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<html>
<head>
<title>Tutorial #4</title>
</head>
<body bgcolor="#ffffff">
<hr>
<h1>ABINIT, fourth lesson of the tutorial: </h1>
<h2>Aluminum, the bulk and the surface. </h2>
<hr>

<p>This lesson aims at showing how to get
the following physical properties, for a metal, and for a surface :
<ul>
<li>the total energy
<li>the lattice parameter
<li>the relaxation of surface atoms
<li>the surface energy
</ul>
You will learn about the smearing of the Brillouin zone integration,
and also a bit about preconditioning the SCF cycle.

<p>This lesson should take about 1 hour and 30 minutes.

<h5>Copyright (C) 2000-2014 ABINIT group (XG,RC)
<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/doc/developers/contributors.txt .
</h5>

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

<br>&nbsp;
<h3><b>Contents of lesson 4</b></h3>

<ul>
  <li><a href="lesson_base4.html#41">4.1.</a>
    Computing the total energy and lattice parameters of aluminum
    for a fixed smearing and number of k points.</li>
  <li><a href="lesson_base4.html#42">4.2.</a>
    The convergence study with respect to k points.</li>
  <li><a href="lesson_base4.html#43">4.3.</a>
    The convergence study with respect to both number of k points AND
    broadening factor (tsmear).</li>
  <li><a href="lesson_base4.html#44">4.4.</a>
    Determination of the surface energy of aluminum (100):
    changing the orientation of the unit cell.</li>
  <li><a href="lesson_base4.html#45">4.5.</a>
    Determination of the surface energy :
    a (3 aluminum layer + 1 vacuum layer) slab calculation.</li>
  <li><a href="lesson_base4.html#46">4.6.</a>
    Determination of the surface energy :
    increasing the number of vacuum layers.</li>
  <li><a href="lesson_base4.html#47">4.7.</a>
    Determination of the surface energy :
    increasing the number of aluminum layers.</li>
</ul>

<hr>




<a name="41">&nbsp;</a>
<h3><b>
4.1.
Computing the total energy and lattice parameters of aluminum
for a fixed smearing and number of k points.
</b></h3>

<p><i>Before beginning, you might consider to work in a different
subdirectory as for lesson 1, 2 or 3 . Why not "Work4" ?
</i>


<p>The file ~abinit/tests/tutorial/Input/tbase4_x.files lists the file
  names and root names. You can copy it in the Work4 directory (and change it, as usual).
  You can also copy the file ~abinit/tests/tutorial/Input/tbase4_1.in</a> in Work4.
  This is your input file. You
  should edit it, read it carefully, and have a look at the following "new" input
  variables, and their explanation :
<ul>
  <li><a href="../input_variables/varbas.html#occopt" target="kwimg">occopt</a>
  <li><a href="../input_variables/vargs.html#tsmear" target="kwimg">tsmear</a>
</ul>

Note also the following :
<br>
1) You will work at fixed <a href="../input_variables/varbas.html#ecut" target="kwimg">ecut</a>
(=6Ha). It is implicit that in "real life", you should do a convergence test with
respect to <a href="../input_variables/varbas.html#ecut" target="kwimg">ecut</a> ... Here, a suitable
<a href="../input_variables/varbas.html#ecut" target="kwimg">ecut</a> is given to you. It will allow
to obtain 0.2% relative accuracy on lattice parameters. Note that this is the
softer pseudopotential of those that we have used until now : the 01h.pspgth for
H needed 30 Ha (it was rather hard), the 14si.pspnc for Si needed 8 Ha. <br>
2) The input variable diemac has been suppressed. Aluminum is a metal,
and the default is taylored for that case.


<p> When you have read the input file, you can run the code, as usual
(it will take a few seconds).
Then, read the output file quietly.
<br>You should note that the Fermi energy and occupation numbers have been
computed automatically :
<pre>
 Fermi (or HOMO) energy (hartree) =   0.26847   Average Vxc (hartree)=  -0.34746
 Eigenvalues (hartree) for nkpt=   2  k points:
 kpt#   1, nband=  3, wtk=  0.75000, kpt= -0.2500  0.5000  0.0000 (reduced coord)
   0.09425   0.25438   0.41909
      occupation numbers for kpt#   1
   2.00003   1.33307   0.00014
 kpt#   2, nband=  3, wtk=  0.25000, kpt= -0.2500  0.0000  0.0000 (reduced coord)
  -0.07038   0.41093   0.68874
      occupation numbers for kpt#   2
   2.00000   0.00030   0.00000
</pre>
You should also note that the components of the total energy
include an entropy term :
<pre>
 Components of total free energy (in Hartree) :

    Kinetic energy  =  8.70954992983644E-01
    Hartree energy  =  3.84986378597834E-03
    XC energy       = -8.08434348776921E-01
    Ewald energy    = -2.72948290133614E+00
    PspCore energy  =  3.78721667879406E-02
    Loc. psp. energy=  8.26684805111510E-02
    NL   psp  energy=  4.52588274660814E-01
    >>>>> Internal E= -2.08998347138353E+00

    -kT*entropy     = -7.99729047045124E-03
    >>>>>>>>> Etotal= -2.09798076185399E+00

 Other information on the energy :
    Total energy(eV)= -5.70889611844117E+01 ; Band energy (Ha)=   3.6059825212E-01
</pre>

<hr>

<a name="42">&nbsp;</a>
<h3><b>
4.2.
The convergence study with respect to k points
</b></h3>

<p>There is of course a convergence study associated to the sampling
of the Brillouin zone. You should examine different grids,
of increasing resolution. You might try the following series
of grids :
<pre>
  ngkpt1  2 2 2
  ngkpt2  4 4 4
  ngkpt3  6 6 6
  ngkpt4  8 8 8
</pre>
with the associated <a href="../input_variables/varbas.html#nkpt" target="kwimg">nkpt</a> :
<pre>
  nkpt1  2
  nkpt2 10
  nkpt3 28
  nkpt4 60
</pre>

<p>The input file ~abinit/tests/tutorial/Input/tbase4_2.in is an example,
  while ~abinit/tests/tutorial/Refs/tbase4_2.out is a reference output file.
  The run might take about thirty seconds on a PC 3 GHz.

<p>You will see that, FOR THE PARTICULAR VALUE OF <a href="../input_variables/vargs.html#tsmear" target="kwimg">tsmear</a>=0.05
Ha, the lattice parameter is already converged with <a href="../input_variables/varbas.html#nkpt" target="kwimg">nkpt</a>=10
:
<pre>
     acell1   7.5588967139E+00  7.5588967139E+00  7.5588967139E+00 Bohr
     acell2   7.5070434097E+00  7.5070434097E+00  7.5070434097E+00 Bohr
     acell3   7.5016879972E+00  7.5016879972E+00  7.5016879972E+00 Bohr
     acell4   7.4992665325E+00  7.4992665325E+00  7.4992665325E+00 Bohr
</pre>
Note that there is usually a STRONG cross-convergence effect between the number
of k points and the value of the broadening, <a href="../input_variables/vargs.html#tsmear" target="kwimg">tsmear</a>.
<br>
The right procedure is : for each value of <a href="../input_variables/vargs.html#tsmear" target="kwimg">tsmear</a>,
to get the convergence with respect to the number of k points, then to compare
the k-point converged values for different values of <a href="../input_variables/vargs.html#tsmear" target="kwimg">tsmear</a>.
<p>In what follows, we will restrict ourselves to the grids with <a href="../input_variables/varbas.html#nkpt" target="kwimg">nkpt</a>=2,
  10 and 28.

<p>

<hr>


<p> <a name="43">&nbsp;</a>
<h3><b>
4.3.
The convergence study with respect to both number of k points AND
broadening factor (tsmear).
</b></h3>

<p>The theoretical convergence rate for <a href="../input_variables/vargs.html#tsmear" target="kwimg">tsmear</a>
  ending to 0, in the case of <a href="../input_variables/varbas.html#occopt" target="kwimg">occopt</a>=4,
  is quartic. This is obtained in the hypothesis of infinitely dense k point grid.
  We will check the evolution of <a href="../input_variables/varbas.html#acell" target="kwimg">acell</a>
  as a function of <a href="../input_variables/vargs.html#tsmear" target="kwimg">tsmear</a>, for
  the following values of <a href="../input_variables/vargs.html#tsmear" target="kwimg">tsmear</a>
  : 0.01, 0.02, 0.03 and 0.04. Use the double-loop capability of the multi-dataset
  mode, with the <a href="../input_variables/vargs.html#tsmear" target="kwimg">tsmear</a> changes
  in the INNER loop. This will saves CPU time, as the wavefunctions of the previous
  dataset will be excellent (no transfer to different k points).
<p>The input file ~abinit/tests/tutorial/Input/tbase4_3.in is an example,
  while ~abinit/tests/tutorial/Refs/tbase4_3.out is a reference output file.
<p>From the output file, here is the evolution of <a href="../input_variables/varbas.html#acell" target="kwimg">acell</a>
  :
<pre>
     acell11  7.5587661088E+00  7.5587661088E+00  7.5587661088E+00 Bohr
     acell12  7.5587695984E+00  7.5587695984E+00  7.5587695984E+00 Bohr
     acell13  7.5587696422E+00  7.5587696422E+00  7.5587696422E+00 Bohr
     acell14  7.5587709917E+00  7.5587709917E+00  7.5587709917E+00 Bohr
     acell21  7.5055170494E+00  7.5055170494E+00  7.5055170494E+00 Bohr
     acell22  7.5056782904E+00  7.5056782904E+00  7.5056782904E+00 Bohr
     acell23  7.5018338126E+00  7.5018338126E+00  7.5018338126E+00 Bohr
     acell24  7.5041512153E+00  7.5041512153E+00  7.5041512153E+00 Bohr
     acell31  7.4963468646E+00  7.4963468646E+00  7.4963468646E+00 Bohr
     acell32  7.4957101136E+00  7.4957101136E+00  7.4957101136E+00 Bohr
     acell33  7.4969518585E+00  7.4969518585E+00  7.4969518585E+00 Bohr
     acell34  7.4993530520E+00  7.4993530520E+00  7.4993530520E+00 Bohr
</pre>
These data should be analyzed properly.
<p>For <a href="../input_variables/vargs.html#tsmear" target="kwimg">tsmear</a>=0.01, the converged
  value, contained in acell31, must be compared to acell11 and acell21 :
  between acell21
  and acell31, the difference
  is below 0.2%. acell31 can be
  considered to be converged with respect to the number of k-points, at fixed
  <a href="../input_variables/vargs.html#tsmear" target="kwimg">tsmear</a>. <br>
  This tsmear being the lowest one, it is usually the most difficult to converge,
  and the values acell31,32,33
  and 34 are indeed well-converged with respect to the k-point number. <br>
  The use of the largest <a href="../input_variables/vargs.html#tsmear" target="kwimg">tsmear</a>(=0.04),
  giving acell34, induces only
  a small error in the lattice parameter.
  For that particular value of <a href="../input_variables/vargs.html#tsmear" target="kwimg">tsmear</a>,
  one can use the second k-point grid,
  giving acell24.
<p>So to <b>summarize</b> :<br>
  we can choose to work with a 10 k-point grid in the irreducible Brillouin zone,
  and the associated <a href="../input_variables/vargs.html#tsmear" target="kwimg">tsmear</a>=0.04,
  with less than 0.1% error on the lattice parameter. <br>
  NOTE that this error due to the Brillouin zone sampling could add to the error
  due to the choice of <a href="../input_variables/varbas.html#ecut" target="kwimg">ecut</a> (that
  was mentioned previously to be on the order of 0.2%).
<p>In what follows, we will stick to these values of
  <a href="../input_variables/varbas.html#ecut" target="kwimg">ecut</a>
  and <a href="../input_variables/vargs.html#tsmear" target="kwimg">tsmear</a>, and try to use
  k-point grids with a similar resolution.
<p>Our final value for the aluminum lattice parameter, in the LDA,
using the 13al.981214.fhi pseudopotential, is thus 7.5041 Bohr.
Note : for historical reasons (consistency with older versions of the tutorial),
we will work on the following, with a slightly
different value, of 7.5056 Bohr, that is 3.9718 Angstrom.
<br>The experimental value at 25 degree Celsius is 4.04958 Angstrom.

<p>The associated total energy and accuracy can be deduced from
<pre>
    etotal11 -2.0916027819E+00
    etotal12 -2.0931968906E+00
    etotal13 -2.0947909992E+00
    etotal14 -2.0963851177E+00
    etotal21 -2.0969713557E+00
    etotal22 -2.0975525285E+00
    etotal23 -2.0978233733E+00
    etotal24 -2.0979980153E+00
    etotal31 -2.0983520905E+00
    etotal32 -2.0983215368E+00
    etotal33 -2.0983305960E+00
    etotal34 -2.0984218116E+00
</pre>
<p><b>etotal</b>24 is -2.0979980153E+00 Ha, with an accuracy of 0.0005 Ha .</p>

<hr>

<p> <a name="44">&nbsp;</a> </p>
<h3><b>
4.4.
Determination of the surface energy of aluminum (100):
changing the orientation of the unit cell.
</b></h3>

<p>In order to study the Aluminum (100) surface, we will have to set up
a supercell representing a slab. This supercell should be chosen
as to be compatible with the primitive surface unit cell.
<br>The corresponding directions are <code>[-1 1 0]</code> and <code>[1 1 0]</code>. The direction
perpendicular to the surface is <code>[0 0 1]</code>. There is no primitive
cell of bulk aluminum based on these vectors, but a doubled
cell. We will first compute the total energy associated with
this doubled cell. This is not strictly needed, but it is a
valuable intermediate step towards the study of the surface.

<p>You might start from tbase4_3.in. <br>
  You have to change <a href="../input_variables/varbas.html#rprim" target="kwimg">rprim</a>. Still,
  try to keep <a href="../input_variables/varbas.html#acell" target="kwimg">acell</a> at the values
  of bulk aluminum that were determined previously. But it is not all : the most
  difficult part in the passage to this doubled cell is the definition of the
  k-point grid. Of course, one could just take a homogeneous simple cubic grid
  of k points, but this will not correspond exactly to the k-point grid used in
  the primitive cell in tbase4_3.in . This would not be a big problem, but you
  would miss some error cancellation.
<p>The answer to this problem is given in the
  input file ~abinit/tests/tutorial/Input/tbase4_4.in.
  <br>
  The procedure to do the exact translation of the k-point grid will not be explained
  here (sorry for this). If you do not see how to do it, just use homogeneous
  simple cubic grids, with about the same resolution as for the primitive cell
  case. There is a simple rule to estimate ROUGHLY whether two grids for different
  cells have the same resolution : simply multiply the linear dimensions of the
  k-point grids, by the number of sublattices, by the number of atoms in the cell.
  For example, the corresponding product for the usual 10 k-point grid is <code>4x4x4
  x 4 x 1 = 256</code> . In the file tbase4_4.in,
  one has <code>4x4x4 x 2 x 2 = 256</code> . The grids of k points should not
  be too anisotropic for this rough estimation to be valid.
<p>Note also the input variables <a href="../input_variables/varbas.html#rprim" target="kwimg">rprim</a>
  and <a href="../input_variables/vargs.html#chkprim" target="kwimg">chkprim</a> in this input
  file.
<p>So, you run tbase4_4.in (only a few seconds, the reference file is ~abinit/tests/tutorial/Refs/tbase4_4.out),
  and you find the following total energy :
<pre>
    etotal   -4.1962972614E+00
</pre>
<p>It is not exactly twice the total energy for the primitive cell, mentioned
  above, but the difference is less than 0.0005 Ha. It is due to the different
  FFT grids used in the two runs, and affect the exchange-correlation energy.
  These grids are always homogeneous primitive 3D grids, so that changing the
  orientation of the lattice will give mutually incompatible lattices. Increasing
  the size of the FFT grid would improve the agreement.</p>

<hr>

<p> <a name="45">&nbsp;</a> </p>
<h3><b>
4.5.
Determination of the surface energy :
a (3 aluminum layer + 1 vacuum layer) slab calculation.
</b></h3>

<p>We will first compute the total energy associated with only three layers of
  aluminum, separated by only one layer of vacuum. This is kind of a minimal slab
  :
  <ul>
    <li>one surface layer
    <li>one "bulk" layer
    <li>one surface layer
    <li>one vacuum layer
    <li>...
  </ul>
  It is convenient to take the vacuum region as having a multiple of the width
  of the aluminum layers, but this is not mandatory. The supercell to use is the
  double of the previous cell (that had two layers of Aluminum atoms along the
  <code>[0 0 1]</code> direction). Of course, the relaxation of the surface might
  give an important contribution to the total energy.
<p>You should start from tbase4_4.in . <br>
  You have to modify <a href="../input_variables/varbas.html#rprim" target="kwimg">rprim</a> (double
  the cell along <code>[0 0 1]</code>), the atomic positions, as well as the k
  point mesh. For the latter, it is supposed that the electrons cannot propagate
  from one slab to its image in the <code>[0 0 1]</code> direction, so that the
  k_z component of the special k points can be taken 0 : only one layer of k points
  is needed along the z-direction. You should also allow the relaxation of atomic
  positions, but not the relaxation of lattice parameters (the lattice parameters
  along x or y must be considered fixed to the bulk value, while, for the z direction,
  there is no interest to allow the vacuum region to collapse !
<p>The input file ~abinit/tests/tutorial/Input/tbase4_5.in is an example,
  while ~abinit/tests/tutorial/Refs/tbase4_5.out is a reference output
  file. The run might last one minute.
<p>The total energy after the first SCF cycle, when the
atomic positions are equal to their starting values, is :
<pre>
 ETOT  6  -6.2619731679890
</pre>
Note that the total energy of three aluminum atoms in the bulk,
(from section 4.3, etotal24 multiplied by three) is
<pre>
   -6.293994 Ha
</pre>
so that the non-relaxed surface energy, per surface unit cell (there
are two surfaces in our simulation cell !) is
<pre>
  0.016010 Ha = 0.436 eV .
</pre>
<p>The total energy after the Broyden relaxation is :
<pre>
    etotal   -6.2622251351E+00
</pre>
so that the relaxed surface energy, per surface unit cell is
<pre>
0.015885 Ha = 0.432eV .
</pre>
<p>It seems that the relaxation energy is very small, compared to the surface
  energy, but we need to do the convergence studies.</p>
<p>

<hr>


<p> <a name="46">&nbsp;</a> </p>
<h3><b>
4.6.
Determination of the surface energy :
increasing the number of vacuum layers.
</b></h3>

<p>One should now increase the number of vacuum layers : 2 and 3 layers
instead of only 1.
<br>It is preferable to define atomic positions in cartesian coordinates.
The same coordinates will work for both 2 and 3 vacuum layers, while
this is not the case for reduced coordinates, as the cell size
increases.

<p>The input file ~abinit/tests/tutorial/Input/tbase4_6.in is an example
  input file, while ~abinit/tests/tutorial/Refs/tbase4_6.out is a
  reference output file. The run is on the order of thirty seconds on a PC 3 GHz.
<p>In the Broyden step 0 of the first dataset, you will notice
the WARNING :
<pre>
 scprqt:  WARNING -
  nstep=    6 was not enough SCF cycles to converge;
  maximum force difference=  6.359E-05 exceeds toldff=  5.000E-05
</pre>
The input variable <a href="../input_variables/varbas.html#nstep" target="kwimg">nstep</a>
was intentionally set to the rather low value of 6,
to warn you about possible convergence difficulties.
The SCF convergence might indeed get more and more difficult with cell size. This is because the default
preconditioner (see the notice of the input variable <a href="../input_variables/vargs.html#dielng" target="kwimg">dielng</a>)
is not very good for the metal+vacuum case. <br>
For the interpretation of the present run, this is not critical,
as the convergence criterion was close of being fulfilled,
but one should keep this in mind, as you will see ...

<p>For the 2 vacuum layer case, one has the non-relaxed total energy :
<pre>
 ETOT  6  -6.2539524080105
</pre>
giving the unrelaxed surface energy
<pre>
  0.0200 Ha = 0.544 eV ;
</pre>and for the relaxed case :
<pre>
    etotal1  -6.2547006703E+00
</pre>
(this one is converged to the required level)
giving the relaxed surface energy
<pre>  0.0196 Ha = 0.533 eV</pre>
Note that the difference between unrelaxed and relaxed case is a bit larger than
in the case of one vacuum layer. This is because there was some interaction between
slabs of different supercells.
<p>For the 3 vacuum layer case, the self-consistency is slightly
more difficult than with 2 vacuum layers :
the Broyden step 0 is not sufficiently converged
(one might set nstep to a larger value, but the best is to change the
preconditioner, as described below)...
<br>However, for the Broyden steps number 2 and beyond, because one
takes advantage of the previous wavefunctions, a sufficient convergence
is reached. The total energy, in the relaxed case, is :
<pre>
    etotal2  -6.2559102400E+00
</pre>
giving the relaxed surface energy <code>0.0190 Ha = 0.515 eV</code> There is a
rather small 0.018 eV difference with the 2 vacuum layer case.
<p>For the next run, we will keep the 2 vacuum layer case, and we know that the
  accuracy of the coming calculation cannot be better than 0.016 eV. One might
  investigate the 4 vacuum layer case, but this is not worth, in the present tutorial
  ...

<p>
<hr>

<p> <a name="47">&nbsp;</a>
<h3><b>
4.7.
Determination of the surface energy :
increasing the number of aluminum layers.
</b></h3>

<p>One should now increase the number of aluminum layers, while keeping 2 vacuum
  layers. We will consider 4 and 5 aluminum layers. This is rather straightforward
  to set up, but will also change the preconditioner. One
  could use an effective dielectric constant of about 3 or 5, with a rather small
  mixing coefficient, on the order of 0.2. However, there is also another possibility,
  using an estimation of the dielectric matrix governed by <a href="../input_variables/vargs.html#iprcel" target="kwimg">iprcel</a>=45
  . For comparison with the previous treatment of SCF, one can recompute the result
  with 3 aluminum layers.
<p>The input file ~abinit/tests/tutorial/Input/tbase4_7.in is an example,
  while ~abinit/tests/tutorial/Refs/tbase4_7.out is a reference output
  file. This run might take about one minute, and is the longer of the four
basic lessons.
  You should start it now.
<p>You can monitor its evolution by editing from time to time
the tbase4_7_STATUS file that the code updates regularly.
The status file, that refer to the skeleton of the code,
is described in the ~abinit/doc/developers/programmer_guide.txt .
You might take advantage of the time of the run to explore
the files contained in the  ~abinit/doc/users directory and the
~abinit/doc/developers directory. The README files
provided interesting entry points in the documentation of the code.

<p>Coming back to the file tbase4_7.out
  ... <br>
  You will notice that the SCF convergence is rather satisfactory, for all the
cases (3, 4 or 5 metal layers).

<p>For the 3 aluminum layer case, one has the non-relaxed total energy :
<pre>
 ETOT  6  -6.2539524363099
</pre>(this quantity is converged, unlike in test 4.6)
<br>giving the unrelaxed surface energy 0.0200 Ha = 0.544 eV ;
and for the relaxed case :
<pre>
    etotal1  -6.2547008083E+0
</pre>
(by contrast the difference with test 4.6 is less than
                       1 microHa)
<br>giving the relaxed surface energy 0.0196 Ha = 0.533 eV .

<p>For the 4 aluminum layer case, one has the non-relaxed total energy :
<pre>
 ETOT  6  -8.3546873356951
</pre>
giving the unrelaxed surface energy 0.0186Ha = 0.506 eV ;
<br>and for the relaxed case :
<pre>
    etotal2  -8.3565593181E+00
</pre>
giving the relaxed surface energy 0.0183 Ha = 0.498 eV .

<p>For the 5 aluminum layer case, one has the non-relaxed total energy :
<pre>
 ETOT  6  -10.453642176454
</pre>
giving the unrelaxed surface energy 0.0183Ha = 0.498 eV ;
<br>and for the relaxed case :
<pre>
    etotal3  -1.0454160417E+01
</pre>
giving the relaxed surface energy 0.0180 Ha = 0.490 eV .

<p>The relative difference in the surface energy of the 4 and 5 layer cases
is on the order of 1.5%.
<p>In the framework of this tutorial, we will
not pursue this investigation, which is a simple application
of the concepts already explored.

<p>Just for your information, and as an additional warning, when the work accomplished
  until now is completed with 6 and 7 layers without relaxation
  (see ~abinit/tests/tutorial/Input/tbase4_8.in and ~abinit/tests/tutorial/Refs/tbase4_8.out
  where 5, 6 and 7 layers are treated), this non-relaxed energy surface energy
  behaves as follows : <br>
  <br>
<table cellspacing="2" cellpadding="2" border="1" width="360">
  <tr>
    <td>
      <div align="center">number of<br>
        aluminum layers </div>
    </td>
    <td>
      <div align="center">surface energy </div>
    </td>
  </tr>
  <tr>
    <td>
      <div align="center">3 </div>
    </td>
    <td>
      <div align="center">0.544 eV </div>
    </td>
  </tr>
  <tr>
    <td>
      <div align="center">4 </div>
    </td>
    <td>
      <div align="center">0.506 eV </div>
    </td>
  </tr>
  <tr>
    <td>
      <div align="center">5 </div>
    </td>
    <td>
      <div align="center">0.498 eV </div>
    </td>
  </tr>
  <tr>
    <td>
      <div align="center">6 </div>
    </td>
    <td>
      <div align="center">0.449 eV </div>
    </td>
  </tr>
  <tr>
    <td>
      <div align="center">7 </div>
    </td>
    <td>
      <div align="center">0.463 eV </div>
    </td>
  </tr>
</table>

<p>So, the surface energy convergence is rather difficult to reach.
<br>
  Our values, with a <code>4x4x1</code> grid, a smearing of 0.04 Ha, a kinetic
  energy cut-off of 6 Ha, the 13al.981214.fhi pseudopotential, still oscillate
  between 0.45 eV and 0.51 eV. Increasing the k point sampling might decrease slightly
the oscillations, but note that this effect is intrinsic to the computation of properties
of a metallic surface : the electrons are confined inside the slab potential, with
subbands in the direction normal to the surface, and the Fermi energy oscillates
with the width of the slab. This effect might be understood
based on a comparison with the behaviour of a jellium slab.<br>
  An error on the order of 0.016 eV is due to the thin vacuum layer.
Other sources of errors might have to be rechecked, seeing the
kind of accuracy that is needed.

<p>Experimental data give a surface energy around 0.55 eV (sorry,
the reference is to be provided).

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