the valence energy computed for the exact atomic problem and the valence energy computed with the PAW parameters. These two results should be in close agreement!

5. Adjusting partial-waves and projectors

Examine the AE partial-waves \phi_i, PS partial-waves \tphi_i and projectors \tprj_i. These are saved in files named wfni, where i ranges over the number of partial-waves used, so 6 in the present example. Each file contains 4 columns: the radius r in column 1, the AE partial-wave \phi_i(r) in column 2, the PS partial-wave \tphi_i(r) in column 3, and the projector \tprj_i(r) in column 4.
Plot the 3 curves as a function of radius using a plotting tool of your choice.

Below the first s- partial-wave /projector of the Ni example:

\phi_i should meet \tphi_i near or after the last maximum (or minimum). If not, it is preferable to change the value of the matching (pseudization) radius r_c.
The maxima of \tphi_i and \tprj_i functions should roughly have the same order of magnitude. If not, you can try to get this in three ways:
1. Change the matching radius for this partial-wave; but this is not always possible (PAW spheres should not overlap in the solid).
2. Change the pseudopotential scheme (see later).
3. If there are two (or more) partial-waves for the angular momentum l under consideration, decreasing the magnitude of the projector is possible by displacing the references energies. Moving the energies away from each other generally reduces the magnitude of the projectors, but too big a difference between energies can lead to wrong logarithmic derivatives (see following section).

Example: plot the wfn6 file, related to the second d- partial-wave:

This partial-wave has been generated at E_{ref}=0 Ry and orthogonalized with the first d- partial-wave which has an eigenenergy equal to -0.65 Ry (see Ni file). These two energies are too close and orthogonalization process produces "high" partial-waves. Try to replace the reference energy for the additional d- partial-wave. For example, put E_{ref}=1. Ry instead of E_{ref}=0. Ry (line 24 of Ni.atompaw.input1 file).

Run ATOMPAW again and plot wfn6 file:

Now the PS partial-wave and projector have the same order of magnitude!

!!! Important Note again that you should always check the two Valence energy values in Ni file and make sure they are as close as possible. If not, choices for projectors and/or partial-waves are certainly not judicious.

6. Examine the logarithmic derivatives

Examine the logarithmic derivatives, i.e., derivatives of an l-state \frac{d(log(\Psi_l(E))}{dE} computed for the exact atomic problem and with the PAW dataset. They are printed in the logderiv.l files. Each logderiv.l file corresponds to an angular momentum l and contains five columns of data: the energy, the logarithmic derivative of the l-state of the exact atomic problem, the logarithmic derivative of the pseudized problem (and two other colums not relevant for this section). In the following, when you edit a logderiv file, only edit the three first columns.
In our Ni example, l=0, 1 or 2.

The logarithmic derivatives should have the following properties:

The 2 curves should be superimposed as much as possible. By construction, they are superimposed at the 2 energies corresponding to the 2 l partial-waves. If the superimposition is not good enough, the reference energy for the second l partial-wave should be changed.
Generally a discontinuity in the logarithmic derivative curve appears at 0 Ry \le E_0\le 4 Ry. A reasonable choice is to choose the 2 reference energies so that E_0 is in between.
Too close reference energies produce "hard" projector functions (see section 5). But moving reference energies away from each other can damage accuracy of logarithmic derivatives

Here are the three logarithmic derivative curves for the current dataset:

As you can see, except for l=2, exact and PAW logarithmic derivatives do not match!
According to the previous remarks, try other values for the references energies of the s- and p- additional partial-waves.
First, edit again the Ni.atompaw.input1 file and put E_{ref}=3 Ry for the additional s- state (line 18); run ATOMPAW again. Plot the logderiv.0 file. You should get:

Then put E_{ref}=4 Ry for the second p- state (line 21); run ATOMPAW again. Plot again the logderiv.1 file.
You should get:

Now, all PAW logarithmic derivatives match with the exact ones in a reasonable interval.

!!! Note It is possible to change the interval of energies used to plot logarithmic derivatives (default is [-5;5]) and also to compute them at more points (default is 200). Just add the following keywords at the end of the SECOND LINE of the input file if you want ATOMPAW to output logarithmic derivatives for energies in [-10;10] at 500 points: logderivrange -10 10 500

Additional information related to logarithmic derivatives: ghost states

Another possible issue could be the presence of a discontinuity in the PAW logarithmic derivative curve at an energy where the exact logarithmic derivative is continuous. This generally shows the presence of a ghost state.

First, try to change to value of reference energies; this sometimes can make the ghost state disappear.
If not, it can be useful to change the pseudopotential scheme. Norm-conserving pseudopotentials are sometimes too attractive near r=0.
- A 1st solution is to change the quantum number used to generate the norm-conserving pseudopotential. But this is generally not sufficient.
- A 2nd solution is to select a ultrasoft pseudopotential, freeing the norm conservation constraint (simply replace troulliermartins by ultrasoft in the input file).
- A 3rd solution is to select a simple bessel pseudopotential (replace troulliermartins by bessel in the input file). But, in that case, one has to noticeably decrease the matching radius r_{Vloc} if one wants to keep reasonable physical results. Selecting a value of r_{Vloc} between 0.6~r_{PAW} and 0.8~r_{PAW} is a good choice. To change the value of r_{Vloc}, one has to explicitely put all matching radii: r_{PAW}, r_{shape}, r_{Vloc} and r_{core}; see user's guide.
Last solution : try to change the matching radius r_c for one (or both) l partial-wave(s). In some cases, changing r_c can remove ghost states.

In most cases (changing pseudopotential or matching radius), one has to restart the procedure from step 5.

To see an example of ghost state, use the $ABI_HOME/doc/tutorial/paw2_assets/Ni.ghost.atompaw.input file and run it with ATOMPAW.

{% dialog tutorial/paw2_assets/Ni.ghost.atompaw.input %}

Look at the l=1 logarithmic derivatives (logderiv.1 file). They look like:

Now, edit the Ni.ghost.atompaw.input file and replace troulliermartins by ultrasoft.
Run ATOMPAW again... and look at logderiv.1 file. The ghost state has moved!

Edit again the file and replace troulliermartins by bessel (line 28); then change the 17th line 2.0 2.0 2.0 2.0 by 2.0 2.0 1.8 2.0 (decreasing the r_{Vloc} radius from 2.0 to 1.8).
Run ATOMPAW: the ghost state disappears!

Start from the original state of Ni.ghost.atompaw.input file and put 1.6 for the matching radius of p- states (put 1.6 on lines 31 and 32). Run ATOMPAW: the ghost state disappears!

7. Testing the "efficiency" of a PAW dataset

Let's use again our Ni.atompaw.input1 file for Nickel (with all our modifications).
You get a file Ni.GGA-PBE-paw.xml containing the PAW dataset designed for ABINIT.

To test the efficiency of the generated PAW dataset, we finally will use ABINIT!
You are about to run a DFT computation and determine the size of the plane wave basis needed to reach a given accuracy. If the cut-off energy defining the plane waves basis is too high (higher than 20 Hartree), some changes have to be made in the input file.

Copy $ABI_TESTS/tutorial/Input/tpaw2_1.abi in your working directory. Edit it, and activate the 8 datasets (uncomment the line ndtset 8).

{% dialog tests/tutorial/Input/tpaw2_1.abi %}

Run 'ABINIT'. It computes the total energy of ferromagnetic FCC Nickel for several values of ecut.
At the end of output file, you get this:

  ecut1     8.00000000E+00 Hartree
  ecut2     1.00000000E+01 Hartree
  ecut3     1.20000000E+01 Hartree
  ecut4     1.40000000E+01 Hartree
  ecut5     1.60000000E+01 Hartree
  ecut6     1.80000000E+01 Hartree
  ecut7     2.00000000E+01 Hartree
  ecut8     2.20000000E+01 Hartree
etotal1    -3.9299840066E+01
etotal2    -3.9503112955E+01
etotal3    -3.9582704516E+01
etotal4    -3.9613343901E+01
etotal5    -3.9622927015E+01
etotal6    -3.9626266739E+01
etotal7    -3.9627470087E+01
etotal8    -3.9627833090E+01

etotal convergence (at 1 mHartree) is achieve for 18 \le e_{cut} \le 20 Hartree.
etotal convergence (at 0,1 mHartree) is achieve for e_{cut} \ge 22 Hartree.

This is not a good result for a PAW dataset; let's try to optimize it.

1st possibility: use vanderbilt projectors instead of bloechl ones.
Vanderbilt's projectors generally are more localized in reciprocal space than Bloechl's ones . Keyword bloechl has to be replaced by vanderbilt in the ATOMPAW input file and r_c values have to be added at the end of the file (one for each PS partial-wave). See this input file: $ABI_HOME/doc/tutorial/paw2_assets/Ni.atompaw.input.vanderbilt.

{% dialog tutorial/paw2_assets/Ni.atompaw.input.vanderbilt %}

2nd possibility: use RRKJ pseudization scheme for projectors.
Use this input file for ATOMPAW: $ABI_HOME/doc/tutorial/paw2_assets/Ni.atompaw.input2. As you can see bloechl has been changed by custom rrkj and six r_c values have been added at the end of the file, each one corresponding to the matching radius of one PS partial-wave. Repeat the entire procedure (ATOMPAW + ABINIT)... and get a new ABINIT output file.
Note: You have check again log derivatives.

{% dialog tutorial/paw2_assets/Ni.atompaw.input2 %}

  ecut1     8.00000000E+00 Hartree
  ecut2     1.00000000E+01 Hartree
  ecut3     1.20000000E+01 Hartree
  ecut4     1.40000000E+01 Hartree
  ecut5     1.60000000E+01 Hartree
  ecut6     1.80000000E+01 Hartree
  ecut7     2.00000000E+01 Hartree
  ecut8     2.20000000E+01 Hartree
etotal1    -3.9599860476E+01
etotal2    -3.9626919903E+01
etotal3    -3.9627249378E+01
etotal4    -3.9627836846E+01
etotal5    -3.9628304332E+01
etotal6    -3.9628429611E+01
etotal7    -3.9628436662E+01
etotal8    -3.9628455467E+01

etotal convergence (at 1 mHartree) is achieve for 12 \le e_{cut} \le 14 Hartree.
etotal convergence (at 0,1 mHartree) is achieve for 16 \le e_{cut} \le 18 Hartree.

This is a reasonable result for a PAW dataset!

3rd possibility: use enhanced polynomial pseudization scheme for projectors.
Edit Ni.atompaw.input2 and replace custom rrkj by custom polynom2 7 10. It may sometimes improve the ecut convergence.

Optional exercise

Let's go back to Vanderbilt projectors.
Repeat the procedure (ATOMPAW\ + ABINIT) with the previous \Ni.atompaw.input.vanderbilt file.

{% dialog tutorial/paw2_assets/Ni.atompaw.input.vanderbilt %}

Let's try to change the pseudization scheme for the local pseudopotential.
Try to replace the troulliermartins keyword by ultrasoft. Repeat the procedure (ATOMPAW + ABINIT).
ABINIT can now reach convergence!

Results are below:

  ecut1     8.00000000E+00 Hartree
  ecut2     1.00000000E+01 Hartree
  ecut3     1.20000000E+01 Hartree
  ecut4     1.40000000E+01 Hartree
  ecut5     1.60000000E+01 Hartree
  ecut6     1.80000000E+01 Hartree
  ecut7     2.00000000E+01 Hartree
  ecut8     2.20000000E+01 Hartree
etotal1    -3.9608001348E+01
etotal2    -3.9613479343E+01
etotal3    -3.9616615528E+01
etotal4    -3.9620665403E+01
etotal5    -3.9622873734E+01
etotal6    -3.9623393021E+01
etotal7    -3.9623440787E+01
etotal8    -3.9623490997E+01

etotal convergence (at 1 mHartree) is achieve for 14 \le e_{cut} \le 16 Hartree.
etotal convergence (at 0,1 mHartree) is achieve for 20 \le e_{cut} \le 22 Hartree.

!!! note You could have tried the bessel keyword instead of ultrasoft one.

Summary of convergence results

!!! Final_remarks

* The localization of projectors in reciprocal space can (generally) be predicted by a look at `tprod.i` files.
  Such a file contains the curve of as a function of $q$ (reciprocal space variable).
  $q$ is given in $Bohr^{-1}$ units; it can be connected to ABINIT plane waves cut-off energy (in Hartree units) by:
  $e_{cut}=\frac{q_{cut}^2}{4}$. These quantities are only calculated for the bound states,
  since the Fourier transform of an extended function is not well-defined.

* Generating projectors with Blochl's scheme often gives the guaranty to have stable calculations.
  `ATOMPAW` ends without any convergence problem and DFT calculations run without any divergence
  (but they need high plane wave cut-off). Vanderbilt projectors (and even more `custom` projectors)
  sometimes produce instabilities during the PAW dataset generation process and/or the DFT calculations
  but are more efficient.

* In most cases, after having changed the projector generation scheme,
  one has to restart the procedure from step 5.

8 Testing against physical quantities

The last step is to examine carefully the physical quantities obtained with our PAW dataset.

Copy $ABI_TESTS/tutorial/Input/tpaw2_2.abi in your working directory. Edit it, activate the 7 datasets (ubcomment the 'ndtset 7` line), and use $ABI_HOME/doc/tutorial/paw2_assets/Ni.PBE-paw.rrkj.xml PAW dataset obtained from Ni.atompaw.input2 file, with a minor change of name (suppression of the indication GGA). Run ABINIT (this may take a while...).

{% dialog tests/tutorial/Input/tpaw2_2.abi %}

{% dialog tutorial/paw2_assets/Ni.PBE-paw.rrkj.xml %}

ABINIT computes the converged ground state of ferromagnetic FCC Nickel for several volumes around equilibrium. Plot the etotal vs acell curve:

{width=70%}

From this graph and output file, you can extract some physical quantities:

Equilibrium cell parameter:     a0 = 3.523 angstrom
Bulk modulus:                    B = 199 GPa
Magnetic moment at equilibrium: mu = 0.60

Compare these results with published results:

all-electron GGA-FLAPW from cite:Kresse1999:

    a0 = 3.52 angstrom
     B = 200 GPa
    mu = 0.60

GGA-PAW with VASP code from cite:Kresse1999:

    a0 = 3.52 angstrom
     B = 194 GPa
    mu = 0.61

Experimental results from from cite:Dewaele2008:

    a0 = 3.52 angstrom
     B = 183 GPa

You should always compare results with all-electron ones (or other PAW computations). Not with experimental ones!

Additional remark: It can be useful to test the sensitivity of results to some ATOMPAW input parameters (see user's guide for details on keywords):

The analytical form and the cut-off radius r_{shape} of the shape function used in compensation charge density definition, By default a sinc function is used but a gaussian shape can have an influence on results. Bessel shapes are efficient and generally need a smaller cut-off radius (r_{shape}=0.8~r_{PAW}).
The matching radius r_{core} used to generate the pseudo core density from atomic core density,
The inclusion of additional ("semi-core") states in the set of valence electrons,
The pseudization scheme used to get pseudopotential Vloc(r).

All these parameters have to be meticulously checked, especially if the PAW dataset is used for non-standard solid structures or thermodynamical domains.

!!! Optional_exercise

Let's add 3s and 3p semi-core states in PAW dataset!  
Repeat the procedure (`ATOMPAW` \+ `ABINIT`) with
<span style="color:green">\$ABI_HOME/doc/tutorial/paw2_assets/Ni.atompaw.input.semicore</span>
file. The execution time is a bit longer as more electrons have to be treated by `ABINIT`.  
Look at $a_0$, $B$ or $\mu$ variation.  
Note: this new PAW dataset has a smaller $r_{PAW}$ radius (because semi-core states are localized).
````
    a0 = 3.518 angstrom
    B = 194 GPa
    mu = 0.60
````

9 The Real Space Optimization (RSO) - for experienced users

In this section, an additional optimization of the atomic data is presented. It can contribute, in some cases, to a speedup of the convergence on ecut. This optimization is not essential to produce efficient PAW datasets but can be useful.
We advise experienced users to try it.

The idea is quite simple: when expressing the different atomic radial functions (\phi_i, \tphi_i, \tprj_i) on the plane wave basis, the number of plane waves depends on the "locality" of these radial functions in reciprocal space.

In cite:KingSmith1991 a method to enforce the locality (in reciprocal space) of projectors \tprj_i is presented; the projectors expressed in reciprocal space \tprj_i(g) are modified according to the following scheme: The reciprocal space is divided in 3 regions:

If g \lt g_{max}, \tprj_i(g) is unchanged
If g \gt \gamma, \tprj_i(g) is set to zero
If g_{max} \le g \le \gamma, \tprj_i(g) is modified so that the contribution of \tprj_i(r) is conserved with an error W (as small as possible).

{width=50%}

The above transformation of \tprj_i(g) is only possible if \tprj_i(r) is defined outside the spherical augmentation region up to a radius R_0, with R_0 > r_c. In practice we have to:

Impose an error W (W is the maximum error admitted on total energy)
Adjust g_{max} according to E_{cut} (g_{max} \le E_{cut})
Choose \gamma so that 2 g_{max} \lt \gamma \lt 3 g_{max}

and let the ATOMPAW code apply the transformation to \tprj_i and deduce R_0 radius.

You can test it now. In your working directory, use the dataset $ABI_HOME/doc/tutorial/paw2_assets/Ni.atompaw.input3 (Bloechl's projectors).

{% dialog tutorial/paw2_assets/Ni.atompaw.input3 %}

Replace the XML options (penultimate line):

    XMLOUT
    default

with:

    XMLOUT
    rsoptim 8. 2 0.0001

8., 2 and 0.0001 are the values for g_{max},\frac{\gamma}{g_{max}} and W.

Run ATOMPAW.
You get a new PAW dataset file for ABINIT. Run ABINIT with it using the tpaw2_1.abi file.
Compare the results with those obtained in section 7.

You can try several values for g_{max} (keeping \frac{\gamma}{g_{max}} and W constant) and compare the efficiency of the atomic data; do not forget to test physical properties again.

!!! Note How to choose the RSO parameters?
\frac{\gamma}{g_{max}} = 2 and 0.0001 \lt W \lt 0.001 is a good choice. g_{max} has to be adjusted. The lower g_{max} the faster the convergence is but too low g_{max} can produce unphysical results.

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