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<meta http-equiv="Content-Type" content="text/html; charset=iso-8859-1"><title>Tutorial PAW 1</title></head><body style="background-color: rgb(255, 255, 255);"><hr><h1>ABINIT,
lesson PAW1 :</h1>
<h2>Projector Augmented-Wave technique : how to use it ?</h2>
<hr><p>This lesson aims at showing how to perform a calculation in
the
frame of the PAW method. </p>
<ul></ul>You will learn how to launch a PAW calculation and what are
the main
input
variables that govern convergency and numerical efficiency.<br>
It is supposed that you already know how to use ABINIT in the
norm-conserving pseudopotential case.<br>
<p>This lesson should take about 1.5 hour. </p>
<h5>Copyright (C) 2000-2014 ABINIT group (MT, FJ) <br>
This file is distributed under the terms of the GNU General Public
License, see ~ABINIT/COPYING or <a href="http://www.gnu.org/copyleft/gpl.txt">
http://www.gnu.org/copyleft/gpl.txt </a>. <br>
For the initials of contributors, see ~ABINIT/Infos/contributors . </h5>

<script type="text/javascript" src="list_internal_links.js"> </script>

<h3><b><b>Contents of lesson
PAW1 :</b></b></h3>
<ul>
<li><a href="lesson_paw1.html#0">0.</a> Summary of the PAW method</li>
<li><a href="lesson_paw1.html#1">1.</a> Using PAW with ABINIT</li>
<li><a href="lesson_paw1.html#2">2.</a> The convergence on plane-wave cut-off (<a href="../input_variables/varbas.html#ecut" target="kwimg">ecut</a>)</li>
<li><a href="lesson_paw1.html#3">3.</a> The convergence on double FFT grid cut-off (<a href="../input_variables/varpaw.html#pawecutdg" target="kwimg">pawecutdg</a>)&nbsp;</li>
<li><a href="lesson_paw1.html#4">4.</a> Plotting PAW contributions to the DOS</li>
<li><a href="lesson_paw1.html#5">5.</a> Testing the completeness of the PAW partial wave basis</li>
<li><a href="lesson_paw1.html#6">6.</a> Checking the validity of PAW results</li>
<li><a href="lesson_paw1.html#7">7.</a> Additional comments about PAW in ABINIT</li>
</ul>
<hr>

<h3><a name="0"></a><b><br>0. Summary of the PAW method</b></h3>
<p>The PAW (Projector Augmented-Wave) method has been
introduced by Peter Blchl in 1994. As he says, "<span style="font-style: italic;">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</span>".<br>
It is based on a linear and invertible transformation (the PAW
transformation) that connects the "true" wavefunctions <span style="font-weight: bold;">&#936;</span><sub style="font-weight: bold;">n</sub>
with
"auxiliary"
(or "pseudo") soft wavefunctions<span style="position: relative; top: -8pt; left: 8pt; font-weight: bold;">~</span><span style="font-weight: bold;">&#936;</span><sub style="font-weight: bold;">n</sub>
: </p>
<div style="margin-left: 80px;">|<span style="font-weight: bold; color: rgb(204, 0, 0);">&#936;</span><sub style="font-weight: bold; color: rgb(204, 0, 0);">n&gt;
&nbsp;=</sub><span style="position: relative; top: -8pt; left: 8pt; font-weight: bold; color: rgb(204, 0, 0);">~</span><span style="font-weight: bold; color: rgb(204, 0, 0);">|&#936;</span><sub style="font-weight: bold; color: rgb(204, 0, 0);">n</sub><span style="font-weight: bold; color: rgb(204, 0, 0);">&gt;+&nbsp;
&#931;&nbsp;(&#966;</span><sub style="font-weight: bold; color: rgb(204, 0, 0);">i&nbsp;
</sub><span style="font-weight: bold; color: rgb(204, 0, 0);">-</span><span style="position: relative; top: -5pt; left: 6pt; font-weight: bold; color: rgb(204, 0, 0);">~</span><span style="font-weight: bold; color: rgb(204, 0, 0);">&#966;</span><sub style="font-weight: bold; color: rgb(204, 0, 0);">i</sub><span style="font-weight: bold; color: rgb(204, 0, 0);">)
&lt;</span><span style="position: relative; top: -5pt; left: 6pt; font-weight: bold; color: rgb(204, 0, 0);">~</span><span style="font-weight: bold; color: rgb(204, 0, 0);">p</span><sub style="font-weight: bold; color: rgb(204, 0, 0);">i&nbsp;</sub><span style="font-weight: bold; color: rgb(204, 0, 0);"> |</span><span style="position: relative; top: -8pt; left: 8pt; font-weight: bold; color: rgb(204, 0, 0);">~</span><span style="font-weight: bold; color: rgb(204, 0, 0);">&#936;</span><sub style="font-weight: bold; color: rgb(204, 0, 0);">n </sub><span style="font-weight: bold; color: rgb(204, 0, 0);">&gt;</span>
</div><p><span style="font-style: italic;"></span>This
relation
is based on
the definition of atomic spheres (<span style="font-style: italic;">augmentation
regions</span>) of radius&nbsp; <span style="font-style: italic; font-weight: bold;">r</span><sub style="font-style: italic; font-weight: bold;">c</sub>,
around the atoms of the
system in which the <span style="font-style: italic;">partial
waves</span>&nbsp; <span style="position: relative; top: -5pt; left: 6pt;"></span><span style="font-weight: bold;">| &#966;</span><sub style="font-weight: bold;">i</sub><span style="font-weight: bold;">&gt;</span>
form a basis
of atomic wavefunctions; <span style="font-weight: bold;">|</span><span style="position: relative; top: -5pt; left: 6pt; font-weight: bold;">~</span><span style="font-weight: bold;">&#966;</span><sub style="font-weight: bold;">i</sub><span style="font-weight: bold;">&gt;</span>
are <span style="font-style: italic;">"pseudized" partial
waves</span> (obtained from <span style="font-weight: bold;">|
</span><span style="position: relative; top: -5pt; left: 6pt; font-weight: bold;"></span><span style="font-weight: bold;">&#966;</span><sub style="font-weight: bold;">i</sub><span style="font-weight: bold;">&gt;</span>), and <span style="position: relative; top: -5pt; left: 6pt; font-weight: bold;">~</span><span style="font-weight: bold;">p</span><sub style="font-weight: bold;">i</sub>
are dual functions
of
the&nbsp;<span style="font-weight: bold;">|</span><span style="position: relative; top: -5pt; left: 6pt; font-weight: bold;">~</span><span style="font-weight: bold;">&#966;</span><sub style="font-weight: bold;">i</sub><span style="font-weight: bold;">&gt;</span> called
projectors.<br>
It is therefore possible to write every quantity depending on&nbsp;<span style="font-weight: bold;">&#936;</span><sub style="font-weight: bold;">n</sub>
(density, energy, Hamiltonian) as a function of<span style="position: relative; top: -8pt; left: 8pt; font-weight: bold;">~</span><span style="font-weight: bold;">&#936;</span><sub style="font-weight: bold;">n</sub>
and to find<span style="position: relative; top: -8pt; left: 8pt; font-weight: bold;">~</span><span style="font-weight: bold;">&#936;</span><sub style="font-weight: bold;">n</sub>
by solving self-consistent equations.
</p><p>The PAW method has two main advantages:<br>
&nbsp; - From<span style="position: relative; top: -8pt; left: 8pt; font-weight: bold;">~</span><span style="font-weight: bold;">&#936;</span><sub style="font-weight: bold;">n</sub>,
it is always
possible to obtain the true&nbsp;"all electron"
wavefunction<span style="position: relative; top: -8pt; left: 8pt; font-weight: bold;"> </span><span style="font-weight: bold;">&#936;</span><sub style="font-weight: bold;">n</sub>.<br>
&nbsp; - The convergency is comparable
to an ultrasoft pseudopotential one.
</p><p>From a practical point of view (user's 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
(<span style="font-style: italic; font-weight: bold;">PAW dataset</span>)
that contains the&nbsp;&nbsp;<span style="font-weight: bold;"></span><span style="position: relative; top: -5pt; left: 6pt; font-weight: bold;"></span><span style="font-weight: bold;">&#966;</span><sub style="font-weight: bold;">i</sub><span style="font-weight: bold;">,</span><span style="font-weight: bold;"></span><span style="position: relative; top: -5pt; left: 6pt; font-weight: bold;">~</span><span style="font-weight: bold;">&#966;</span><sub style="font-weight: bold;">i</sub><span style="font-weight: bold;"> and</span> <span style="font-style: italic;"></span><span style="font-weight: bold;"></span><span style="position: relative; top: -5pt; left: 6pt; font-weight: bold;">~</span><span style="font-weight: bold;">p</span><sub style="font-weight: bold;">i</sub>
and that plays
the same role as a pseudopotential file.
</p><p>It is highly recommended to read the following papers to
understand
correctly the basic concepts of the PAW method:</p>
<span style="font-weight: bold;">[1]</span> "<span style="font-style: italic;">Projector augmented-wave
method</span>", P.E.
Blochl, Physical Review B <span style="font-weight: bold;">50</span>,
17953
(1994)<br>
<span style="font-weight: bold;">[2]</span> "<span style="font-style: italic;">From ultrasoft pseudopotentials
to
the projector augmented-wave method</span>", G. Kresse and D.
Joubert, Physical Review
B <span style="font-weight: bold;">59</span>, 1758
(1999)<br>
<p>The implementation of the PAW method in <span style="font-weight: bold;">ABINIT</span> is detailed
in the following paper, describing specific notations and formulations:</p>
<p style="color: rgb(153, 0, 0);"><span style="font-weight: bold;">[3]</span>
"<span style="font-style: italic;">Implementation of the
projector augmented-wave method in the ABINIT code: Application to the
study of iron under pressure</span>", M. Torrent, F. Jollet, F.
Bottin, G. Zerah, and X. Gonze, Computational Materials Science <span style="font-weight: bold;">42</span>, 337
(2008)&nbsp;&nbsp;</p>

<h3><hr> <a name="1"></a>&nbsp;<b><b><br>
1.&nbsp; Using PAW with ABINIT</b></b></h3>
<p><i>Before continuing, you might
consider to work in a different
subdirectory as for the other lessons. Why not "Work_paw1" ?
</i><i>In what follows, the name of files are
mentioned as if
you were in this subdirectory.<br>
All the input files can be found in the </i><i><span style="color: rgb(0, 102, 0);">~abinit/tests/tutorial/Input</span>
directory.</i></p><p><i>You can compare your results with reference
output files located in </i><i><span style="color: rgb(0, 102, 0);">~abinit/tests/tutorial/Refs</span>
and </i><i><span style="color: rgb(0, 102, 0);">~abinit/tests/tutorial/Refs/tpaw1_addons
</span></i><i>directories (for the present tutorial they are named
tpaw1_*.out).</i></p>
<p>The input file&nbsp;<span style="color: rgb(0, 102, 0);">tpaw1_1.in</span>
is an example of a file
that contains data for computing the total energy for&nbsp;<span style="font-weight: bold; color: rgb(153, 0, 0);">diamond</span>
at the experimental volume (within the LDA exchange-correlation
functional).
You might use the file&nbsp;<span style="color: rgb(0, 102, 0);">tpaw1_1.files</span>
(with a standard
norm-conserving pseudopotential)
as a "files" file, and get the corresponding output file
(it is available as <span style="color: rgb(0, 102, 0);">../Refs/tpaw1_1.out</span>).
<br>
Copy the files&nbsp;<span style="color: rgb(0, 102, 0); font-weight: bold;">tpaw1_1.in</span>
and&nbsp;<span style="color: rgb(0, 102, 0); font-weight: bold;">tpaw1_1.files</span>
in your work
directory, and run ABINIT:
</p><pre style="margin-left: 40px;"><span style="color: rgb(153, 0, 0);">abinit &lt; tpaw1_1.files &gt; tmp-log</span>
</pre>In the meantime, you can read the input file and see that there
is no
PAW input
variable.<br>
ABINIT should run very quickly...<br>
<br>Now, open the&nbsp;<span style="color: rgb(0, 102, 0);">tpaw1_1.files</span>
and modify the last line; replace the&nbsp;<span style="font-weight: bold; color: rgb(0, 102, 0);">6c.pspnc</span>
file by&nbsp;<span style="font-weight: bold; color: rgb(0, 102, 0);">6c.lda.atompaw</span>.
<p>Run ABINIT again:</p>
<pre style="margin-left: 40px;"><span style="color: rgb(153, 0, 0);">abinit &lt; tpaw1_1.files &gt; tmp-log</span></pre><p><span style="text-decoration: underline;">Your
run should stop before end !</span> &nbsp;The input file
is missing a mandatory argument:&nbsp;<a href="../input_variables/varpaw.html#pawecutdg" target="kwimg">pawecutdg</a><span style="font-weight: bold;"></span>
!!</p>
<p>Add the line "<span style="font-weight: bold;">pawecutdg
50.</span>" in the &nbsp;<span style="color: rgb(0, 102, 0);">tpaw1_1.in&nbsp;</span>file
and run ABINIT again.<br>
Now ABINIT runs to the end.<span style="font-style: italic;"></span></p><p><span style="font-style: italic;">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 </span><a style="font-style: italic;" href="../input_variables/varbas.html#ecut" target="kwimg" onclick="return (false);">ecut</a><span style="font-style: italic;">,
PAW requires more computational
resources:</span><span style="font-style: italic;">&nbsp;&nbsp;&nbsp; -
the "on-site"
contributions have to be computed,</span><span style="font-style: italic;">&nbsp;&nbsp;&nbsp;
<br>
-&nbsp;the nonlocal contribution of the PAW dataset uses 2 projectors
per angular momentum, while the nonlocal contribution of the present
norm-conserving pseudopotential&nbsp; uses only one.</span><span style="font-style: italic;"><br>
However,
as the plane wave cut-off energy required by PAW is much
smaller&nbsp;than the cut-off needed for the norm-conserving
pseudopotential (see next section), a PAW calculation will actually
require less CPU time.</span></p>
<p>Let's open the output file and have a look inside (be
careful, it is the last output file of the tpaw1_1 series).<br>
Compared to an output file for a norm-conserving pseudopotential run,
an
output file for PAW contains
the following specific topics:</p>
<p><span style="font-weight: bold; text-decoration: underline;">At
the beginning of the file:</span>
</p><div style="margin-left: 40px;">- Some specific default
PAW input variables (<a href="../input_variables/varpaw.html#ngfftdg" target="kwimg">ngfftdg</a>,
<a href="../input_variables/varpaw.html#pawecutdg" target="kwimg">pawecutdg</a>,
and <a href="../input_variables/vardev.html#useylm" target="kwimg">useylm</a>)
are mentionned in the
section:
</div><p style="color: rgb(0, 153, 0); margin-left: 80px;"><font size="-1">-outvars:
echo values
of preprocessed input variables --------</font></p>
<p style="margin-left: 40px;">- The use of two FFT grids,
mentioned as:</p>
<p style="font-family: monospace; margin-left: 80px; color: rgb(0, 153, 0);">&nbsp;Coarse
grid specifications (used for wave-functions):<br>
<br>&nbsp;getcut: wavevector=&nbsp; 0.0000&nbsp;
0.0000&nbsp; 0.0000&nbsp; ngfft=&nbsp; 18&nbsp;
18&nbsp; 18<br>
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;
ecut(hartree)=&nbsp;&nbsp;&nbsp;&nbsp;
15.000&nbsp;&nbsp; =&gt; boxcut(ratio)=&nbsp;&nbsp;
2.17276<br>
<br>&nbsp;Fine grid specifications (used for densities):<br>
<br>&nbsp;getcut: wavevector=&nbsp; 0.0000&nbsp;
0.0000&nbsp; 0.0000&nbsp; ngfft=&nbsp; 32&nbsp;
32&nbsp; 32<br>
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;
ecut(hartree)=&nbsp;&nbsp;&nbsp;&nbsp;
50.000&nbsp;&nbsp; =&gt; boxcut(ratio)=&nbsp;&nbsp;
2.10918
</p><div style="margin-left: 40px;">- A specific description
of the <span style="font-style: italic;">PAW dataset</span>
(you might
follow
the tutorial PAW2, devoted to the building of the PAW atomic data, for
a complete understanding of the file):
</div><div style="margin-left: 80px;"><big><font style="font-family: monospace; color: rgb(0, 153, 0);" size="-1"><big><br>
&nbsp;Pseudopotential
format is: paw4<br>
&nbsp;basis_size (lnmax)=&nbsp; 4 (lmn_size=&nbsp; 8),
orbitals=&nbsp;&nbsp; 0&nbsp;&nbsp;
0&nbsp;&nbsp; 1&nbsp;&nbsp; 1<br>
&nbsp;Spheres core radius: rc_sph= 1.50000000<br>
&nbsp;4 radial meshes are used:<br>
&nbsp; - mesh 1: r(i)=AA*[exp(BB*(i-1))-1], size= 505 , AA=
0.21824E-02 BB= 0.13095E-01<br>
&nbsp; - mesh 2: r(i)=AA*[exp(BB*(i-1))-1], size= 500 , AA=
0.21824E-02 BB= 0.13095E-01<br>
&nbsp; - mesh 3: r(i)=AA*[exp(BB*(i-1))-1], size= 530 , AA=
0.21824E-02 BB= 0.13095E-01<br>
&nbsp; - mesh 4: r(i)=AA*[exp(BB*(i-1))-1], size= 644 , AA=
0.21824E-02 BB= 0.13095E-01<br>
&nbsp;Shapefunction is SIN type:
shapef(r)=[sin(pi*r/rshp)/(pi*r/rshp)]**2<br>
&nbsp;Radius for shape functions = sphere core radius<br>
&nbsp;Radial grid used for partial waves is grid 1<br>
&nbsp;Radial grid used for projectors is grid 2<br>
&nbsp;Radial grid used for (t)core density is grid 3<br>
&nbsp;Radial grid used for Vloc is grid 4<br>
&nbsp;Radial grid used for pseudo valence density is grid 4</big></font></big>
</div><p><span style="font-weight: bold; text-decoration: underline;">After
the SCF cycle section:</span>
</p><div style="margin-left: 40px;">- The value of the
integrated compensation charge evaluated by two different numerical methodologies
(remember: PAW atomic dataset are not norm-conserving pseudopotentials); it is
given&nbsp;calculated in the augmentation regions (PAW spheres) on
the
"spherical" grid and also in the whole simulation cell on the "FFT"
grid. A discussion on these two values will be done in a
forthcoming section.
</div><div style="margin-left: 80px;"><big><font style="font-family: monospace; color: rgb(0, 153, 0);" size="-1"><big>&nbsp;PAW
TEST:<br>
&nbsp;==== Compensation charge inside spheres ============<br>
&nbsp;The following values must be close to each other ...<br>
&nbsp;Compensation charge over spherical meshes
=&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; 0.413178580356274<br>
&nbsp;Compensation charge over fine fft
grid&nbsp;&nbsp;&nbsp;
=&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; 0.413177280314290
</big></font></big></div><div style="margin-left: 40px;"><br>- &nbsp;
Information concerning the non-local term
(<span style="font-style: italic;">pseudopotential
strength</span> <span style="font-style: italic;">Dij</span>
) and the spherical density matrix (<span style="font-style: italic;">augmentation
wave
occupancies Rhoij</span>)
</div><p style="margin-left: 80px; font-family: monospace; color: rgb(0, 153, 0);"><big><font size="-1"><big><small>&nbsp;==== Results
concerning PAW augmentation regions ====<br>
&nbsp;<br>
&nbsp;Total pseudopotential strength Dij (hartree):<br>
&nbsp;Atom #&nbsp; 1<br>
&nbsp;&nbsp; ...<br>
&nbsp;Atom #&nbsp; 2<br>
&nbsp;&nbsp; ...</small>
</big></font></big></p><p style="margin-left: 80px;"><big style="color: rgb(0, 153, 0);"><font style="font-family: monospace;" size="-1"><big>Augmentation
waves occupancies Rhoij:<br>
&nbsp;Atom #&nbsp; 1<br>
&nbsp; ...<br>
&nbsp;Atom #&nbsp; 2<br>
&nbsp; ...</big></font></big>
</p><p><span style="font-weight: bold; text-decoration: underline;">At
the end of the file:</span>
</p><div style="margin-left: 40px;">- The decomposition of
the total energy both by direct calculation and
double counting calculation:
</div><div style="margin-left: 80px;"><big><font style="font-family: monospace; color: rgb(0, 153, 0);" size="-1"><big>--------------------------------------------------------------------------------<br>
&nbsp;Components of total free energy (in Hartree) :<br>
<br>&nbsp;&nbsp;&nbsp; Kinetic energy&nbsp; =&nbsp;
6.40164318808980E+00<br>
&nbsp;&nbsp;&nbsp; Hartree energy&nbsp; =&nbsp;
9.63456708252837E-01<br>
&nbsp;&nbsp;&nbsp; XC
energy&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; =
-3.53223656186138E+00<br>
&nbsp;&nbsp;&nbsp; Ewald
energy&nbsp;&nbsp;&nbsp; = -1.27864121210521E+01<br>
&nbsp;&nbsp;&nbsp; PspCore energy&nbsp; =&nbsp;
5.41017918797015E-01<br>
&nbsp;&nbsp;&nbsp; Loc. psp. energy= -5.27003595856857E+00<br>
&nbsp;&nbsp;&nbsp; Spherical terms =&nbsp;
2.15689044331394E+00<br>
&nbsp;&nbsp;&nbsp; &gt;&gt;&gt;&gt;&gt;
Internal E= -1.15256763830284E+01<br>
<br>&nbsp;"Double-counting" decomposition of free energy:<br>
&nbsp;&nbsp;&nbsp; Band
energy&nbsp;&nbsp;&nbsp;&nbsp; =&nbsp;
6.87331579398577E-01<br>
&nbsp;&nbsp;&nbsp; Ewald
energy&nbsp;&nbsp;&nbsp; = -1.27864121210521E+01<br>
&nbsp;&nbsp;&nbsp; PspCore energy&nbsp; =&nbsp;
5.41017918797015E-01<br>
&nbsp;&nbsp;&nbsp; Dble-C XC-energy=&nbsp;
1.22161340385476E-01<br>
&nbsp;&nbsp;&nbsp; Spherical terms = -8.97688814082645E-02<br>
&nbsp;&nbsp;&nbsp; &gt;&gt;&gt;&gt;&gt;
Internal E= -1.15256701638793E+01<br>
<br>&nbsp; &gt;Total energy in
eV&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;
= -3.13629604304723E+02<br>
&nbsp;&nbsp;&gt;Total DC energy in
eV&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;
= -3.13629435073068E+02</big></font></big>
</div><p>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)<i> energies</i>
between different PAW potentials or
different codes.</p>
<h3><hr> <a name="2"></a>&nbsp;<b><b><br>
2.&nbsp; The convergence on plane-wave cut-off (<a target="kwimg" href="../input_variables/varbas.html#ecut">ecut</a>)</b></b></h3>
As in the usual case, the critical convergence parameter is the cut-off
defining the size of the plane-wave basis...
<p><b>1.a</b><b> Computing the convergence in
ecut for&nbsp;<span style="color: rgb(153, 0, 0);">diamond</span>
in
the norm-conserving case</b></p>
The input file&nbsp;<span style="color: rgb(0, 102, 0); font-weight: bold;">tpaw1_2.in</span>
contains data for computing the convergence in ecut for&nbsp;<span style="font-weight: bold; color: rgb(153, 0, 0);">diamond</span>
(at experimental volume).
There are 9 datasets, for which&nbsp;<a href="../input_variables/varbas.html#ecut" target="kwimg">ecut</a>
increases from 8 Ha to 24 Ha
by step of 2 Ha.<br>
You might use the&nbsp;<span style="color: rgb(0, 102, 0); font-weight: bold;">tpaw1_2.files</span>
file (with a standard
norm-conserving pseudopotential), and run ABINIT:<br>
<pre style="margin-left: 40px;"><span style="color: rgb(153, 0, 0);">abinit &lt; tpaw1_2.files &gt; tmp-log</span>
</pre>You
should obtain the values (output file <span style="color: rgb(0, 102, 0);">tpaw1_2.out</span>) :
<p style="margin-left: 80px;"><big><font style="font-family: monospace;" size="-1"><big><span style="color: rgb(0, 153, 0);">&nbsp;&nbsp;&nbsp;
etotal1&nbsp; -1.1628880677E+01</span><br>
<span style="color: rgb(0, 153, 0);">&nbsp;&nbsp;&nbsp;
etotal2&nbsp; -1.1828052470E+01</span><br>
<span style="color: rgb(0, 153, 0);">&nbsp;&nbsp;&nbsp;
etotal3&nbsp; -1.1921833945E+01</span><br>
<span style="color: rgb(0, 153, 0);">&nbsp;&nbsp;&nbsp;
etotal4&nbsp; -1.1976374633E+01</span><br>
<span style="color: rgb(0, 153, 0);">&nbsp;&nbsp;&nbsp;
etotal5&nbsp; -1.2017601960E+01</span><br>
<span style="color: rgb(0, 153, 0);">&nbsp;&nbsp;&nbsp;
etotal6&nbsp; -1.2046855404E+01</span><br>
<span style="color: rgb(0, 153, 0);">&nbsp;&nbsp;&nbsp;
etotal7&nbsp; -1.2062173253E+01</span><br>
<span style="color: rgb(0, 153, 0);">&nbsp;&nbsp;&nbsp;
etotal8&nbsp; -1.2069642342E+01</span><br>
<span style="color: rgb(0, 153, 0);">&nbsp;&nbsp;&nbsp;
etotal9&nbsp; -1.2073328672E+01</span></big></font></big>
</p><p>You can check that the
<span style="font-weight: bold; font-style: italic;">
etotal</span>
convergence (at the 1 mHartree level) is not achieved for&nbsp;<span style="font-style: italic; font-weight: bold;">ecut</span>=24
Hartree.</p>
<p><b>1.b</b><b> Computing the convergence in
ecut for <span style="color: rgb(153, 0, 0);">diamond</span>
in
the PAW case</b></p>
<p>Use the same input files as in section 1.a.<br>
Again, modify the last line of&nbsp;<span style="color: rgb(0, 102, 0);">tpaw1_2.files</span>,
replacing the&nbsp;<span style="font-weight: bold; color: rgb(0, 102, 0);">6c.pspnc</span>
file by <span style="font-weight: bold; color: rgb(0, 102, 0);">6c.lda.atompaw</span>.<br>
Run ABINIT again and open the output file (it should be <span style="color: rgb(0, 102, 0);">tpaw1_2.outA</span>)<br>
You should obtain the values:</p>
<div style="margin-left: 80px;"><big><font style="color: rgb(0, 153, 0);" size="-1"><big><span style="font-family: monospace;">&nbsp;&nbsp;&nbsp;
etotal1&nbsp; -1.1474828697E+01<br>
&nbsp;&nbsp;&nbsp; etotal2&nbsp; -1.1518675625E+01<br>
&nbsp;&nbsp;&nbsp; etotal3&nbsp; -1.1524581240E+01<br>
&nbsp;&nbsp;&nbsp; etotal4&nbsp; -1.1525548758E+01<br>
&nbsp;&nbsp;&nbsp; etotal5&nbsp; -1.1525741818E+01<br>
&nbsp;&nbsp;&nbsp; etotal6&nbsp; -1.1525865084E+01<br>
&nbsp;&nbsp;&nbsp; etotal7&nbsp; -1.1525926864E+01<br>
&nbsp;&nbsp;&nbsp; etotal8&nbsp; -1.1525947400E+01<br>
&nbsp;&nbsp;&nbsp; etotal9&nbsp; -1.1525954817E+01</span></big></font><font size="-1"><big><span style="font-family: monospace;"></span></big></font></big>
</div><p>You can check that:<br>
The <span style="font-weight: bold; font-style: italic;">etotal</span>
convergence (at 1 mHartree) is achieved for 12&lt;=<span style="font-style: italic; font-weight: bold;">ecut</span>&lt;=14
Hartree (etotal4 is within 1 mHartree of the final value); <br>The <span style="font-weight: bold; font-style: italic;">
etotal</span>
convergence (at 0.1 mHartree) is achieved for&nbsp;16&lt;=<span style="font-style: italic; font-weight: bold;">ecut</span>&lt;=18
Hartree (etotal6 is within 0.1 mHartree of the final value).</p>
<p>So with the same input, a PAW calculation for
diamond needs a lower cutoff, compared to a norm-conserving
pseudopotential
calculation. </p>

<hr>

<h3><a name="3">&nbsp;</a><b><b><br>
3. The convergence on the double grid FFT cut-off (<a href="../input_variables/varpaw.html#pawecutdg" target="kwimg">pawecutdg</a>)
</b></b></h3><p>In a<span style="font-style: italic;">
norm-conserving pseudopotential</span> calculation, the (plane wave) density
grid is (at least)
twice
bigger than the wavefunctions grid, in each direction. In
a <span style="font-style: italic;">PAW</span>
calculation, the (plane wave) density grid is tunable thanks to the
input variable <a href="../input_variables/varpaw.html#pawecutdg" target="kwimg">pawecutdg</a>
(PAW: ECUT for Double Grid). This is needed because of the mapping of
&nbsp;objects (densities, potentials) located in
the augmentation regions (PAW spheres) onto the global FFT grid.<br>
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&nbsp; <a href="../input_variables/varpaw.html#pawecutdg" target="kwimg">pawecutdg</a>. An
alternative
is to use directly the input variable <a href="../input_variables/varpaw.html#ngfftdg" target="kwimg">ngfftdg</a>.
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).</p>
<p>Use now the input file&nbsp;<span style="color: rgb(0, 102, 0); font-weight: bold;">tpaw1_3.in</span>
and the associated <span style="color: rgb(0, 102, 0); font-weight: bold;">tpaw1_3.files</span>
file.<br>
The only difference with the <span style="color: rgb(0, 102, 0);">tpaw1_2.in</span>
file is that <a href="../input_variables/varbas.html#ecut" target="kwimg">ecut</a>
is fixed to 12 Ha,
while <a href="../input_variables/varpaw.html#pawecutdg" target="kwimg">pawecutdg</a>
runs from 12 to 39 Ha.
</p><p>Launch ABINIT with these files; you should obtain the values
(file <span style="color: rgb(0, 102, 0);">tpaw1_3.out</span>):
</p><div style="margin-left: 80px;"><big><font style="font-family: monospace; color: rgb(0, 153, 0);" size="-1"><big>&nbsp;&nbsp;&nbsp;
etotal1&nbsp; -1.1524629595E+01<br>
&nbsp;&nbsp;&nbsp; etotal2&nbsp; -1.1524595840E+01<br>
&nbsp;&nbsp;&nbsp; etotal3&nbsp; -1.1524585370E+01<br>
&nbsp;&nbsp;&nbsp; etotal4&nbsp; -1.1524580630E+01<br>
&nbsp;&nbsp;&nbsp; etotal5&nbsp; -1.1524584720E+01<br>
&nbsp;&nbsp;&nbsp; etotal6&nbsp; -1.1524583573E+01<br>
&nbsp;&nbsp;&nbsp; etotal7&nbsp; -1.1524582786E+01<br>
&nbsp;&nbsp;&nbsp; etotal8&nbsp; -1.1524582633E+01<br>
&nbsp;&nbsp;&nbsp; etotal9&nbsp; -1.1524582213E+01<br>
&nbsp;&nbsp;&nbsp; etotal10 -1.1524582316E+01</big></font></big>
</div><p>We see that the variation of the energy wit respect to this parameter is well below
the 1 mHa level. In principle, it should be sufficient to choose <span style="font-weight: bold;">pawecutdg</span>=12 Ha in order to obtain an energy change lower than 1 <span style="font-style: italic;">mHa</span>. In practice, it is better to keep a security margin. Here, for <span style="font-weight: bold;">pawecutdg</span>=24 Ha
(5th
dataset), the energy change is lower than 0.001 <span style="font-style: italic;">mHa</span><code></code>: this choice will be more than enough.</p>
<p>Note the steps
in the convergency. They are due to the sudden (integer) changes in the grid size
(see the output values for&nbsp; <a href="../input_variables/varpaw.html#ngfftdg" target="kwimg">ngfftdg</a>) which do not occur
for each increase of&nbsp;<a href="../input_variables/varpaw.html#pawecutdg" target="kwimg">pawecutdg</a>. To avoid troubles due to these steps, it is better to choose a value of pawecutdg slightly higher.<br>
</p>

<p>The&nbsp;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:</p>
<div style="margin-left: 80px;"><big><font style="font-family: monospace; color: rgb(0, 153, 0);" size="-1"><big>&nbsp;PAW
TEST:<br>
&nbsp;==== Compensation charge inside spheres ============</big></font></big><big><font style="font-family: monospace; color: rgb(0, 153, 0);" size="-1"><big><br>&nbsp;The
following values must be close...</big></font></big><br><big><font style="font-family: monospace; color: rgb(0, 153, 0);" size="-1"><big>&nbsp;Compensation
charge over spherical meshes
=&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; 0.409392121335747</big></font></big><br><big><font style="font-family: monospace; color: rgb(0, 153, 0);" size="-1"><big>&nbsp;</big></font></big><big><font style="font-family: monospace; color: rgb(0, 153, 0);" size="-1"><big>Compensation
charge over fine fft
grid&nbsp;&nbsp;&nbsp;
=&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; 0.409392418241149</big></font></big><big><font style="font-family: monospace; color: rgb(0, 153, 0);" size="-1"><big>
</big></font></big></div><p>The two values of the integrated
compensation charge
density
must be close to each other.<br>
<span style="font-style: italic;">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).</span></p>
<span style="text-decoration: underline;">Additional test:</span><br>
We want now to check the convergence in <a href="../input_variables/varbas.html#ecut" target="kwimg">ecut</a>
with a fixed value
of 24 Ha for <a href="../input_variables/varpaw.html#pawecutdg" target="kwimg">pawecutdg</a>.
Modify the file<span style="color: rgb(0, 102, 0);">
tpaw1_2.in</span>, setting pawecutdg to 24 Ha, and launch ABINIT
again...<br>
You should obtain the values :<br>
<small style="color: rgb(0, 153, 0);"></small><div style="margin-left: 80px;"><big><small style="color: rgb(0, 153, 0);"><span style="font-family: monospace;"><span style="font-family: mon;">&nbsp;&nbsp;&nbsp;
etotal1&nbsp; -1.1474831477E+01<br>
&nbsp;&nbsp;&nbsp; etotal2&nbsp; -1.1518678975E+01<br>
&nbsp;&nbsp;&nbsp; etotal3&nbsp; -1.1524584720E+01<br>
&nbsp;&nbsp;&nbsp; etotal4&nbsp; -1.1525552267E+01<br>
&nbsp;&nbsp;&nbsp; etotal5&nbsp; -1.1525745330E+01<br>
&nbsp;&nbsp;&nbsp; etotal6&nbsp; -1.1525868591E+01<br>
&nbsp;&nbsp;&nbsp; etotal7&nbsp; -1.1525930368E+01<br>
&nbsp;&nbsp;&nbsp; etotal8&nbsp; -1.1525950904E+01<br>
&nbsp;&nbsp;&nbsp; etotal9&nbsp; -1.1525958319E+01</span></span></small></big>
</div><br>You can check again that:<br>
The <span style="font-weight: bold; font-style: italic;">etotal</span>
convergence (at the 1 mHartree level) is achieved for 12&lt;=<span style="font-style: italic; font-weight: bold;">ecut</span>&lt;=14
Hartree ;
<br>
The
<span style="font-weight: bold; font-style: italic;">
etotal</span>
convergence (at the 0.1 mHartree level) is achieved for&nbsp;16&lt;=<span style="font-style: italic; font-weight: bold;">ecut</span>&lt;=18
Hartree.
<br>
<span style="text-decoration: underline;"><br>Note 1:</span><br>
Associated with the input variable <a href="../input_variables/varpaw.html#pawecutdg" target="kwimg">pawecutdg</a>
is the input variable <a href="../input_variables/varpaw.html#ngfftdg" target="kwimg">ngfftdg</a>:
it
gives the size of the FFT grid associated with <a href="../input_variables/varpaw.html#pawecutdg" target="kwimg">pawecutdg</a>.
Note that <a href="../input_variables/varpaw.html#pawecutdg" target="kwimg">pawecutdg</a>
is only useful to define the FFT grid for the density in a convenient
way. You can therefore tune directly <a href="../input_variables/varpaw.html#ngfftdg" target="kwimg">ngfftdg</a>
to define the
size of the FFT
grid for the density.<br>
<br><span style="text-decoration: underline;">Note 2:</span><br>
Although <a href="../input_variables/varpaw.html#pawecutdg" target="kwimg">pawecutdg</a>
should always be checked, in practice, a common use it to put it bigger than&nbsp;<a href="../input_variables/varbas.html#ecut" target="kwimg">ecut</a>
and keep it constant during
all calculations. Increasing&nbsp;<span style="text-decoration: underline;"></span><a href="../input_variables/varpaw.html#pawecutdg" target="kwimg">pawecutdg</a>
slightly changes the CPU execution time, but above all it is
memory-consuming.<br>
Note that, if <a href="../input_variables/varbas.html#ecut" target="kwimg">ecut</a> is already high,
there is no need for a high&nbsp;<a href="../input_variables/varpaw.html#pawecutdg" target="kwimg">pawecutdg</a>.<span style="text-decoration: underline;"></span><br>
Last warning: when testing <a href="../input_variables/varbas.html#ecut" target="kwimg">ecut</a>
convergency, <span style="text-decoration: underline;"></span><a href="../input_variables/varpaw.html#pawecutdg" target="kwimg">pawecutdg</a>
has to remain constant to obtain consistent results.<br>
<br><hr><h3><p><b><a name="4"><br>4.</a> Plotting PAW contributions to
the Density of States (DOS)</b></p>
</h3><p>We use now the input file&nbsp;<span style="color: rgb(0, 102, 0); font-weight: bold;">tpaw1_4.in</span>
and the associated <span style="color: rgb(0, 102, 0); font-weight: bold;">tpaw1_4.files</span>
file.<br>
ABINIT is now asked to compute the Density Of State (DOS) (see
the&nbsp; <a href="../input_variables/varfil.html#prtdos" target="kwimg">prtdos</a>&nbsp;keyword in the
input file). Also note that more k-points are used in order to increase
the accuracy of the DOS. <a href="../input_variables/varbas.html#ecut" target="kwimg">ecut</a>
is set to 12 Ha,
while <a href="../input_variables/varpaw.html#pawecutdg" target="kwimg">pawecutdg</a> is 24 Ha.
</p><p>Launch ABINIT with these files; you should obtain the <span style="color: rgb(0, 102, 0);">tpaw1_4.out</span> and
the DOS file (<span style="color: rgb(0, 153, 0); font-weight: bold;">tpaw1_4o_DOS</span>):</p>
<pre style="margin-left: 40px;"><span style="color: rgb(153, 0, 0);">abinit &lt; tpaw1_4.files &gt; tmp-log</span>
</pre><p>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,<span style="text-decoration: line-through;"></span><span style="text-decoration: line-through;"></span>
launch:</p>
<pre style="margin-left: 40px;"><span style="color: rgb(153, 0, 0);">xmgrace -block tpaw1_4o_DOS -bxy 1:2</span></pre><p>At
this stage, you have the usual plot for a DOS; nothing
specific to PAW.</p>
<p>Now, edit the&nbsp;<span style="color: rgb(0, 102, 0); font-weight: bold;">tpaw1_4.in</span>
file,&nbsp;comment the "<span style="font-family: monospace; color: rgb(0, 153, 0);"></span><a href="../input_variables/varfil.html#prtdos" target="kwimg">prtdos</a>
1", and uncomment (or add):</p>
<div style="margin-left: 80px;"><span style="font-family: monospace; color: rgb(0, 153, 0);">prtdos
3&nbsp; pawprtdos 1</span>
<span style="font-family: monospace; color: rgb(0, 153, 0);">natsph
1 iatsph 1 ratsph 1.5</span>
</div><p>The "&nbsp;<a href="../input_variables/varfil.html#prtdos" target="kwimg">prtdos</a>
3<span style="font-family: monospace;"></span>"
statement now requires the output of the projected DOS; "<a href="../input_variables/vargs.html#natsph" target="kwimg">natsph</a>
1&nbsp; <a href="../input_variables/vargs.html#iatsph" target="kwimg">iatsph</a> 1&nbsp; <a href="../input_variables/vargs.html#ratsph" target="kwimg">ratsph</a>
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).<br>
The "<a href="../input_variables/varpaw.html#pawprtdos" target="kwimg">pawprtdos</a> 1" is&nbsp;specific
to PAW. With this option, ABINIT is asked to compute all the
contributions to the projected DOS.<br>
Let's remember that:</p>
<div style="margin-left: 80px;">|<span style="font-weight: bold; color: rgb(204, 0, 0);">&#936;</span><sub style="font-weight: bold; color: rgb(204, 0, 0);">n&gt;
&nbsp;=</sub><span style="position: relative; top: -8pt; left: 8pt; font-weight: bold; color: rgb(204, 0, 0);">~</span><span style="font-weight: bold; color: rgb(204, 0, 0);">|&#936;</span><sub style="font-weight: bold; color: rgb(204, 0, 0);">n</sub><span style="font-weight: bold; color: rgb(204, 0, 0);">&gt;+&nbsp;
&#931;&nbsp;(&#966;</span><sub style="font-weight: bold; color: rgb(204, 0, 0);">i&nbsp;
</sub><span style="font-weight: bold; color: rgb(204, 0, 0);">-</span><span style="position: relative; top: -5pt; left: 6pt; font-weight: bold; color: rgb(204, 0, 0);">~</span><span style="font-weight: bold; color: rgb(204, 0, 0);">&#966;</span><sub style="font-weight: bold; color: rgb(204, 0, 0);">i</sub><span style="font-weight: bold; color: rgb(204, 0, 0);">)
&lt;</span><span style="position: relative; top: -5pt; left: 6pt; font-weight: bold; color: rgb(204, 0, 0);">~</span><span style="font-weight: bold; color: rgb(204, 0, 0);">p</span><sub style="font-weight: bold; color: rgb(204, 0, 0);">i&nbsp;</sub><span style="font-weight: bold; color: rgb(204, 0, 0);"> |</span><span style="position: relative; top: -8pt; left: 8pt; font-weight: bold; color: rgb(204, 0, 0);">~</span><span style="font-weight: bold; color: rgb(204, 0, 0);">&#936;</span><sub style="font-weight: bold; color: rgb(204, 0, 0);">n </sub><span style="font-weight: bold; color: rgb(204, 0, 0);">&gt;</span>
</div><p>Within PAW, the total projected DOS has 3 contributions:<br>
1- the smooth plane-waves contribution (from<span style="font-weight: bold; color: black;"></span><span style="position: relative; top: -8pt; left: 8pt; font-weight: bold; color: black;">~</span><span style="font-weight: bold; color: black;">|&#936;</span><sub style="font-weight: bold; color: black;">n</sub><span style="font-weight: bold; color: black;">&gt;</span><span style="color: black;">)</span><span style="font-weight: bold; color: black;"></span><span style="position: relative; top: -5pt; left: 6pt; font-weight: bold; color: black;"></span><span style="font-weight: bold; color: black;"></span><span style="position: relative; top: -5pt; left: 6pt; font-weight: bold; color: black;"></span><span style="font-weight: bold; color: black;"></span><span style="position: relative; top: -8pt; left: 8pt; font-weight: bold; color: black;"></span><span style="color: black;"><br>
2- the all-electron on-site contribution (from&nbsp;</span><span style="font-weight: bold; color: black;">&#966;</span><sub style="font-weight: bold; color: black;">i&nbsp;&nbsp;</sub><span style="font-weight: bold; color: black;"></span><span style="position: relative; top: -5pt; left: 6pt; font-weight: bold; color: black;"></span><span style="font-weight: bold; color: black;">&lt;</span><span style="position: relative; top: -5pt; left: 6pt; font-weight: bold; color: black;">~</span><span style="font-weight: bold; color: black;">p</span><sub style="font-weight: bold; color: black;">i&nbsp;</sub><span style="font-weight: bold;"> |</span><span style="position: relative; top: -8pt; left: 8pt; font-weight: bold;">~</span><span style="font-weight: bold;">&#936;</span><sub style="font-weight: bold;">n
</sub><span style="font-weight: bold;">&gt;</span>)<br>
3- the pseudo on-site contribution (from<span style="font-weight: bold;"></span><span style="position: relative; top: -8pt; left: 8pt; font-weight: bold;"></span><span style="font-weight: bold;"></span><span style="font-weight: bold;"></span><span style="font-weight: bold;"></span><span style="position: relative; top: -5pt; left: 6pt; font-weight: bold;">~</span><span style="font-weight: bold;">&#966;</span><sub style="font-weight: bold;">i</sub><span style="font-weight: bold;">
&lt;</span><span style="position: relative; top: -5pt; left: 6pt; font-weight: bold;">~</span><span style="font-weight: bold;">p</span><sub style="font-weight: bold;">i&nbsp;</sub><span style="font-weight: bold;"> |</span><span style="position: relative; top: -8pt; left: 8pt; font-weight: bold;">~</span><span style="font-weight: bold;">&#936;</span><sub style="font-weight: bold;">n
</sub><span style="font-weight: bold;">&gt;</span>).
</p>Launch ABINIT again (with the modified input file). You get a new
DOS
file, named <span style="color: rgb(0, 153, 0); font-weight: bold;">tpaw1_4o_DOS_AT0001</span>.<br>
You
can edit it and look inside; it contains the 3 PAW contributions
(mentioned above) for each angular momentum. In the diamond case, only
<span style="font-style: italic;">l=0</span> and <span style="font-style: italic;">l=1</span> momenta are
treated.<br>
Now, plot the file, using the 7th, 12th and 17th columns with respect to
the 2nd one; it plots the 3 PAW contributions for <span style="font-weight: bold; font-style: italic;">l=0</span>
(the total DOS is the sum of the three contributions).<br>
If you use the "xmgrace" tool,<span style="text-decoration: line-through;"></span> launch:
<pre style="margin-left: 40px;"><span style="color: rgb(153, 0, 0);">xmgrace -block tpaw1_4o_DOS_AT0001 -bxy 1:7 -bxy 1:12 -bxy 1:17</span></pre>You
should get this:<br>
<img style="height: 446px; width: 434px;" alt="Projected DOS - 4 proj" src="lesson_paw1/DOS-4proj.jpg"><br>As you can see, the smooth PW
contribution and the PS on-site
contribution are close.<br>
So, in a first approach, they cancel; we could approximate the DOS by
the AE on-site
part taken alone.<br>
That is exactly what is done when&nbsp;<a href="../input_variables/varpaw.html#pawprtdos" target="kwimg">pawprtdos</a>=2;
in that case, only the AE on-site contribution is computed and given
as&nbsp;a
good approximation of the total projected DOS. The main advantage of
this option is the&nbsp;decrease of the CPU time needed to
compute
the DOS (it is instantaneously computed).<br>
<br><span style="font-weight: bold;">But, as you will see in
the next section, this approximation is only valid when:</span>
<div style="margin-left: 40px;"><span style="font-weight: bold;"> (1) the</span><span style="position: relative; top: -8pt; left: 8pt; font-weight: bold;"></span><span style="font-weight: bold;"></span><span style="position: relative; top: -5pt; left: 6pt; font-weight: bold;">~</span><span style="font-weight: bold;">&#966;</span><sub style="font-weight: bold;">i</sub><span style="font-weight: bold;"> basis is complete enough <br></span>
<span style="font-weight: bold;">(2) the electronic density
is mainly contained in the sphere defined by </span><a style="font-weight: bold;" href="../input_variables/vargs.html#ratsph" target="kwimg">ratsph</a><span style="font-weight: bold;">.
</span></div><br><hr><h3><p><b><a name="5"><br>5.</a> Testing the
completeness of the PAW
partial wave basis</b></p>
</h3>Note that, in the previous section, we used a "standard" PAW
dataset,
with 2 partial waves per angular momentum. It is generally the best
compromise beween 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).<br>
Let's have a look at the<span style="color: rgb(0, 153, 0);">
~abinit/tests/Psps_for_tests/6c.lda.atompaw</span> file. The
sixth line indicates the number of partial waves and their <span style="font-style: italic;">l</span> angular momentum.
In the present file, "0 0 1 1" means "<span style="font-style: italic;">two
l=0 partial waves, two l=1
partial waves</span>".<br>
Now, let's open the <span style="color: rgb(0, 153, 0);">
~abinit/tests/Psps_for_tests/6c.lda.test-2proj.atompaw</span>&nbsp;and
<span style="color: rgb(0, 153, 0);">~abinit/tests/Psps_for_tests/6c.lda.test-6proj.atompaw</span>
files. In the first file, only one partial wave per <span style="font-style: italic;">l</span> is present; in
the second one, 3 partial waves per <span style="font-style: italic;">l</span>
are present. In
other words, the completeness of the partial wave basis increases when
you use <span style="color: rgb(0, 153, 0);">6c.lda.test-2proj.atompaw</span>,
<span style="color: rgb(0, 153, 0);">6c.lda.atompaw</span>&nbsp;and&nbsp;<span style="color: rgb(0, 153, 0);">6c.lda.test-6proj.atompaw</span>.<br>
<br>Now, let's plot the DOS for the two new PAW datasets.<br>
<div style="margin-left: 40px;">- First, save the existing
<span style="color: rgb(0, 153, 0);">tpaw1_4o_DOS</span><span style="color: rgb(0, 153, 0);">_AT0001</span> file,
naming it, for instance,&nbsp;<span style="color: rgb(0, 153, 0); font-weight: bold;">tpaw1_4o_4proj_DOS_AT0001</span>.<br>
- Open the <span style="color: rgb(0, 102, 0); font-weight: bold;">tpaw1_4.files</span>
file and modify it in order to use the&nbsp;<span style="color: rgb(0, 153, 0);">6c.lda.test-2proj.atompaw</span>
PAW dataset.<br>
- Launch ABINIT again.<br>
- Save the new <span style="color: rgb(0, 153, 0);">tpaw1_4o_DOS_AT0001</span>
file, naming it, for instance,&nbsp;<span style="color: rgb(0, 153, 0); font-weight: bold;">tpaw1_4o_2proj_DOS</span><span style="color: rgb(0, 153, 0); font-weight: bold;">_AT0001</span>.<br>
- Open the <span style="color: rgb(0, 102, 0); font-weight: bold;">tpaw1_4.files</span>
file and modify it in order to use the&nbsp;<span style="color: rgb(0, 153, 0);">6c.lda.test-6proj.atompaw</span>
PAW dataset.<br>
- Launch ABINIT again.<br>
- Save the new <span style="color: rgb(0, 153, 0);">tpaw1_4o_DOS</span><span style="color: rgb(0, 153, 0);">_AT0001</span> file,
naming it, for instance,&nbsp;<span style="color: rgb(0, 153, 0); font-weight: bold;">tpaw1_4o_6proj_DOS</span><span style="color: rgb(0, 153, 0); font-weight: bold;">_AT0001</span>.
</div><br>Then, plot the contributions to the projected DOS for the two
new DOS
files. You should get:<br>
<img style="width: 434px; height: 446px;" alt="Projected DOS - 2 proj" src="lesson_paw1/DOS-2proj.jpg"><img style="width: 434px; height: 446px;" alt="Projected DOS - 6 proj" src="lesson_paw1/DOS-6proj.jpg"><br>Adding the DOS obtained in the
previous section to the comparison, you
immediately see that <span style="text-decoration: underline;">the
superposition
of the Smooth part DOS and the PS on-site DOS depends on the
completeness of the partial wave basis</span> !<br>
<br>Now, you can have a look at the 3 output files (one for each PAW
dataset)... for instance in a comparison tool.<br>
A way to estimate the completeness of the partial wave basis is to
compare derivatives of total energy; if you look at the <span style="font-weight: bold; text-decoration: underline;">stress
stensor</span>:<br>
<br><div style="margin-left: 40px;">For the 2 partial-wave
bases:<span style="color: rgb(0, 153, 0); font-family: monospace;">
&nbsp;-1.0866668849E-03 -1.0866668849E-03
-1.0866668849E-03&nbsp; 0.&nbsp; 0.&nbsp; 0.</span><br>
For the 4 partial-wave basis:<span style="font-family: monospace; color: rgb(0, 153, 0);">
&nbsp; 4.1504385879E-04&nbsp; 4.1504385879E-04&nbsp;
4.1504385879E-04 &nbsp;0.&nbsp; 0.&nbsp; 0.</span><br>
For the 6 partial-wave basis:<span style="font-family: monospace; color: rgb(0, 153, 0);">
&nbsp; 4.1469803037E-04&nbsp; 4.1469803037E-04&nbsp;
4.1469803037E-04&nbsp; 0.&nbsp; 0.&nbsp; 0.</span>
</div><br>The 2 partial-wave basis is clearly not complete; the 4
partial-wave
basis results are correct...<br>Such a test is useful to estimate the
precision we can expect on the stress tensor
(at least due to the partial wave basis).<br>
<br>You can compare other results in the 3 output files: <span style="font-style: italic;">total energy</span>,
&nbsp;<span style="font-style: italic;">eigenvalues</span>, <span style="font-style: italic;">occupations</span>...<br>
Also notice that the dimensions of the PAW on-site quantities change:
have a look at "<span style="font-family: monospace; color: rgb(0, 153, 0);">Pseudopotential
strengh Dij</span>" or "<span style="font-family: monospace; color: rgb(0, 153, 0);">Augmentation
waves occupancies Rhoij</span>" sections...<br>
<br><span style="font-style: italic;">Note: if you want to
learn how to generate PAW datasets with different partial wave bases,
you might
follow
the tutorial PAW2.</span>
<br><hr><h3><p><b><a name="6"><br>6.</a> Checking the validity of PAW
results</b></p>
</h3>As usual, the validity of a "pseudopotential" (PAW dataset) has to
be
checked by comparison, on known structures, with known results. In the
case of <span style="font-weight: bold; color: rgb(153, 0, 0);">diamond</span>,
lots of computations and experimental results exist.<br><span style="font-weight: bold;">
Very important remark: the validity (<span style="font-style: italic;">completeness
of plane wave basis and partial wave basis</span>) 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.</span><br>
As the PAW method has the same accuracy than all-electron methods,
results should be very close.<br>
<br>Concerning <span style="color: rgb(153, 0, 0); font-weight: bold;">diamond</span>,
all-electron results can be found (for instance) in <span style="font-style: italic;">PRB 55, 2005 (1997)</span>.<br>
With the famous <span style="font-weight: bold;">WIEN2K</span>
code (which uses the <span style="font-style: italic;">FP-LAPW</span>
method), all-electron equilibrium parameters for diamond (for LDA) are:<br>
<br><div style="margin-left: 40px;"><span style="color: rgb(153, 0, 0); font-weight: bold;">a<sub>0</sub>
= 3.54 angstrom</span>
<span style="color: rgb(153, 0, 0); font-weight: bold;">B
= 470 GPa</span>
</div><br>Experiments give:<br>
<br><div style="margin-left: 40px;">a<sub>0</sub>
= 3.56 angstrom<br>
B = 443 GPa
</div><br>Let's test with ABINIT.<br>
We use now the input file&nbsp;<span style="color: rgb(0, 102, 0); font-weight: bold;">tpaw1_5.in</span>
and the associated <span style="color: rgb(0, 102, 0); font-weight: bold;">tpaw1_5.files</span>
file.<br>
ABINIT is now asked to compute values of etotal for several cell
parameters around 3.54 angstrom, using the standard PAW dataset.<br>
Launch ABINIT with these files; you should obtain the <span style="color: rgb(0, 102, 0);">tpaw1_5.out.</span>
<pre style="margin-left: 40px;"><span style="color: rgb(153, 0, 0);">abinit &lt; tpaw1_5.files &gt; tmp-log</span>
</pre>From
the <span style="color: rgb(0, 102, 0);">tpaw1_5.out</span>
file, you can extract the 7 values of <span style="font-style: italic;">acell</span>
and 7 values
of <span style="font-style: italic;">etotal</span>,
then put them into a file and plot it with a graphical tool. You should
get:<br>
<img style="width: 590px; height: 411px;" alt="diamond: etotal vs acell" src="lesson_paw1/etotal-acell.jpg"><br>From this curve, you can
extract the cell values of <span style="font-weight: bold;">a<sub>0</sub></span>
and <span style="font-weight: bold;">B</span> (with
the method of your choice, for example by a <span style="font-style: italic;">Birch-Murnhagan</span>
spline fit).<br>
You get:<br>
<div style="margin-left: 40px;"><span style="color: rgb(153, 0, 0); font-weight: bold;">a<sub>0</sub>
= 3.535 angstrom</span>
<span style="color: rgb(153, 0, 0); font-weight: bold;">B
= 465 GPa</span>
</div><br>These results are in excellent agreement with FP-LAPW ones !<br>
<br>

<hr>

<h3><p><b><a name="7"><br>7.</a>&nbsp;Additional comments about
PAW in ABINIT</b></p>
</h3>

<p>

<b>7.a</b><b> Mixing scheme for
the Self-Consistent cycle; decomposition of the total energy.</b></p>
The use of an efficient mixing scheme in the self-consistent loop is a
crucial point to minimize
the number of steps to achieve convergency. This mixing can be done on
the potential or on the valence 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.<br>
The mixing scheme can be controlled by the <a href="../input_variables/varbas.html#iscf" target="kwimg">iscf</a>
variable (please, read again
the different options of this input variable).<br>
By default,&nbsp;<a href="../input_variables/varbas.html#iscf" target="kwimg">iscf</a>=<span style="font-weight: bold;">7</span> for norm-conserving
pseudopotentials, while
<a href="../input_variables/varbas.html#iscf" target="kwimg">iscf</a>=
<span style="font-weight: bold;">17</span> for PAW...<br>
To compare both schemes, you can edit the <span style="color: rgb(0, 153, 0); font-weight: bold;">tpaw1_1.in</span>
file and try <a href="../input_variables/varbas.html#iscf" target="kwimg">iscf</a>=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.<br>
<br>Now, have a look at the end of the file and focus on the "<big><font style="font-family: monospace; color: rgb(0, 153, 0);" size="-1"><big>Components
of total free energy"</big></font></big>;
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.<br>
<br>You can try (using the <span style="color: rgb(0, 153, 0);">tpaw1_1.in</span>&nbsp;file)
to decrease the values of&nbsp;<a href="../input_variables/varbas.html#tolvrs" target="kwimg">tolvrs</a>
and look at the difference between both values of energy.<br>
<br>Also note that PAW <span style="font-style: italic; font-weight: bold;">&#961;</span><sub style="font-style: italic; font-weight: bold;">ij</sub> 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.<br><br><p><b>7.b</b><b> Overlap of
PAW spheres</b></p><p>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.<br>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&nbsp;<a href="../input_variables/varpaw.html#pawovlp" target="kwimg">pawovlp</a>&nbsp;input keyword. The overlap
check can even be by-passed with&nbsp;<a href="../input_variables/varpaw.html#pawovlp" target="kwimg">pawovlp</a>=-1.</p><p><span style="text-decoration: underline;">Important warning</span>:
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 <span style="font-style: italic; font-weight: bold;">r</span><sub style="font-style: italic; font-weight: bold;">shape</sub>
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...</p><p>Also note that you can control
the compensation charge radius and shape function while generating the
PAW dataset (see tutorial PAW2).</p><b></b><p><b>7.c</b><b>
Printing volume for PAW</b></p>If you want to get more detailed output
concerning the PAW computation, you can use the&nbsp;<a href="../input_variables/varpaw.html#pawprtvol" target="kwimg">pawprtvol</a> input keyword. See its
description in the user's manual...<br>It is particullary useful to
print details about pseudopotential strengh (<span style="font-style: italic; font-weight: bold;">D</span><sub style="font-style: italic; font-weight: bold;">ij</sub>) or partial
waves occupancies (<span style="font-style: italic; font-weight: bold;">&#961;</span><sub style="font-style: italic; font-weight: bold;">ij</sub>).
<br><p><b>7.d</b><b>
Additional PAW input variables</b></p>Looking at the <a href="../input_variables/varbas.html" target="kwimg">~abinit/doc/input_variables/varbas.html</a>
file, you can find input ABINIT keywords specific to PAW. They are to
be used when tuning the computation, in order to gain accuracy or save
CPU time.<br>
<span style="font-weight: bold;">Warning : in a standard computation, these
variables should not be modified !</span>

<br>
<br>Variables that can be used to gain accuracy (in ascending order of
importance):<br>
<div style="margin-left: 40px;"><a href="../input_variables/varpaw.html#pawxcdev" target="kwimg">pawxcdev</a>: control the accuracy of
exchange-correlation on-site potentials (try pawxcdev=2 to increase
accuracy).<br> <a href="../input_variables/varpaw.html#mqgriddg" target="kwimg">mqgriddg</a>: control the
accuracy of spline fits to transfer densities/potentials from FFT grid
to spherical grid.<br> <a href="../input_variables/varpaw.html#pawnzlm" target="kwimgk">pawnzlm</a>: control the
computation of moments of spherical densities that should be zero by
symmetry. </div>

<br> Variables that can be used to save memory (in
ascending order of importance):<br>
<div style="margin-left: 40px;"><a target="kwimg" href="../input_variables/varpaw.html#pawstgylm">pawstgylm</a>: control the storage of
spherical harmonics computed around atoms. </div><div style="margin-left: 40px;"><a target="kwimg" href="../input_variables/varpaw.html#pawmixdg">pawmixdg</a>: control on which grid the
potential/density is mixed during SCF cycles.<br> <a href="../input_variables/varpaw.html#pawlcutd" target="kwimg">pawlcutd</a>: control the number of angular
momenta to take into account in on-site densities.<br> <a href="../input_variables/varpaw.html#pawlmix" target="kwimg">pawlmix</a>: control the number of <span style="font-style: italic;">&#961;</span><sub style="font-style: italic;">ij</sub> to be mixed during SCF cycle.
</div>

<br> Variables that can be used to save
CPU time (in ascending order of importance):
<div style="margin-left: 40px;">
 <a target="kwimg" href="../input_variables/varpaw.html#pawnhatxc">pawnhatxc</a>: control the numerical
treatment of gradients of compensation charge density in case of GGA<br>
<a target="kwimg" href="../input_variables/varpaw.html#pawstgylm">pawstgylm</a>: control the storage of
spherical harmonics computed around atoms.<br><a href="../input_variables/varpaw.html#pawlcutd" target="kwimg">pawlcutd</a>: control the number of angular
momenta to take into account in on-site densities.<br> <a href="../input_variables/varpaw.html#pawlmix" target="kwimg">pawlmix</a>: control the number of <span style="font-style: italic;">&#961;</span><sub style="font-style: italic;">ij</sub> to be mixed during SCF cycle.<br>
<a href="../input_variables/varpaw.html#bxctmindg" target="kwimg">bxctmindg</a>: can be used to decrease the
size of fine FFT grid for a given value of pawecutdg. </div><div style="margin-left: 40px;"> </div><p>The above list is not exhaustive. several other keywords can be used to tune ABINIT PAW calculations.</p>
<p><b><br>
</b></p>
<p><b>7.e</b><b> PAW+U</b></p>
If
the system under study contains strongly correlated electrons, the&nbsp;<span style="font-style: italic; font-weight: bold;">LDA+U</span> method can
be useful. It is controlled by the&nbsp;<a href="../input_variables/varpaw.html#usepawu" target="kwimg">usepawu</a>,&nbsp;<a href="../input_variables/varpaw.html#lpawu" target="kwimg">lpawu</a>, <a href="../input_variables/varpaw.html#upawu" target="kwimg">upawu</a> and <a href="../input_variables/varpaw.html#jpawu" target="kwimg">jpawu</a> input keywords. Note that the
formalism implemented in ABINIT is approximate, i.e. it is only valid if:
<br>
<div style="margin-left: 40px;">(1) the<span style="position: relative; top: -8pt; left: 8pt;"></span><span style="position: relative; top: -5pt; left: 6pt;">~</span>&#966;<sub>i</sub>
basis is complete enough ;
<br> (2) the electronic density is mainly
contained in the PAW sphere.
<br> <span style="font-weight: bold;"></span></div>
The approximation done here is the same as the one explained in the 5th
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.<br>
<br>Converging a Self-Consistent Cycle, or ensuring the global minimum
is reached, with PAW+U is sometimes difficult. Using <a href="../input_variables/varpaw.html#usedmatpu" target="kwimg">usedmatpu</a> and <a href="../input_variables/varpaw.html#dmatpawu" target="kwimg">dmatpawu</a> can help...<br>

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