# First tutorial on the Projector Augmented-Wave (PAW) method¶

## Projector Augmented-Wave approach, how to use it?¶

This tutorial aims at showing how to perform a calculation within the **Projector Augmented-Wave** (PAW) method.

You will learn how to launch a PAW calculation and what are the main input
variables that govern convergence and numerical efficiency.
You are supposed to know how to use ABINIT with *Norm-Conserving PseudoPotentials* (NCPP).

This tutorial should take about 1.5 hour.

Note

Supposing you made your own installation of ABINIT, the input files to run the examples
are in the *~abinit/tests/* directory where *~abinit* is the **absolute path** of the abinit top-level directory.
If you have NOT made your own install, ask your system administrator where to find the package,
especially the executable and test files.

In case you work on your own PC or workstation, to make things easier, we suggest you define some handy environment variables by executing the following lines in the terminal:

export ABI_HOME=Replace_with_absolute_path_to_abinit_top_level_dir # Change this line export PATH=$ABI_HOME/src/98_main/:$PATH # Do not change this line: path to executable export ABI_TESTS=$ABI_HOME/tests/ # Do not change this line: path to tests dir export ABI_PSPDIR=$ABI_TESTS/Psps_for_tests/ # Do not change this line: path to pseudos dir

Examples in this tutorial use these shell variables: copy and paste
the code snippets into the terminal (**remember to set ABI_HOME first!**) or, alternatively,
source the `set_abienv.sh`

script located in the *~abinit* directory:

```
source ~abinit/set_abienv.sh
```

The ‘export PATH’ line adds the directory containing the executables to your PATH
so that you can invoke the code by simply typing *abinit* in the terminal instead of providing the absolute path.

To execute the tutorials, create a working directory (`Work*`

) and
copy there the input files of the lesson.

Most of the tutorials do not rely on parallelism (except specific tutorials on parallelism). However you can run most of the tutorial examples in parallel with MPI, see the topic on parallelism.

## 1. Summary of the PAW method¶

The **Projector Augmented-Wave** approach has been introduced by Peter Blochl in 1994:

The Projector Augmented-Wave method is an extension of augmented wave methods and the pseudopotential approach, which combines their traditions into a unified electronic structure method”. It is based on a linear and invertible transformation (the PAW transformation) that connects the “true” wavefunctions \Psi with “auxiliary” (or “pseudo”) soft wavefunctions \tPsi:

This relation is based on the definition of *augmentation regions*
(atomic spheres of radius r_c), around the atoms in which the partial
waves |\phi_i\rangle form a basis of atomic wavefunctions; |\tphi_i\rangle are *pseudized*
partial waves (obtained from |\phi_i\rangle), and |\tprj_i\rangle are dual functions of
the |\tphi_i\rangle called projectors.
It is therefore possible to write every quantity depending on \Psi_n (density,
energy, Hamiltonian) as a function of \tPsi_n and to find \tPsi_n by solving self-consistent equations.

The PAW method has two main advantages:

- From \tPsi, it is always possible to obtain the true
*all electron*wavefunction \Psi, - The convergence is comparable to an
*UltraSoft PseudoPotential*(USPP) one.

From a practical point of view, a PAW calculation is
rather similar to a Norm-Conserving PseudoPotential one.
Most noticeably, one will have to use a special atomic data file (**PAW dataset**) that contains the
\phi_i, \tphi_i and \tprj_i and that plays the same role as a pseudopotential file.

Tip

It is highly recommended to read the following papers to understand correctly the basic concepts of the PAW method: [Bloechl1994] and [Kresse1999]. The implementation of the PAW method in ABINIT is detailed in [Torrent2008], describing specific notations and formulations

## 2. Using PAW with ABINIT¶

*Before continuing, you might consider to work in a different subdirectory as
for the other tutorials. Why not Work_paw1?
In what follows, the name of files are mentioned as if you were in this subdirectory.
All the input files can be found in the $ABI_TESTS/tutorial/Input directory.*

Important

You can compare your results with reference output files located in
`$ABI_TESTS/tutorial/Refs`

and `$ABI_TESTS/tutorial/Refs/tpaw1_addons`

directories (for the present tutorial they are named `tpaw1_*.abo`

).

The input file *tpaw1_1.abi* is an example of a file to be used to compute
the total energy of diamond at the experimental volume (within the
*LDA exchange-correlation functional*). Copy the *tpaw1_1.abi* file in your work directory,

cd $ABI_TESTS/tutorial/Input mkdir Work_paw1 cd Work_paw1 cp ../tpaw1_1.abi .

and execute ABINIT:

abinit tpaw1_1.abi >& log

The code should run very quickly. In the meantime, you can read the input file and see that there is no PAW input variable.

# Input for PAW1 tutorial # Diamond at experimental volume #Definition of the unit cell acell 3*3.567 angstrom # Lengths of the primitive vectors (exp. param. in angstrom) rprim # 3 orthogonal primitive vectors (FCC lattice) 0.0 1/2 1/2 1/2 0.0 1/2 1/2 1/2 0.0 nsym 0 # Automatic detection of symetries #Definition of the atom types and pseudopotentials ntypat 1 # There is only one type of atom znucl 6 # Atomic number of the possible type(s) of atom. Here carbon. pp_dirpath "$ABI_PSPDIR" # Path to the directory were # pseudopotentials for tests are stored pseudos "PseudosTM_pwteter/6c.pspnc" # Name and location of the pseudopotential #pseudos "Pseudodojo_paw_pw_standard/C.xml" # To be uncommented later #Definition of the atoms natom 2 # There are two atoms typat 1 1 # They both are of type 1, that is, Carbon xred # Location of the atoms: 0.0 0.0 0.0 # Triplet giving the reduced coordinates of atom 1 1/4 1/4 1/4 # Triplet giving the reduced coordinates of atom 2 #Definition of bands and occupation numbers nband 6 # Compute 6 bands (4 occupied, 2 empty) occopt 1 # Automatic generation of occupation numbers, as a semiconductor #Numerical parameters of the calculation : planewave basis set and k point grid ecut 15.0 # Maximal plane-wave kinetic energy cut-off, in Hartree ecutsm 0.5 # Introduce a smooth PW cutoff within an 0.5 Ha region kptopt 1 # Automatic generation of k points, taking into account the symmetry ngkpt 6 6 6 # This is a 6x6x6 grid based on the primitive vectors nshiftk 4 # of the reciprocal space, repeated four times, shiftk # with different shifts: 0.5 0.5 0.5 0.5 0.0 0.0 0.0 0.5 0.0 0.0 0.0 0.5 #Parameters for the SCF procedure nstep 20 # Maximal number of SCF cycles tolvrs 1.0d-10 # Will stop when, twice in a row, the difference # between two consecutive evaluations of potential residual # differ by less than tolvrs #Miscelaneous parameters prtwf 0 # Do not print wavefunctions prtden 0 # Do not print density prteig 0 # Do not print eigenvalues ############################################################## # This section is used only for regression testing of ABINIT # ############################################################## #%%<BEGIN TEST_INFO> #%% [setup] #%% executable = abinit #%% [files] #%% files_to_test = #%% tpaw1_1.abo, tolnlines= 0, tolabs= 0.000e+00, tolrel= 0.000e+00 #%% output_file = "tpaw1_1.abo" #%% [paral_info] #%% max_nprocs = 4 #%% [extra_info] #%% authors = M. Torrent #%% keywords = PAW #%% description = #%% Input for PAW1 tutorial #%% Diamond at experimental volume #%%<END TEST_INFO>

Now, open the *tpaw1_1.abi* file and change the line with pseudopotential information:
replace the *PseudosTM_pwteter/6c.pspnc* file with *Pseudodojo_paw_pw_standard/C.xml*.
Run the code again:

abinit tpaw1_1.abi >& log

Your run should stop almost immediately! The input file, indeed, is missing the mandatory argument pawecutdg!!

Add the line:

pawecutdg 50

to *tpaw1_1.abi* and run it again. Now the code completes successfully.

Note

The time needed for the PAW run is greater than the time needed for the Norm-Conserving PseudoPotential run; indeed, at constant value of plane wave cut-off energy ecut PAW requires more computational resources:

- the
*on-site*contributions have to be computed, - the nonlocal contribution of the PAW dataset uses 2 projectors per angular momentum, while the nonlocal contribution of the Present Norm-Conserving Pseudopotential uses only one.

However, as the plane wave cut-off energy required by PAW is much smaller than
the cut-off needed for the Norm-Conserving PseudoPotential (see next section),
**a PAW calculation will actually require less CPU time**.

Let’s open the output file (*tpaw1_1.abo*) and have a look inside (remember:
you can compare with a reference file in *$ABI_TESTS/tutorial/Refs/*).
Compared to an output file for a Norm-Conserving PseudoPotential run, an
output file for PAW contains the following specific topics:

- At the beginning of the file, some specific default PAW input variables (ngfftdg, pawecutdg, and useylm), mentioned in the section:

-outvars: echo values of preprocessed input variables --------

- The use of two FFT grids, mentioned in:

Coarse grid specifications (used for wave-functions): getcut: wavevector= 0.0000 0.0000 0.0000 ngfft= 18 18 18 ecut(hartree)= 15.000 => boxcut(ratio)= 2.17276 Fine grid specifications (used for densities): getcut: wavevector= 0.0000 0.0000 0.0000 ngfft= 32 32 32 ecut(hartree)= 50.000 => boxcut(ratio)= 2.10918

- A specific description of the PAW dataset (you might follow the tutorial PAW2, devoted to the building of the PAW atomic data, for a complete understanding of the file):

--- Pseudopotential description ------------------------------------------------ - pspini: atom type 1 psp file is /Users/torrentm/WORK/ABINIT/GIT/beauty/tests//Psps_for_tests/Pseudodojo_paw_pw_standard/C.xml - pspatm: opening atomic psp file /Users/torrentm/WORK/ABINIT/GIT/beauty/tests//Psps_for_tests/Pseudodojo_paw_pw_standard/C.xml - pspatm : Reading pseudopotential header in XML form from /Users/torrentm/WORK/ABINIT/GIT/beauty/tests//Psps_for_tests/Pseudodojo_paw_pw_standard/C.xml Pseudopotential format is: paw10 basis_size (lnmax)= 4 (lmn_size= 8), orbitals= 0 0 1 1 Spheres core radius: rc_sph= 1.50736703 1 radial meshes are used: - mesh 1: r(i)=AA*[exp(BB*(i-1))-1], size=2001 , AA= 0.94549E-03 BB= 0.56729E-02 Shapefunction is SIN type: shapef(r)=[sin(pi*r/rshp)/(pi*r/rshp)]**2 Radius for shape functions = 1.30052589 mmax= 2001 Radial grid used for partial waves is grid 1 Radial grid used for projectors is grid 1 Radial grid used for (t)core density is grid 1 Radial grid used for Vloc is grid 1 Radial grid used for pseudo valence density is grid 1 Mesh size for Vloc has been set to 1756 to avoid numerical noise. Compensation charge density is not taken into account in XC energy/potential pspatm: atomic psp has been read and splines computed

- After the SCF cycle section:
The value of the integrated compensation charge evaluated by two different
numerical methodologies:

1. computed in the*augmentation regions*on the “spherical” grid,

2. computed in the whole simulation cell on the “FFT” grid…

A discussion on these two values will be done in a forthcoming section.

PAW TEST: ==== Compensation charge inside spheres ============ The following values must be close to each other ... Compensation charge over spherical meshes = 0.263866295499286 Compensation charge over fine fft grid = 0.263862122639226

- Information related the non-local term (pseudopotential intensity D_{ij}) and the spherical density matrix (augmentation wave occupancies \rho_{ij}):

==== Results concerning PAW augmentation regions ==== Total pseudopotential strength Dij (hartree): Atom # 1 ... Atom # 2 ... Augmentation waves occupancies Rhoij: Atom # 1 ... Atom # 2 ...

- At the end of the file we find the decomposition of the total energy both by direct calculation and double counting calculation:

--- !EnergyTerms iteration_state : {dtset: 1, } comment : Components of total free energy in Hartree kinetic : 6.90447470595323E+00 hartree : 9.62706609299964E-01 xc : -4.29580260772849E+00 Ewald energy : -1.27864121210521E+01 psp_core : 9.19814512188249E-01 local_psp : -4.66849481780168E+00 spherical_terms : 1.43754510318459E+00 total_energy : -1.15261686159562E+01 total_energy_eV : -3.13642998643870E+02 ... --- !EnergyTermsDC iteration_state : {dtset: 1, } comment : '"Double-counting" decomposition of free energy' band_energy : 3.08710945092562E-01 Ewald energy : -1.27864121210521E+01 psp_core : 9.19814512188249E-01 xc_dc : -7.38241276339915E-01 spherical_terms : 7.69958781192718E-01 total_energy_dc : -1.15261691589185E+01 total_energy_dc_eV : -3.13643013418624E+02 ...

Note

The PAW total energy is not the equal to the one obtained in the Norm-Conserving PseudoPotential case.
This is due to the arbitrary modification of the energy reference during the pseudopotential
construction.

In PAW, comparing total energies computed with different atomic datasets is more meaningful: most of
the parts of the energy are calculated exactly, and in general you should be
able to compare numbers for (valence) *energies* between different PAW potentials or different codes.

## 3. Convergence with respect to the plane-wave basis cut-off¶

As in the usual Norm-Conserving PseudoPotential case, the critical convergence parameter is the cut-off energy defining the size of the plane-wave basis.

### 3.a. Norm-Conserving PseudoPotential case¶

The input file *tpaw1_2.abi* contains data to be used to compute the convergence in ecut
for diamond (at experimental volume). There are 9 datasets, with increasing ecut values
from 8 Ha to 24 Ha.
You might use the *tpaw1_2.abi* file (with a standard Norm-Conserving
PseudoPotential), and run:

abinit tpaw1_1.abi >& log

You should obtain the following *total energy* values (see *tpaw1_2.abo*):

etotal1 -1.1628880677E+01 etotal2 -1.1828052470E+01 etotal3 -1.1921833945E+01 etotal4 -1.1976374633E+01 etotal5 -1.2017601960E+01 etotal6 -1.2046855404E+01 etotal7 -1.2062173253E+01 etotal8 -1.2069642342E+01 etotal9 -1.2073328672E+01

You can check that the etotal convergence (at the 1 mHartree level) is not achieved for e_{cut} = 14 Hartree.

### 3.b. Projector Augmented-Wave case¶

Use the same input file as in section **3.a**.

Again, modify the last line of *tpaw1_2.abi*, replacing the *PseudosTM_pwteter/6c.pspnc* file
by *Pseudodojo_paw_pw_standard/C.xml*.

Run the code again and open the ABINIT output file (*.abo*). You should obtain the values:

etotal1 -1.1404460615E+01 etotal2 -1.1496598029E+01 etotal3 -1.1518754947E+01 etotal4 -1.1524981521E+01 etotal5 -1.1526736707E+01 etotal6 -1.1527011746E+01 etotal7 -1.1527027274E+01 etotal8 -1.1527104066E+01 etotal9 -1.1527236307E+01

You can check that:

The *etotal* convergence (at 1 mHartree) is achieved for *14 \le e_{cut} \le 16 Hartree*
(*etotal5* is within 1 mHartree of the final value);

With the same input parameters, for diamond, **a PAW calculation needs a lower cutoff,
compared to a calculation with NCPPs**.

## 4. Convergence with respect to the double grid FFT cut-off¶

In a NCPP calculation, the *plane wave* density grid should be (at least) twice bigger
than the wavefunctions grid, in each direction.
In a PAW calculation, the *plane wave* density grid is tunable
thanks to the input variable pawecutdg (PAW: ECUT for Double Grid).

This is mainly needed to allow the mapping of densities and potentials, located
in the augmentation regions (spheres), onto the global FFT grid.

The number of points of the Fourier grid located in the spheres must be large
enough to preserve a minimal accuracy. It is determined from the cut-off energy
pawecutdg.

One of the most sensitive objects affected by this “grid
transfer” is the compensation charge density; its integral over the
augmentation regions (on spherical grids) must cancel with its integral over
the whole simulation cell (on the FFT grid).

Use now the input file *tpaw1_3.abi*.
The only difference with the *tpaw1_2.abi* file is that ecut is fixed to 12
Ha, while pawecutdg runs from 12 to 39 Ha.

# Input for PAW1 tutorial # Diamond at experimental volume # Convergence with respect to the Double Grid plane-wave cut-off #Define the different datasets ndtset 10 # 10 datasets pawecutdg: 12. # The starting values of the Double Grid plane-wave cut-off energy pawecutdg+ 3. # The increment of pawecutdg from one dataset to the other getwfk -1 # The starting wave-functions are those of the previous dataset #------------------------------------------------------------------------------- #The rest of this file is similar to the tpaw1_1.abi file # except the ecut value (12 Ha) deduced from the convergence study #Definition of the unit cell acell 3*3.567 angstrom # Lengths of the primitive vectors (exp. param. in angstrom) rprim # 3 orthogonal primitive vectors (FCC lattice) 0.0 1/2 1/2 1/2 0.0 1/2 1/2 1/2 0.0 nsym 0 # Automatic detection of symetries #Definition of the atom types and pseudopotentials ntypat 1 # There is only one type of atom znucl 6 # Atomic number of the possible type(s) of atom. Here carbon. pp_dirpath "$ABI_PSPDIR" # Path to the directory were # pseudopotentials for tests are stored pseudos "Pseudodojo_paw_pw_standard/C.xml" # Name and location of the pseudopotential #Definition of the atoms natom 2 # There are two atoms typat 1 1 # They both are of type 1, that is, Carbon xred # Location of the atoms: 0.0 0.0 0.0 # Triplet giving the reduced coordinates of atom 1 1/4 1/4 1/4 # Triplet giving the reduced coordinates of atom 2 #Definition of bands and occupation numbers nband 6 # Compute 6 bands (4 occupied, 2 empty) occopt 1 # Automatic generation of occupation numbers, as a semiconductor #Numerical parameters of the calculation : planewave basis set and k point grid ecut 12.0 # Maximal plane-wave kinetic energy cut-off, in Hartree ecutsm 0.5 # Introduce a smooth PW cutoff within an 0.5 Ha region (not used here) pawecutdg 50. # Max. plane-wave kinetic energy cut-off, in Ha, for the PAW double grid kptopt 1 # Automatic generation of k points, taking into account the symmetry ngkpt 6 6 6 # This is a 6x6x6 grid based on the primitive vectors nshiftk 4 # of the reciprocal space, repeated four times, shiftk # with different shifts: 0.5 0.5 0.5 0.5 0.0 0.0 0.0 0.5 0.0 0.0 0.0 0.5 #Parameters for the SCF procedure nstep 20 # Maximal number of SCF cycles tolvrs 1.0d-10 # Will stop when, twice in a row, the difference # between two consecutive evaluations of potential residual # differ by less than tolvrs #Miscelaneous parameters prtwf 1 # Print wavefunctions (re-used from one dataset to the other) prtden 0 # Do not print density prteig 0 # Do not print eigenvalues ############################################################## # This section is used only for regression testing of ABINIT # ############################################################## #%%<BEGIN TEST_INFO> #%% [setup] #%% executable = abinit #%% [files] #%% files_to_test = #%% tpaw1_3.abo, tolnlines= 0, tolabs= 0.000e+00, tolrel= 0.000e+00, fld_options=-easy #%% output_file = "tpaw1_3.abo" #%% [paral_info] #%% max_nprocs = 4 #%% [extra_info] #%% authors = M. Torrent #%% keywords = PAW #%% description = #%% Input for PAW1 tutorial #%% Diamond at experimental volume #%% Convergence with respect to the Double Grid plane-wave cut-off #%%<END TEST_INFO>

Launch ABINIT with these files; you should obtain the values (file *tpaw1_3.abo*):

etotal1 -1.1518201683E+01 etotal2 -1.1518333032E+01 etotal3 -1.1518666700E+01 etotal4 -1.1518802859E+01 etotal5 -1.1518785111E+01 etotal6 -1.1518739739E+01 etotal7 -1.1518716669E+01 etotal8 -1.1518723268E+01 etotal9 -1.1518738376E+01 etotal10 -1.1518752752E+01

We see that the variation of the energy with respect to the pawecutdg parameter is well
below the 1 mHa level.

In principle, it should be sufficient to choose
pawecutdg = 12 Ha in order to obtain an energy change lower than 1 mHa.
In practice, it is better to keep a security margin. Here, for pawecutdg = 24 Ha
(5^{th} dataset), the energy change is lower than 0.001 mHa: this choice will be more than enough.

Note

Note the steps in the convergence. They are due to sudden changes in the grid size (see the output values for ngfftdg) which do not occur for each increase of pawecutdg. To avoid troubles due to these steps, it is better to choose a value of pawecutdg slightly higher.

The convergence of the compensation charge has a similar behaviour; it is possible to check it in the output file, just after the SCF cycle by looking at:

PAW TEST: ==== Compensation charge inside spheres ============ The following values must be close to each other ... Compensation charge over spherical meshes = 0.252499599273249 Compensation charge over fine fft grid = 0.252497392737764

The two values of the integrated compensation charge density must be close to each other. Note that, for numerical reasons, they cannot be exactly the same (integration over a radial grid does not use the same scheme as integration over a FFT grid).

*Additional test*:
We want now to check the convergence with respect to ecut with a fixed value pawecutdg = 24 Ha.
Let’s modify *tpaw1_2.abi* file, setting pawecutdg to 24 Ha, and let’s launch ABINIT again.

You should obtain the values:

etotal1 -1.1628880677E+01 etotal2 -1.1828052470E+01 etotal3 -1.1921833945E+01 etotal4 -1.1976374633E+01 etotal5 -1.2017601960E+01 etotal6 -1.2046855404E+01 etotal7 -1.2062173253E+01 etotal8 -1.2069642342E+01 etotal9 -1.2073328672E+01

You can check again that the *etotal* convergence (at the 1 mHartree level)
is achieved for 14 \le e_{cut} \le 16 Hartree.

Note

Although pawecutdg should always be checked, in practice, a common use it to put it bigger than ecut and keep it constant during all calculations. Increasing pawecutdg slightly changes the CPU execution time, but above all it is memory-consuming. Note that, if ecut is already high, there is no need for a high pawecutdg.

Important

When testing ecut convergency, pawecutdg has to remain constant to obtain consistent results.

## 5. Plotting PAW contributions to the Density of States (DOS)¶

We now use the input file *tpaw1_4.abi* file.
ABINIT is used to compute the Density Of State (DOS)
(see the prtdos keyword in the input file).
Also note that more k-points are used in order to increase the accuracy of the DOS.
ecut is set to 12 Ha, while pawecutdg is 24 Ha.

# Input for PAW1 tutorial # Diamond at experimental volume # Computation of Density of States (DOS) #Specific input parameters related to DOS printing #Case 1: Total DOS prtdos 1 # Flag to activate the total DOS output #Case 2: projected DOS (to be uncommented later) #prtdos 3 # Flag to activate the projected DOS output #pawprtdos 1 # Flag to activate the output of all PAW contributions #natsph 1 # Number of atom(s) around which the projected DOS has to computed #iatsph 1 # Index of these atoms #ratsph 1.51 # Radius defining the projection area around the atom(s) #For this calculation we take a metallic scheme for the occupation numbers #This is mandatory for a DOS calculation occopt 7 # Automatic generation of occupation numbers, as a metal tsmear 0.005 # Smearing temperature for the metallic occupation scheme (Hartree) #For this calculation we take more k-points to have a smoother DOS curve ngkpt 10 10 10 # This is a 10x10x10 grid based on the primitive vectors #------------------------------------------------------------------------------- #The rest of this file is similar to the tpaw1_1.abi file # using the ecut and pawecutdg values deduced from the convergence studies #Definition of the unit cell acell 3*3.567 angstrom # Lengths of the primitive vectors (exp. param. in angstrom) rprim # 3 orthogonal primitive vectors (FCC lattice) 0.0 1/2 1/2 1/2 0.0 1/2 1/2 1/2 0.0 nsym 0 # Automatic detection of symetries #Definition of the atom types and pseudopotentials ntypat 1 # There is only one type of atom znucl 6 # Atomic number of the possible type(s) of atom. Here carbon. pp_dirpath "$ABI_PSPDIR" # Path to the directory were # pseudopotentials for tests are stored pseudos "Pseudodojo_paw_pw_standard/C.xml" # Name and location of the pseudopotential #pseudos "C_paw_pw_2proj.xml" # Uncomment this line later #pseudos "C_paw_pw_2proj.xml" # Uncomment this line later #Definition of the atoms natom 2 # There are two atoms typat 1 1 # They both are of type 1, that is, Carbon xred # Location of the atoms: 0.0 0.0 0.0 # Triplet giving the reduced coordinates of atom 1 1/4 1/4 1/4 # Triplet giving the reduced coordinates of atom 2 #Definition of bands and occupation numbers nband 6 # Compute 6 bands (4 occupied, 2 empty) #Numerical parameters of the calculation : planewave basis set and k point grid ecut 12.0 # Maximal plane-wave kinetic energy cut-off, in Hartree ecutsm 0.5 # Introduce a smooth PW cutoff within an 0.5 Ha region (not used here) pawecutdg 24. # Max. plane-wave kinetic energy cut-off, in Ha, for the PAW double grid kptopt 1 # Automatic generation of k points, taking into account the symmetry nshiftk 4 # of the reciprocal space, repeated four times, shiftk # with different shifts: 0.5 0.5 0.5 0.5 0.0 0.0 0.0 0.5 0.0 0.0 0.0 0.5 #Parameters for the SCF procedure nstep 20 # Maximal number of SCF cycles tolvrs 1.0d-10 # Will stop when, twice in a row, the difference # between two consecutive evaluations of potential residual # differ by less than tolvrs #Miscelaneous parameters prtwf 0 # Do not print wavefunctions prtden 0 # Do not print density prteig 0 # Do not print eigenvalues ############################################################## # This section is used only for regression testing of ABINIT # ############################################################## #%%<BEGIN TEST_INFO> #%% [setup] #%% executable = abinit #%% [files] #%% files_to_test = #%% tpaw1_4.abo, tolnlines= 0, tolabs= 0.000e+00, tolrel= 0.000e+00, fld_options = -easy #%% output_file = "tpaw1_4.abo" #%% [paral_info] #%% max_nprocs = 4 #%% [extra_info] #%% authors = M. Torrent #%% keywords = PAW #%% description = #%% Input for PAW1 tutorial #%% Diamond at experimental volume #%% Computation of Density of States #%%<END TEST_INFO>

Launch the code with these files; you should obtain the *tpaw1_4.abo* and the DOS file (*tpaw1_4o_DOS*):

abinit tpaw1_4.abi >& log

You can plot the DOS file; for this purpose, use a graphical tool
and plot column 3 with respect to column 2.

Example: if you use the xmgrace tool, launch:

xmgrace -block tpaw1_4o_DOS -bxy 1:2

At this stage, you have a usual Density of State plot; nothing specific to PAW.

Now, edit the *tpaw1_4.abi* file, comment the “prtdos 1” line, and uncomment (or add):

prtdos 3 pawprtdos 1 natsph 1 iatsph 1 ratsph 1.51

prtdos=3 requires the output of the projected DOS;
natsph=1, iatsph=1 select the first carbon atom as
the center of projection, and ratsph=1.51 sets the radius of the projection area to 1.51
atomic units (this is exactly the radius of the PAW augmentation regions:
generally the best choice).

pawprtdos=1 is specific to PAW.
With this option, ABINIT should compute all the contributions to the projected DOS.

Let us remember that:

Within PAW, the total projected DOS has 3 contributions:

- The smooth plane-waves (PW) contribution (from |\tPsi\rangle),
- The all-electron on-site (AE) contribution (from \langle \tprj^a_i|\tPsi\rangle |\phi_i^a\rangle),
- The pseudo on-site (PS) contribution (from \langle \tprj^a_i|\tPsi\rangle |\tphi_i^a\rangle).

Execute ABINIT again (with the modified input file).
You get a new DOS file, named *tpaw1_4o_DOS_AT0001*.
You can edit it and look inside; it contains the 3 PAW contributions
(mentioned above) for each angular momentum.
*In the diamond case, only l = 0 and l = 1 momenta are to be considered*.

Now, plot the file, using the 7^{th}, 12^{th} and 17^{th} columns with respect to the
2^{nd} one; it plots the 3 PAW contributions for l = 0 (the total DOS is the sum of
the three contributions).
If you use the xmgrace tool, launch:

xmgrace -block tpaw1_4o_DOS_AT0001 -bxy 1:7 -bxy 1:12 -bxy 1:17

You should get this:

As you can see, the smooth PW contribution and the PS on-site contribution are close. At basis completeness,
they should cancel; we could approximate the DOS by the AE on-site part taken alone.

That’s exactly the purpose of the pawprtdos = 2 option: in that case, only the AE
on-site contribution is computed and given as a good approximation of the
total projected DOS. The main advantage of this option is that the computing time is
greatly reduced (the DOS is instantaneously computed).

However, as you will see in the next section, this approximation is only valid when:

- The \tphi_i basis is complete enough
- The electronic density is mainly contained in the sphere defined by ratsph.

## 6. Testing the completeness of the PAW partial wave basis¶

In the previous section we used a “standard” PAW dataset, with 2
*partial waves* per angular momentum. It is generally the best compromise
between the completeness of the partial wave basis and the efficiency of the
PAW dataset (the more *partial waves* you have, the longer the CPU time used by ABINIT is).

Let’s have a look at the *$ABI_PSPDIR/Pseudodojo_paw_pw_standard/C.xml* file.
The tag

Now, let’s open the *$ABI_PSPDIR/C_paw_pw_2proj.xml*
and *$ABI_PSPDIR/C_paw_pw_6proj.xml* PAW dataset files.
The first dataset is based on only **one** *partial wave* per l; the second one is based on
**three** *partial waves* per l.
The completeness of the partial wave basis increases when you use
*C_paw_pw_2proj.xml*,
*Pseudodojo_paw_pw_standard/C.xml*
and *C_paw_pw_6proj.xml*.

Now, let us plot the DOS using the two new PAW datasets.

- Save the existing
*tpaw1_4o_DOS_AT0001*file, naming it f.i.*tpaw1_4o_4proj_DOS_AT0001*. - Open the
*tpaw1_4.abi*file and modify it in order to use the*C_paw_pw_2proj.xml*PAW dataset. - Run ABINIT.
- Save the new
*tpaw1_4o_DOS_AT0001*file, naming it f.i.*tpaw1_4o_2proj_DOS_AT0001*. - Open the
*tpaw1_4.abi*file and modify it in order to use the*C_paw_pw_2proj.xml*PAW dataset. - Run ABINIT again.
- Save the new
*tpaw1_4o_DOS_AT0001*file, naming it f.i.*tpaw1_4o_6proj_DOS_AT0001*.

Then, plot the contributions to the projected DOS for the two new DOS files. You should get:

Adding the DOS obtained in the previous section to the comparison, you immediately see that the superposition of the plane wave part DOS (PW) and the PS on-site DOS depends on the completeness of the partial wave basis!

Now, you can have a look at the 3 output files (one for each PAW dataset).

A way to estimate the completeness of the partial wave basis is to compare
derivatives of total energy; if you look at the stress tensor:

For the 2 `partial-wave` basis: 2.85307799E-04 2.85307799E-04 2.85307799E-04 0. 0. 0. For the 4 `partial-wave` basis: 4.97872484E-04 4.97872484E-04 4.97872484E-04 0. 0. 0. For the 6 `partial-wave` basis: 5.38392693E-04 5.38392693E-04 5.38392693E-04 0. 0. 0.

The *2 partial-wave* basis is clearly not complete; the *4 partial-wave basis* results are correct.
Such a test is useful to estimate the precision we can expect on the stress tensor
(at least due to the partial wave basis completeness).

You can compare other results in the 3 output files: total energy, eigenvalues, occupations…

Note

The CPU time increases with the size of the partial wave basis. This is why it is
not recommended to use systematically high-precision PAW datasets.

If you want to learn how to generate PAW datasets with different *partial wave* basis,
you might follow the tutorial on generating PAW datasets (PAW2).

## 7. Checking the validity of PAW results¶

The validity of our computation has to be checked
by comparison, on known structures, with known results.

In the case of diamond, lots of computations and experimental results exist.

Important

The validity of PAW calculations (**completeness of plane wave basis and
partial wave basis**) should always be checked by comparison
with **all-electrons** computations or with other existing PAW results;
it should not be done by comparison with experimental results.
As the PAW method has the same accuracy than all-electron methods, results should be very close.

In the case of diamond, all-electron results can be found f.i. in [Holzwarth1997].
All-electron equilibrium parameters for diamond (within *Local Density Approximation*) obtained
with the FP-LAPW WIEN2K code are:

a0 = 3.54 angstrom B = 470 GPa

Experiments give:

a0 = 3.56 angstrom B = 443 GPa

Let’s test with ABINIT.
We use now the input file *tpaw1_5.abi* file and we run
ABINIT to compute values of *etotal* for several cell parameters
around 3.54 angstrom, using the standard PAW dataset.

# Input for PAW1 tutorial # Diamond: etotal vs acell curve around equilibrium #Define the different datasets ndtset 7 # 7 datasets acell: 3*3.515 angstrom # The starting values of the primitive vector lengths acell+ 3*0.005 angstrom # The increment of acell from one dataset to the other getwfk -1 # The starting wave-functions are those of the previous dataset #------------------------------------------------------------------------------- #The rest of this file is similar to the tpaw1_4.abi file # with ecut=20, pawecutdg=50 and ngkpt=3*10 for higher precision #Definition of the unit cell acell 3*3.567 angstrom # Lengths of the primitive vectors (not used here) rprim # 3 orthogonal primitive vectors (FCC lattice) 0.0 1/2 1/2 1/2 0.0 1/2 1/2 1/2 0.0 nsym 0 # Automatic detection of symetries #Definition of the atom types and pseudopotentials ntypat 1 # There is only one type of atom znucl 6 # Atomic number of the possible type(s) of atom. Here carbon. pp_dirpath "$ABI_PSPDIR" # Path to the directory were # pseudopotentials for tests are stored pseudos "Pseudodojo_paw_pw_standard/C.xml" # Name and location of the pseudopotential #Definition of the atoms natom 2 # There are two atoms typat 1 1 # They both are of type 1, that is, Carbon xred # Location of the atoms: 0.0 0.0 0.0 # Triplet giving the reduced coordinates of atom 1 1/4 1/4 1/4 # Triplet giving the reduced coordinates of atom 2 #Definition of bands and occupation numbers nband 6 # Compute 6 bands (4 occupied, 2 empty) occopt 1 # Automatic generation of occupation numbers, as a semiconductor #Numerical parameters of the calculation : planewave basis set and k point grid ecut 20.0 # Maximal plane-wave kinetic energy cut-off, in Hartree ecutsm 0.5 # Introduce a smooth PW cutoff within an 0.5 Ha region (not used here) pawecutdg 50. # Max. plane-wave kinetic energy cut-off, in Ha, for the PAW double grid kptopt 1 # Automatic generation of k points, taking into account the symmetry ngkpt 10 10 10 # This is a 10x10x10 grid based on the primitive vectors nshiftk 4 # of the reciprocal space, repeated four times, shiftk # with different shifts: 0.5 0.5 0.5 0.5 0.0 0.0 0.0 0.5 0.0 0.0 0.0 0.5 #Parameters for the SCF procedure nstep 10 # Maximal number of SCF cycles tolvrs 1.0d-10 # Will stop when, twice in a row, the difference # between two consecutive evaluations of potential residual # differ by less than tolvrs #Miscelaneous parameters prtwf 1 # Print wavefunctions (re-used from one dataset to the other) prtden 0 # Do not print density prteig 0 # Do not print eigenvalues ############################################################## # This section is used only for regression testing of ABINIT # ############################################################## #%%<BEGIN TEST_INFO> #%% [setup] #%% executable = abinit #%% [files] #%% files_to_test = #%% tpaw1_5.abo, tolnlines= 0, tolabs= 0.000e+00, tolrel= 0.000e+00, fld_options = -easy #%% output_file = "tpaw1_5.abo" #%% [paral_info] #%% max_nprocs = 4 #%% [extra_info] #%% authors = M. Torrent #%% keywords = PAW #%% description = #%% Input for PAW1 tutorial #%% Diamond: etotal vs acell curve around equilibrium #%%<END TEST_INFO>

abinit tpaw1_1.abi >& log

From the *tpaw1_5.abo* file, you can extract the 7 values of *acell* and 7 values
of *etotal*, then put them into a file and plot it with a graphical tool.
You should get:

From this curve, you can extract the cell values of a_0 and B (with the method of your choice, for example by a Birch-Murnhagan spline fit). You get:

a0 = 3.53 angstrom B = 469.5 GPa

These results are in excellent agreement with FP-LAPW ones!

## 8. Additional comments about PAW in ABINIT¶

### 8.a. Overlap of PAW augmentation regions¶

In principle, the PAW formalism is only valid for non-overlapping augmentation spherical regions. But, in usual cases, a small overlap between spheres is acceptable. By default, ABINIT checks that the distances between atoms are large enough to avoid overlap; a “small” voluminal overlap of 5% is accepted by default. This value can be tuned with the pawovlp input keyword. The overlap check can even be by-passed with pawovlp=-1 (not recommended!).

Warning

While a small overlap can be acceptable for the augmentation regions, an overlap of the compensation charge densities has to be avoided. The compensation charge density is defined by a radius (named r_{shape} in the PAW dataset) and an analytical shape function. The overlap related to the compensation charge radius is checked by ABINIT and a WARNING is eventually printed.

Also note that you can control the compensation charge radius and shape function while generating the PAW dataset (see tutorial on generating PAW datasets (PAW2)).

### 8.b. Mixing scheme for the Self-Consistent cycle; decomposition of the total energy¶

The use of an efficient **mixing scheme** in the self-consistent loop is a crucial
point to minimize the number of steps to achieve convergence.
This mixing can be done on the potential or on the density.

By default, in a Norm-Conserving PseudoPotential calculation, the mixing is done on the potential;
but, for technical reasons, this choice is not optimal for PAW calculations.
Thus, by default, the mixing is done on the density when PAW is activated.

The mixing scheme can be controlled by the iscf variable (see the different options of this input variable).
To compare both schemes, you can edit the *tpaw1_1.abi* file and try iscf = 7
or 17 and compare the behaviour of the SC cycle in both cases; as you can see, the
final *total energy* is the same but the way to reach it is completely different.

Now, have a look at the end of the file and focus on the `Components of totalfree energy`

; the total energy is decomposed according to two different schemes (`direct`

and `double counting`

);
at very high convergence of the SCF cycle the potential/density
residual is very small and these two values should be the same.

But it has been observed that
the converged value was reached more rapidly by the `direct`

energy, when the
mixing is on the potential, and by the `double counting`

energy when the mixing
is on the density. Thus, by default, in the output file is to print the direct
energy when the mixing is on the potential, and the double counting energy
when the mixing is on the density.

Also note that PAW *partial waves occupancies* \rho_{ij} also are
mixed during the SC cycle; by default, the mixing is done in the same way as the density.

### 8.c. PAW+U for correlated materials¶

If the system under study contains strongly correlated electrons, the `DFT+U`

method can be useful. It is controlled by the usepawu, lpawu,
upawu and jpawu input keywords.
Note that the formalism implemented in ABINIT is approximate, i.e. it is only valid if:

- The \tphi^a_i basis is complete enough;
- The electronic density is mainly contained in the PAW sphere.

The approximation done here is the same as the one explained in the 5^{th}
section of this tutorial: considering that smooth PW contributions and PS
on-site contributions are closely related, only the AE on-site contribution is
computed; it is indeed a very good approximation.

Converging a *Self-Consistent Cycle*, or ensuring the global minimum is reached,
with PAW+U is sometimes difficult. Using usedmatpu and dmatpawu can help.
See tutorial on DFT+U.

### 8.d. Printing volume for PAW¶

If you want to get more detailed output concerning the PAW computation, you
can use the pawprtvol input keyword.
It is particularly useful to print details about *pseudopotential strength*
D_{ij} or *partial waves occupancies* \rho_{ij}.

### 8.e. Additional PAW input variables¶

Looking at the PAW variable set, you can find the description of additional input keywords related to PAW. They are to be used when tuning the computation, in order to gain accuracy or save CPU time.

See also descriptions of these variables and input file examples in the PAW topic page.

Warning

In a standard computation, these variables should not be modified!