# ABINIT, lesson PAW1 :

## Projector augmented-wave technique : how to use it ?

This lesson aims at showing how to perform a calculation in the frame of the PAW method.

It is supposed that you already know how to use ABINIT in the norm-conserving pseudopotential case.

This lesson should take about 1.5 hour.

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**Contents of lesson
PAW1 :**

**Contents of lesson PAW1 :**

- 0. Summary of the PAW method
- 1. Using PAW with ABINIT
- 2. The convergence on plane-wave cut-off (ecut)
- 3. The convergence on double FFT grid cut-off (pawecutdg)
- 4. Plotting PAW contributions to the DOS
- 5. Testing the completeness of the PAW partial wave basis
- 6. Checking the validity of PAW results
- 7. Additional comments about PAW in ABINIT

0. Summary of the PAW method

0. Summary of the PAW method

The PAW (Projector Augmented-Wave) method has been
introduced by Peter Blöchl in 1994. As he says, "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 Ψ_{n}
with
"auxiliary"
(or "pseudo") soft wavefunctions~Ψ_{n}
:

_{n> =}~|Ψ

_{n}>+ Σ (φ

_{i }-~φ

_{i}) <~p

_{i }|~Ψ

_{n }>

This
relation
is based on
the definition of atomic spheres (augmentation
regions) of radius r_{c},
around the atoms of the
system in which the partial
waves | φ_{i}>
form a basis
of atomic wavefunctions; |~φ_{i}>
are "pseudized" partial
waves (obtained from |
φ_{i}>), and ~p_{i}
are dual functions
of
the |~φ_{i}> called
projectors.

It is therefore possible to write every quantity depending on Ψ_{n}
(density, energy, Hamiltonian) as a function of~Ψ_{n}
and to find~Ψ_{n}
by solving self-consistent equations.

The PAW method has two main advantages:

- From~Ψ_{n},
it is always
possible to obtain the true "all electron"
wavefunction Ψ_{n}.

- The convergency is comparable
to an ultrasoft pseudopotential one.

From a practical point of view (user's point of view), a PAW calculation is rather
similar to a norm-conserving pseudopotential. Most noticeably, one will have to use
a special atomic data file
(PAW dataset)
that contains the φ_{i},~φ_{i} and ~p_{i}
and that plays
the same role as a pseudopotential file.

It is highly recommanded to read the following papers to understand correctly the basic concepts of the PAW method:

[1] "Projector augmented-wave method", P.E. Blochl, Physical Review B 50, 17953 (1994)[2] "From ultrasoft pseudopotentials to the projector augmented-wave method", G. Kresse and D. Joubert, Physical Review B 59, 1758 (1999)

The implementation of the PAW method in ABINIT is detailed in the following paper, describing specific notations and formulations:

[3] "Implementation of the projector augmented-wave method in the ABINIT code: Application to the study of iron under pressure", M. Torrent, F. Jollet, F. Bottin, G. Zerah, and X. Gonze, Computational Materials Science 42, 337 (2008)

1. Using PAW with ABINIT

1. Using PAW with ABINIT

1. Using PAW with ABINIT

*Before continuing, you might
consider to work in a different
subdirectory as for the other lessons. 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 *

*~abinit/tests/tutorial/Input directory.*

*You can compare your results with reference
output files located in **~abinit/tests/tutorial/Refs
and **~abinit/tests/tutorial/Refs/tpaw1_addons
**directories (for the present tutorial they are named
tpaw1_*.out).*

The input file tpaw1_1.in
is an example of a file
that contains data for computing the total energy for diamond
at the experimental volume (within the LDA exchange-correlation
functional).
You might use the file tpaw1_1.files
(with a standard
norm-conserving pseudopotential)
as a "files" file, and get the corresponding output file
(it is available as ../Refs/tpaw1_1.out).

Copy the files tpaw1_1.in
and tpaw1_1.files
in your work
directory, and run ABINIT:

```
abinit < tpaw1_1.files > tmp-log
```

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

Now, open the tpaw1_1.files and modify the last line; replace the 6c.pspnc file by 6c.lda.atompaw.

Run ABINIT again:

`abinit < tpaw1_1.files > tmp-log`

Your run should stop before end ! The input file is missing a mandatory argument: pawecutdg !!

Add the line "pawecutdg
50." in the tpaw1_1.in file
and run ABINIT again.

Now ABINIT runs to the end.

Note
that 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 and have a look inside (be
careful, it is the last output file of the tpaw1_1 series).

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:

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

- The use of two FFT grids, mentioned as:

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

basis_size (lnmax)= 4 (lmn_size= 8), orbitals= 0 0 1 1

Spheres core radius: rc_sph= 1.50000000

4 radial meshes are used:

- mesh 1: r(i)=AA*[exp(BB*(i-1))-1], size= 505 , AA= 0.21824E-02 BB= 0.13095E-01

- mesh 2: r(i)=AA*[exp(BB*(i-1))-1], size= 500 , AA= 0.21824E-02 BB= 0.13095E-01

- mesh 3: r(i)=AA*[exp(BB*(i-1))-1], size= 530 , AA= 0.21824E-02 BB= 0.13095E-01

- mesh 4: r(i)=AA*[exp(BB*(i-1))-1], size= 644 , AA= 0.21824E-02 BB= 0.13095E-01

Shapefunction is SIN type: shapef(r)=[sin(pi*r/rshp)/(pi*r/rshp)]**2

Radius for shape functions = sphere core radius

Radial grid used for partial waves is grid 1

Radial grid used for projectors is grid 2

Radial grid used for (t)core density is grid 3

Radial grid used for Vloc is grid 4

Radial grid used for pseudo valence density is grid 4

After the SCF cycle section:

==== Compensation charge inside spheres ============

The following values must be close to each other ...

Compensation charge over spherical meshes = 0.413178580356274

Compensation charge over fine fft grid = 0.413177280314290

- Information concerning the non-local term (pseudopotential strength Dij ) and the spherical density matrix (augmentation wave occupancies Rhoij)

==== 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:

Components of total free energy (in Hartree) :

Kinetic energy = 6.40164318808980E+00

Hartree energy = 9.63456708252837E-01

XC energy = -3.53223656186138E+00

Ewald energy = -1.27864121210521E+01

PspCore energy = 5.41017918797015E-01

Loc. psp. energy= -5.27003595856857E+00

Spherical terms = 2.15689044331394E+00

>>>>> Internal E= -1.15256763830284E+01

"Double-counting" decomposition of free energy:

Band energy = 6.87331579398577E-01

Ewald energy = -1.27864121210521E+01

PspCore energy = 5.41017918797015E-01

Dble-C XC-energy= 1.22161340385476E-01

Spherical terms = -8.97688814082645E-02

>>>>> Internal E= -1.15256701638793E+01

>Total energy in eV = -3.13629604304723E+02

>Total DC energy in eV = -3.13629435073068E+02

Note that the total energy calculated in PAW is not the same
as the one obtained
in the norm-conserving pseudopotential case.
This is normal: in the norm-conserving potential case, the
energy reference has been arbitrarily modified by the pseudopotential construction procedure.
Comparing total energies computed with different
PAW potentials 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.

2. The convergence on plane-wave cut-off (ecut)

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

2. The convergence on plane-wave cut-off (ecut)

2. The convergence on plane-wave cut-off (ecut)

**1.a**** Computing the convergence in
ecut for diamond
in
the norm-conserving case**

You might use the tpaw1_2.files file (with a standard norm-conserving pseudopotential), and run ABINIT:

```
abinit < tpaw1_2.files > tmp-log
```

You
should obtain the values (output file tpaw1_2.out) :
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 ecut=24 Hartree.

**1.b**** Computing the convergence in
ecut for diamond
in
the PAW case**

Use the same input files as in section 1.a.

Again, modify the last line of tpaw1_2.files,
replacing the 6c.pspnc
file by 6c.lda.atompaw.

Run ABINIT again and open the output file (it should be tpaw1_2.outA)

You should obtain the values:

etotal2 -1.1518675625E+01

etotal3 -1.1524581240E+01

etotal4 -1.1525548758E+01

etotal5 -1.1525741818E+01

etotal6 -1.1525865084E+01

etotal7 -1.1525926864E+01

etotal8 -1.1525947400E+01

etotal9 -1.1525954817E+01

You can check that:

The etotal
convergence (at 1 mHartree) is achieved for 12<=ecut<=14
Hartree (etotal4 is within 1 mHartree of the final value);

The
etotal
convergence (at 0.1 mHartree) is achieved for 16<=ecut<=18
Hartree (etotal6 is within 0.1 mHartree of the final value).

So with the same input, a PAW calculation for diamond needs a lower cutoff, compared to a norm-conserving pseudopotential calculation.

###

3. The convergence on the double grid FFT cut-off (pawecutdg)

3. The convergence on the double grid FFT cut-off (pawecutdg)

3. The convergence on the double grid FFT cut-off (pawecutdg)

In a
norm-conserving pseudopotential calculation, the (plane wave) density
grid is (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 needed because of the mapping of
objects (densities, potentials) located in
the augmentation regions (PAW spheres) onto the global FFT grid.

The number of points
of the Fourier grid located in the spheres must be high enough to
preserve the accuracy. It is determined from the cut-off
energy pawecutdg. An
alternative
is to use directly the input variable ngfftdg.
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.in
and the associated tpaw1_3.files
file.

The only difference with the tpaw1_2.in
file is that ecut
is fixed to 12 Ha,
while pawecutdg
runs from 12 to 39 Ha.

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

etotal2 -1.1524595840E+01

etotal3 -1.1524585370E+01

etotal4 -1.1524580630E+01

etotal5 -1.1524584720E+01

etotal6 -1.1524583573E+01

etotal7 -1.1524582786E+01

etotal8 -1.1524582633E+01

etotal9 -1.1524582213E+01

etotal10 -1.1524582316E+01

We see that the variation of the energy wit respect to this parameter is well below
the 1 mHa level. For pawecutdg=24 Ha
(5th
dataset), the
the energy change is lower than 0.001 mHa```
. Note the steps
in the convergency. They are due to the sudden (integer) changes in the grid size
(see the output values for ngfftdg) which do not occur
for each increase of pawecutdg.
```

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:

==== Compensation charge inside spheres ============

The following values must be close...

Compensation charge over spherical meshes = 0.409392121335747

Compensation charge over fine fft grid = 0.409392418241149

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).

We want now to check the convergence in ecut with a fixed value of 24 Ha for pawecutdg. Modify the file tpaw1_2.in, setting pawecutdg to 24 Ha, and launch ABINIT again...

You should obtain the values :

etotal2 -1.1518678975E+01

etotal3 -1.1524584720E+01

etotal4 -1.1525552267E+01

etotal5 -1.1525745330E+01

etotal6 -1.1525868591E+01

etotal7 -1.1525930368E+01

etotal8 -1.1525950904E+01

etotal9 -1.1525958319E+01

You can check again that: The

etotal convergence (at the 1 mHartree level) is achieved for 12<=ecut<=14 Hartree ;

The etotal convergence (at the 0.1 mHartree level) is achieved for 16<=ecut<=18 Hartree.

Note 1:

Associated with the input variable pawecutdg is the input variable ngfftdg: it gives the size of the FFT grid associated with pawecutdg. Note that pawecutdg is only useful to define the FFT grid for the density in a convenient way. You can therefore tune directly ngfftdg to define the size of the FFT grid for the density.

Note 2:

Although pawecutdg should always be checked, in practice, a common use it to put it at least twice 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.

Last warning: when testing ecut convergency, pawecutdg has to remain constant to obtain consistent results.

4. Plotting PAW contributions to
the Density of States (DOS)

We use now the input file tpaw1_4.in
and the associated tpaw1_4.files
file.

ABINIT is now asked 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.

Launch ABINIT with these files; you should obtain the tpaw1_4.out and the DOS file (tpaw1_4o_DOS):

```
abinit < tpaw1_4.files > tmp-log
```

You can plot the DOS file if you want; for this purpose, use a graphical tool and plot column 3 with respect to column 2. If you use the "xmgrace" tool, you first have to edit the tpaw1_4o_DOS file, comment all the lines of the header (adding a "#") and launch:

`xmgrace -block tpaw1_4o_DOS -bxy 1:2`

At this stage, you have the usual plot for a DOS; nothing specific to PAW.

Now, edit the tpaw1_4.in file, comment the "prtdos 1", and uncomment (or add):

The " prtdos
3"
statement now requires the output of the projected DOS; "natsph
1 iatsph 1 ratsph
1.5" selects the first carbon atom as the center of projection, and
sets the
radius of the projection area to 1.5 atomic units (this is exactly the
radius of the PAW augmentation regions: generally the best choice).

The "pawprtdos 1" is specific
to PAW. With this option, ABINIT is asked to compute all the
contributions to the projected DOS.

Let's remember that:

_{n> =}~|Ψ

_{n}>+ Σ (φ

_{i }-~φ

_{i}) <~p

_{i }|~Ψ

_{n }>

Within PAW, the total projected DOS has 3 contributions:

1- the smooth plane-waves contribution (from~|Ψ_{n}>)

2- the all-electron on-site contribution (from φ_{i }<~p_{i } |~Ψ_{n
}>)

3- the pseudo on-site contribution (from~φ_{i}
<~p_{i } |~Ψ_{n
}>).

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 treated.

Now, plot the file, using the 3rd, 8th and 13th columns with respect to the 2nd 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, you first have to edit the file, comment all the lines of the header and launch:

`xmgrace -block tpaw1_4o_DOS_AT0001 -bxy 1:2 -bxy 1:7 -bxy 1:12`

You
should get this:As you can see, the smooth PW contribution and the PS on-site contribution are close.

So, in a first approach, they cancel; we could approximate the DOS by the AE on-site part taken alone.

That is exactly what is done when pawprtdos=2; 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 the decrease of the CPU time needed to compute the DOS (it is instantaneously computed).

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

_{i}basis is complete enough

(2) the electronic density is mainly contained in the sphere defined by ratsph.

5. Testing the
completeness of the PAW
partial wave basis

Let's have a look at the ~abinit/tests/Psps_for_tests/6c.lda.atompaw file. The sixth line indicates the number of partial waves and their l angular momentum. In the present file, "0 0 1 1" means "two l=0 partial waves, two l=1 partial waves".

Now, let's open the ~abinit/tests/Psps_for_tests/6c.lda.test-2proj.atompaw and ~abinit/tests/Psps_for_tests/6c.lda.test-6proj.atompaw files. In the first file, only one partial wave per l is present; in the second one, 3 partial waves per l are present. In other words, the completeness of the partial wave basis increases when you use 6c.lda.test-2proj.atompaw, 6c.lda.atompaw and 6c.lda.test-6proj.atompaw.

Now, let's plot the DOS for the two new PAW datasets.

- Open the tpaw1_4.files file and modify it in order to use the 6c.lda.test-2proj.atompaw PAW dataset.

- Launch ABINIT again.

- Save the new tpaw1_4o_DOS_AT0001 file, naming it, for instance, tpaw1_4o_2proj_DOS_AT0001.

- Open the tpaw1_4.files file and modify it in order to use the 6c.lda.test-6proj.atompaw PAW dataset.

- Launch ABINIT again.

- Save the new tpaw1_4o_DOS_AT0001 file, naming it, for instance, 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 Smooth part DOS 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)... for instance in a comparison tool.

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

For the 4 partial-wave basis: 4.1504385879E-04 4.1504385879E-04 4.1504385879E-04 0. 0. 0.

For the 6 partial-wave basis: 4.1469803037E-04 4.1469803037E-04 4.1469803037E-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).

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

Also notice that the dimensions of the PAW on-site quantities change: have a look at "Pseudopotential strengh Dij" or "Augmentation waves occupancies Rhoij" sections...

Note: if you want to learn how to generate PAW datasets with different partial wave bases, you might follow the tutorial PAW2.

6. Checking the validity of PAW
results

Very important remark: the validity (completeness of plane wave basis and partial wave basis) of PAW calculations should always be checked by comparison with all-electrons computation results (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.

Concerning diamond, all-electron results can be found (for instance) in PRB 55, 2005 (1997).

With the famous WIEN2K code (which uses the FP-LAPW method), all-electron equilibrium parameters for diamond (for LDA) are:

_{0}= 3.54 angstrom B = 470 GPa

Experiments give:

_{0}= 3.56 angstrom

B = 443 GPa

Let's test with ABINIT.

We use now the input file tpaw1_5.in and the associated tpaw1_5.files file.

ABINIT is now asked to compute values of etotal for several cell parameters around 3.54 angstrom, using the standard PAW dataset.

Launch ABINIT with these files; you should obtain the tpaw1_5.out.

```
abinit < tpaw1_5.files > tmp-log
```

From
the tpaw1_5.out
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:

_{0}= 3.535 angstrom B = 465 GPa

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

7. Additional comments about
PAW in ABINIT

**7.a**** Mixing scheme for
the Self-Consistent cycle; decomposition of the total energy.**

The mixing scheme can be controlled by the iscf variable (please, read again the different options of this input variable).

By default, iscf=7 for norm-conserving pseudopotentials, while iscf= 17 for PAW...

To compare both schemes, you can edit the tpaw1_1.in file and try iscf=7 or 17 and compare the behaviour of the SC cycle in both cases; as you can see, 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 total free energy"; the total energy is decomposed according to both schemes; at very high convergence of the SCF cycle (very small potential or density residual), 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.

You can try (using the tpaw1_1.in file) to decrease the values of tolvrs and look at the difference between both values of energy.

Also note that PAW ρ

_{ij}quantities (occupancies of partial waves) also are mixed during the SC cycle; by default, the mixing is done in the same way as the density.

**7.b**** Overlap of
PAW spheres**

In
principle, the PAW formalism is only valid for non-overlapping
augmentation
regions (PAW spheres). 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.

Important 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 file) 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 PAW2).

**7.c****
Printing volume for PAW**

It is particullary useful to print details about pseudopotential strengh (D

_{ij}) or partial waves occupancies (ρ

_{ij}).

**7.d****
Additional PAW input variables**

Warning : in a standard computation, these variables should not be modified !

Variables that can be used to gain accuracy (in ascending order of importance):

mqgriddg: control the accuracy of spline fits to transfer densities/potentials from FFT grid to spherical grid.

pawnzlm: control the computation of moments of spherical densities that should be zero by symmetry.

Variables that can be used to save memory (in ascending order of importance):

pawlcutd: control the number of angular momenta to take into account in on-site densities.

pawlmix: control the number of ρ

_{ij}to be mixed during SCF cycle.

Variables that can be used to save CPU time (in ascending order of importance):

pawstgylm: control the storage of spherical harmonics computed around atoms.

pawlcutd: control the number of angular momenta to take into account in on-site densities.

pawlmix: control the number of ρ

_{ij}to be mixed during SCF cycle.

bxctmindg: can be used to decrease the size of fine FFT grid for a given value of pawecutdg.

**7.e**** PAW+U**

_{i}basis is complete enough ;

(2) the electronic density is mainly contained in the PAW sphere.

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

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