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First lesson on DFPT

Dynamical and dielectric properties of AlAs.

This lesson aims at showing how to get the following physical properties, for an insulator:

• the phonon frequencies and eigenvectors at Gamma
• the dielectric constant
• the Born effective charges
• the LO-TO splitting
• the phonon frequencies and eigenvectors at other q-points in the Brillouin Zone

You will learn to use of density-functional perturbation theory (DFPT) features of ABINIT. In order to learn the use of the associated codes Mrgddb and Anaddb, to produce efficiently phonon band structures and the associated thermodynamical properties, please see the tutorial DFPT 2 (labelled RF2)

This lesson should take about 2 hours.

1 The ground-state geometry of AlAs

Before beginning, you might consider to work in a different subdirectory as for the other lessons. Why not create “Work_rf1” in ~abinit/tests/tutorespfn/Input?

Important

The reference directory that contains the example files for the tutorial is no more ~abinit/tests/tutorial (as for the basic lessons and the specialized, non-DFPT ones), but ~abinit/tests/tutorespfn. This will be the case for all the DFPT based part of the tutorial.

The file ~abinit/tests/tutorespfn/Input/trf1_x.files lists the file names and root names.

View tests/tutorespfn/Input/trf1_x.files

You can copy it in the Work_rf1 directory and, as usual, change its name to trf1_1.files and replace the occurences of trf1_x by trf1_1. Note that two pseudopotentials are mentioned in this “files” file: one for the Aluminum atom, and one for the Arsenic atom. The first to be mentioned, for Al, will define the first type of atom. The second to be mentioned, for As, will define the second type of atom. It might the first time that you encounter this situation (more than one type of atoms) in the tutorials, at variance with the four “basic” lessons.

You can also copy the file ~abinit/tests/tutorespfn/Input/trf1_1.in in Work_rf1.

cp ../trf1_1.in .

This is your input file. You should read it carefully.

View tests/tutorespfn/Input/trf1_1.in

It drives a single self-consistent calculation of the total energy of AlAs, also generating the corresponding self-consistent charge density and wavefunctions, that will be used for later DFPT calculations. Because of the use of two types of atoms, have a look at the following input variables:

• ntypat
• typat

Note that the value of tolvrs is rather stringent. This is because the wavefunctions determined by the present run will be used later as starting point of the DFPT calculation. However, the number of steps, nstep, in this example file has been set to 15, and you will see that this is not enough to reach the target tolvrs. In production runs, you should choose a larger value of nstep, sufficient to reach your target tolvrs. In the present tutorial, due to portability concerns related to automatic testing, we could not allow a larger nstep value. This minor problem with some tutorial examples was mentioned briefly in a side note to the answer to question 1 of lesson 1 - just before this section. So, do not follow blindly all examples in the tutorials: check by yourself the convergence of your runs!

You will work at fixed ecut (3Ha) and k-point grid, defined by kptrlatt (the 8x8x8 Monkhorst-Pack grid). It is implicit that in real life, you should do a convergence test with respect to both parameters. We postpone the discussion of the accuracy of these choices and the choice of pseudopotential to the end of the fifth section of this tutorial. They give acceptable results, not very accurate, but, more important, the speed is reasonable for a tutorial.

You should make the run (a few seconds).

./abinit < trf1_1.files > trf1_1.stdout

where ./abinit has to be replaced by the path to the abinit executable. The resulting main output file, trf1_1.out, should be similar to the one below.

View tests/tutorespfn/Refs/trf1_1.out

This output file is not very long, so you can quickly read it entirely.

Note that one obtains the following value for the energy, in the final echo section:

 etotal   -9.7626837450E+00

However, we will rely later, for the purpose of doing finite differences, on a more accurate (more digits) value of this total energy, that can be found about a dozen of lines before this final echo:

>>>>>>>>> Etotal= -9.76268374500280E+00

The output file also mentions that the forces on both atoms vanish.

The run that you just made will be considered as defining a ground-state configuration, on top of which responses to perturbations will be computed. The main output of this ground-state run is the wavefunction file trf1_1o_WFK, that you can already rename as trf1_1i_WFK. Indeed, it will be used in the next runs, as an input file.

Warning

So, in the corresponding “files” file for all the following runs, at third line, pay attention to keep “trf1_1i”. By contrast, for the second run, you should change the first line from “trf1_1”
to “trf1_2”, and do similarly for the second, fourth and fifth lines of this file.

2 Frozen-phonon calculation of a second derivative of the total energy

We will now aim at computing the second derivative of the total energy with respect to an atomic displacement by different means. For that purpose, you must first read sections 0 and the first paragraph of section 1 of the respfn_help file (the auxiliary help file, that deals specifically with the DFPT features). We will explain later, in more detail, the signification of the different input parameters introduced in section 1 of the respfn_help file.

For now, in order to be able to perform a direct comparison with the result of a DFPT calculation, we choose as a perturbation the displacement of the Al atom along the first axis of the reduced coordinates.

You can copy the file ~abinit/tests/tutorespfn/Input/trf1_2.in in Work_rf1. This is your input file. You should edit it and briefly look at the two changes with respect to the file ~abinit/tests/tutorespfn/Input/trf1_1.in: the change of xred, and the reading of the wavefunction file, using the irdwfk input variable.

Then, you can make the run, following the same command as before, with a different files file, referring to trf1_2.in. The symmetry is lowered with respect to the ground-state geometry, so that the number of k-points increases a lot, and of course, the CPU time.

View tests/tutorespfn/Input/trf1_2.in View tests/tutorespfn/Refs/trf1_2.out

From this run, it is possible to get the values of the total energy, and the value of the gradient of the total energy (dE) with respect to change of reduced coordinate (dt):

 rms dE/dt=  3.5517E-03; max dE/dt=  5.0079E-03; dE/dt below (all hartree)
    1       0.005007937776      0.002526310510      0.002526310510
    2      -0.005007879256     -0.002526283046     -0.002526283046
 ...
    >>>>>>>>> Etotal= -9.76268124105767E+00

The change of reduced coordinate (xred) of the Al atom along the first axis was rather small (1/1000), and we can make an estimate of the second derivative of the total energy with respect to the reduced coordinate thanks to finite-difference formulas.

We start first from the total-energy difference. The total energy is symmetric with respect to that perturbation, so that it has no linear term. The difference between the ground-state value (-9.76268374500280E+00 hartree) of the previous run, and the perturbed value (-9.76268124105767E+00 hartree) of the present one, is thus one half of the square of the coordinate change (1/1000) times the second derivative of total energy (2DTE). From these number, the 2DTE is 5.0078902589 hartree.

Alternatively, we can start from the reduced gradients. The value of the reduced gradient with respect to a displacement of the Al atom along the first reduced axis is 0.005007937776 (hartree). At first order, this quantity is the product of the 2DTE by the reduced coordinate difference. The estimate of the 2DTE is thus 5.007937776 hartree. The agreement with the other estimate is rather good (4.10^-5 Hartree).

However, it is possible to do much better, thanks to the use of a higher-order finite-difference formula. For this purpose, one can perform another calculation, in which the change of reduced coordinate along the first axis is 0.002, instead of 0.001. The doubling of the perturbation allows for a rather simple higher-order estimation, as we will see later. The results of this calculation are as follows:

 rms dE/dt=  7.1249E-03; max dE/dt=  1.0016E-02; dE/dt below (all hartree)
    1       0.010016307568      0.005097516146      0.005097516146
    2      -0.010016187598     -0.005097463261     -0.005097463261
 ...

From these data, taking into account that the perturbation was twice stronger, the same procedure as above leads to the values 5.00800219 hartree (from finite difference of energy) and 5.008153784 hartree (from finite difference of forces, the value 0.010016307568 has to be multiplied by 1000/2). The combination of these data with the previous estimate can be done thanks to an higher-order finite-difference formula, in which the difference of estimations (the largest perturbation minus the smallest one) is divided by three, and then subtracted from the smallest estimation.

As far as the total-energy estimation is concerned, the difference is 0.00011194 Ha, which divided by three, and subtracted from 5.00789025 hartree, gives 5.00785294 hartree. The same higher-order procedure for force estimates gives 5.0078657 hartree. So, the agreement between total-energy estimate and force estimate of the 2DTE can be observed up to the 6^th digit, inclusive.

Before comparing this result with the 2DTE directly computed from the DFPT capabilities of ABINIT, a last comment is in order. One can observe that the action-reaction law is fulfilled only approximately by the system. Indeed, the force created on the second atom, should be exactly equal in magnitude to the force on the first atom. The values of dE/dt, mentioned above show a small, but non-negligible difference between the two atoms. As an example, for the doubled perturbation, there is a difference in the absolute values of the first component of the reduced force, 0.010016307568 and -0.010016187598. Actually, the forces should cancel each other exactly if the translation symmetry were perfect. This is not the case, but the breaking of this symmetry can be shown to arise only from the presence of the exchange-correlation grid of points. This grid does not move when atoms are displaced, and so there is a very small variation of the total energy when the system is moved as a whole. It is easy to restore the action-reaction law, by subtracting from every force component the mean of the forces on all atoms. This is actually done when the gradient with respect to reduced coordinates are transformed into forces, and specified in cartesian coordinates, as can be seen in the output file for the small displacement:

 cartesian forces (hartree/bohr) at end:
    1     -0.00001684560079    -0.00094403841497    -0.00094403841497
    2      0.00001684560079     0.00094403841497     0.00094403841497

This effect will be seen also at the level of 2DTE. The so-called “acoustic sum rule”, imposing that the frequency of three modes (called acoustic modes) tends to zero with vanishing wavevector, will also be slightly broken. In this case also, it will be rather easy to reimpose the acoustic sum rule. In any case, taking a finer XC grid will allow one to reduce this effect.

3 DFPT calculation of a second derivative of the total energy

We now compute the second derivative of the total energy with respect to the same atomic displacement, using the DFPT capabilities of ABINIT.

You can copy the file ~abinit/tests/tutorespfn/Input/trf1_3.in in Work_rf1. This is your input file. You should examine it. The changes with respect to the file ~abinit/tests/tutorespfn/Input/trf1_1.in are all gathered in the first part of this file, before

#######################################################################
#Common input variables

Accordingly, you should get familiarized with the new input variables : rfphon, rfatpol, rfdir. Then, pay attention to the special use of the kptopt input variable. It will be explained in more detail later.

View tests/tutorespfn/Input/trf1_3.in

When you have understood the purpose of the input variable values specified before the “Common input variables” section, you can make the code run, as usual.

Then, we need to analyze the different output files. For that purpose, you should read the content of the section 6 of the respfn_help file. Read it quickly, as we will come back to the most important points hereafter.

ABINIT has created several different files:

• trf1_3.log (the log file)
• trf1_3.out (the output file), possibly also trf1_3o_OUT.nc, an abridged netCDF version
• trf1_3o_1WF1 (the 1^st-order wavefunction file)
• trf1_3o_DEN1 (the 1^st-order density file)
• trf1_3o_POT1 (the 1^st-order potential file)
• trf1_3o_DDB (the derivative database), possibly also trf1_3o_DDB.nc, its netCDF version

Let us have a look at the output file. You can follow the description provided in the section 6.2 of the respfn_help file.

View tests/tutorespfn/Refs/trf1_3.out

You should be able to find the place where the iterations for the minimisation (with respect to the unique perturbation) take place:

      iter   2DEtotal(Ha)       deltaE(Ha) residm    vres2
-ETOT  1   6.5156050845578     -1.464E+01 1.146E-02 1.947E+02
 ETOT  2   5.0216332567384     -1.494E+00 9.267E-04 2.027E+00
 ETOT  3   5.0081675390166     -1.347E-02 5.325E-06 5.660E-02
 ETOT  4   5.0078655420156     -3.020E-04 1.601E-07 2.087E-03
 ETOT  5   5.0078558671678     -9.675E-06 5.588E-09 3.089E-05
 ETOT  6   5.0078557436628     -1.235E-07 9.879E-11 2.297E-07
 ETOT  7   5.0078557427888     -8.741E-10 8.545E-13 2.728E-09

From these data, you can see that the 2DTE determined by the DFPT technique is in excellent agreement with the higher-order finite-difference values for the 2DTE, determined in the previous section: 5.007855 hartree from the energy differences, and 5.007852 hartree from the force differences.

Now, you can read the remaining of the section 6.2 of the respfn_help file. Then, you should also edit the trf1_3o_DDB file, and read the corresponding section 6.5 of the respfn_help file.

Finally, the excellent agreement between the finite-difference formula and the DFPT approach calls for some accuracy considerations. These can be found in section 7 of the respfn_help file.

4 DFPT calculation of the dynamical matrix at Gamma

We are now in the position to compute the full dynamical matrix at $\Gamma$ (q=0). You can copy the file ~abinit/tests/tutorespfn/Input/trf1_4.in in Work_rf1. This is your input file. You should edit it.

As for test rf1_3, the changes with respect to the file ~abinit/tests/tutorespfn/Input/trf1_1.in are all gathered in the first part of this file. Moreover, the changes with respect to trf1_3.in concern only the input variables rfatpol, and rfdir. Namely, all the atoms will be displaced, in all the directions.

View tests/tutorespfn/Input/trf1_3.in

There are six perturbations to consider. So, one might think that the CPU time will raise accordingly. This is not true, as ABINIT is able to determine which perturbations are the symmetric of another perturbation, see section 3 of the respfn_help file.

Now, you can make the run. You edit the file trf1_4.out, and notice that the response to two perturbations were computed explicitly, while the response to the other four could be deduced by using the symmetries.

 The list of irreducible perturbations for this q vector is:
    1)    idir= 1    ipert=   1
    2)    idir= 1    ipert=   2

Nothing mysterious: one of the two irreducible perturbations is for the Al atom, placed in a rather symmetric local site, and the other perturbation is for the As atom.

The phonon frequencies, obtained by diagonalizing the dynamical matrix (where the atomic masses have been taken into account, see amu), are given as follows:

  Phonon wavevector (reduced coordinates) :  0.00000  0.00000  0.00000
 Phonon energies in Hartree :
   2.558878E-06  2.558879E-06  2.558880E-06  1.568559E-03  1.568559E-03
   1.568559E-03
 Phonon frequencies in cm-1    :
-  5.616089E-01  5.616089E-01  5.616092E-01  3.442590E+02  3.442590E+02
-  3.442590E+02

Tip

You might wonder about the dash sign present in the first column of the two lines giving the frequencies in $cm^{-1}$. The first column of the main ABINIT output files is always dedicated to signs needed to automatically treat the comparison with respect to reference files. Except if you become a ABINIT developer, you should ignore these signs. In the present case, they should not be interpreted as a minus sign for the floating numbers that follow them…

There is a good news about this result, and a bad news. The good news is that there are indeed three acoustic modes, with frequency rather close to zero (less than 1 cm^-1, which is rather good !). The bad news comes when the three other frequencies are compared with experimental results, or other theoretical results. Indeed, in the present run, one obtains three degenerate modes, while there should be a (2+1) splitting. This can be seen in the paper Ab initio calculation of phonon dispersions in semiconductors [Giannozzi1991] , especially Fig. 2.

Actually, we have forgotten to take into account the coupling between atomic displacements and the homogeneous electric field, that exists in the case of polar insulators, for so-called “Longitudinal Optic (LO) modes”. A splitting appears between these modes and the “Transverse Optic (TO) modes”. This splitting (Lyddane-Sachs-Teller LO-TO splitting) is presented in simple terms in standard textbooks, and should not be forgotten in doing Ab initio calculations of phonon frequencies.

Thus we have now to treat correctly the homogeneous electric field type perturbation.

5 DFPT calculation of the effect of an homogeneous electric field

The treatment of the homogeneous electric field perturbation is formally much more complex than the treatment of atomic displacements. This is primarily because the change of potential associated with an homogeneous electric field is not periodic, and thus does not satisfy the Born-von Karman periodic boundary conditions.

For the purpose of the present tutorial, one should read the section II.C of the above-mentioned paper [Giannozzi1991]. The reader will find in [Gonze1997] and [Gonze1997a] more detailed information about this perturbation, closely related to the ABINIT implementation. There is also an extensive discussion of the Born effective charges by [Ghosez1998].

In order to compute the response of solids to an homogeneous electric field, as implemented in ABINIT, the remaining sections of the respfn_help file should be read. These sections also present the information needed to compute phonons with non-zero q wavevector, which will be the subject of the next section of the present tutorial. The sections to be read are :

• the part of section 1 that had not yet been read
• section 2
• section 4
• and, for completeness, section 5

You are now in the position to compute the full dynamical matrix at Gamma (q=0), including the coupling with an homogeneous electric field. You can copy the file ~abinit/tests/tutorespfn/Input/trf1_5.in in Work_rf1. This is your input file. You should edit it.

View tests/tutorespfn/Input/trf1_5.in View tests/tutorespfn/Refs/trf1_5.out

As for the other RF tests, the changes with respect to the file ~abinit/tests/tutorespfn/Input/trf1_1.in are all gathered in the first part of this file. Unlike the other tests, however, the multi-dataset mode was used, computing from scratch the ground-state properties, then computing the effect of the ddk perturbation, then the effect of all other perturbations (electric field as well as atomic displacements).

The analysis of the output file is even more cumbersome than the previous ones. Let us skip the first dataset. In the dataset 2 section, one perturbation is correctly selected:

 ==>  initialize data related to q vector <==

 The list of irreducible perturbations for this q vector is:
    1)    idir= 1    ipert=   3

================================================================================

--------------------------------------------------------------------------------
 Perturbation wavevector (in red.coord.)   0.000000  0.000000  0.000000
 Perturbation : derivative vs k along direction   1

The analysis of the output for this particular perturbation is not particularly interesting, except for the f-sum rule ratio

 loper3 : ek2=    1.6833336546E+01
          f-sum rule ratio=    1.0028582975E+00

that should be close to 1, and becomes closer to it when ecut is increased, and the sampling of k points is improved. (In the present status of ABINIT, the f-rule ratio is not computed correctly when ecutsm/=0)

In the third dataset section, three irreducible perturbations are considered:

 ==>  initialize data related to q vector <==

 The list of irreducible perturbations for this q vector is:
    1)    idir= 1    ipert=   1
    2)    idir= 1    ipert=   2
    3)    idir= 1    ipert=   4

Much later, the dielectric tensor is given:

  Dielectric tensor, in cartesian coordinates,
     j1       j2             matrix element
  dir pert dir pert     real part    imaginary part

   1    4   1    4    9.7606052146    -0.0000000000
   1    4   2    4    0.0000000000    -0.0000000000
   1    4   3    4    0.0000000000    -0.0000000000

   2    4   1    4    0.0000000000    -0.0000000000
   2    4   2    4    9.7606052146    -0.0000000000
   2    4   3    4    0.0000000000    -0.0000000000

   3    4   1    4    0.0000000000    -0.0000000000
   3    4   2    4    0.0000000000    -0.0000000000
   3    4   3    4    9.7606052146    -0.0000000000

It is diagonal and isotropic, and corresponds to a dielectric constant of 9.7606052146.

Then, the Born effective charges are given, either computed from the derivative of the wavefunctions with respect to the electric field, or computed from the derivative of the wavefunctions with respect to an atomic displacement, as explained in section II of [Gonze1997a]:

  Effective charges, in cartesian coordinates,
  (from electric field response)
  ...

and

  Effective charges, in cartesian coordinates,
  (from phonon response)
  ...

Namely, the Born effective charge of the Al atom is 2.104, and the one of the As atom is -2.127 . The charge neutrality sum rule is not fulfilled exactly. When ecut is increased, and the sampling of k points is improved, the sum of the two charges goes closer to zero.

Finally, the phonon frequencies are computed:

  Phonon wavevector (reduced coordinates) :  0.00000  0.00000  0.00000
 Phonon energies in Hartree :
   2.558878E-06  2.558879E-06  2.558880E-06  1.568559E-03  1.568559E-03
   1.568559E-03
 Phonon frequencies in cm-1    :
-  5.616089E-01  5.616089E-01  5.616092E-01  3.442590E+02  3.442590E+02
-  3.442590E+02

  Phonon at Gamma, with non-analyticity in the
  direction (cartesian coordinates)  1.00000  0.00000  0.00000
 Phonon energies in Hartree :
   2.558878E-06  2.558879E-06  4.068910E-06  1.568559E-03  1.568559E-03
   1.729575E-03
 Phonon frequencies in cm-1    :
-  5.616089E-01  5.616089E-01  8.930225E-01  3.442590E+02  3.442590E+02
-  3.795979E+02

  Phonon at Gamma, with non-analyticity in the
  direction (cartesian coordinates)  0.00000  1.00000  0.00000
 Phonon energies in Hartree :
   2.558878E-06  2.558880E-06  4.068909E-06  1.568559E-03  1.568559E-03
   1.729575E-03
 Phonon frequencies in cm-1    :
-  5.616089E-01  5.616092E-01  8.930223E-01  3.442590E+02  3.442590E+02
-  3.795979E+02

  Phonon at Gamma, with non-analyticity in the
  direction (cartesian coordinates)  0.00000  0.00000  1.00000
 Phonon energies in Hartree :
   2.558879E-06  2.558880E-06  4.068909E-06  1.568559E-03  1.568559E-03
   1.729575E-03
 Phonon frequencies in cm-1    :
-  5.616089E-01  5.616092E-01  8.930223E-01  3.442590E+02  3.442590E+02
-  3.795979E+02

The first few lines discard any effect of the homogeneous electric field, while the next sections consider it along the three cartesian coordinates.

In the present material, the directionality of the electric field has no influence. We note that there are still three acoustic mode, below 1$cm^{-1}$, while the optic modes have the correct degeneracies: two TO modes at 344.3 $cm^{-1}$, and one LO mode at 379.6 $cm^{-1}$.

These values can be compared to experimental (361 $cm^{-1}$, 402 $cm^{-1}$) as well as theoretical (363 $cm^{-1}$, 400 $cm^{-1}$) values (again [Giannozzi1991]). Most of the discrepancy comes from the too low value of ecut. Using ABINIT with ecut=6 hartree gives (358.8 $cm^{-1}$, 389.8 $cm^{-1}$). The remaining of the discrepancy may come partly from the pseudopotentials, that are particularly soft.

The comparison of Born effective charges is also interesting. After imposition of the neutrality sum rule, the Al Born effective charge is 2.116 . The value from Gianozzi et al is 2.17, the experimental value is 2.18. Increasing ecut to 6 hartree in ABINIT gives 2.168.

For the dielectric tensor, it is more delicate. The value from Gianozzi et al is 9.2, while the experimental value is 8.2 . The agreement is not very good, a fact that can be attributed to the LDA lack of polarization-dependence [Gonze1995a]. Still, the agreement of our calculation with the theoretical result is not very good. With ecut=3 hartree, we have 9.76. Changing it to 6 hartree gives 10.40 . A better k point sampling (8x8x8), with ecut=6 hartree, reduces the value to 9.89 . Changing pseudopotentials finally improves the agreement : with the much harder 13al.pspgth and 33as.psphgh pseudopotentials with adequate ecut=16 hartree and 8x8x8 Monkhorst-Pack sampling, we reach a value of 9.37 . This illustrates that the dielectric tensor is a much more sensitive quantity than the others.

6 DFPT calculation of phonon frequencies at non-zero q

The computation of phonon frequencies at non-zero q is actually simpler than the one at Gamma. One must distinguish two cases. Either the q wavevector connects k points that belong to the same grid, or the wavevector q is general. In any case, the computation within the DFPT formalism is more efficient than using the frozen-phonon technique: the use of supercell is completely avoided. For an explanation of this fact, see for example section IV of [Gonze1997].

You can copy the file ~abinit/tests/tutorespfn/Input/trf1_6.in in Work_rf1. This is your input file. You should edit it.

View tests/tutorespfn/Input/trf1_6.in View tests/tutorespfn/Refs/trf1_6.out

As for the other RF tests, the changes with respect to the file ~abinit/tests/tutorespfn/Input/trf1_1.in are all gathered in the first part of this file. The multi-dataset mode is used, computing from scratch the ground-state properties, then computing different dynamical matrices. The run is about 2 minutes on a 2.8 GHz machine. So, you would better leave your computer running, and either read more of the ABINIT documentation (why not the mrgddb_help and the anaddb_help), or make a walk).

The results of this simulation can be compared to those provided in the Gianozzi et al paper. The agreement is rather good, despite the low cut-off energy, and different pseudopotentials.

At X, they get 95$cm^{-1}$, 216$cm^{-1}$, 337$cm^{-1}$ and 393$cm^{-1}$, while we get 92.5$cm^{-1}$, 204.6$cm^{-1}$, 313.9$cm^{-1}$ and 375.9$cm^{-1}$. With ecut=6 hartree, we get 89.7$cm^{-1}$, 212.3$cm^{-1}$, 328.5$cm^{-1}$ and 385.8$cm^{-1}$.

At L, they get 71$cm^{-1}$, 212$cm^{-1}$, 352$cm^{-1}$ and 372$cm^{-1}$, while we get 69.0$cm^{-1}$, 202.5$cm^{-1}$, 332.6$cm^{-1}$ and 352.3$cm^{-1}$. With ecut=6 hartree, we get 68.1$cm^{-1}$, 208.5$cm^{-1}$, 346.7$cm^{-1}$ and 362.6$cm^{-1}$.

At q=(0.1 0 0), we get 31.6$cm^{-1}$, 63.6$cm^{-1}$, 342.0$cm^{-1}$ and 379.7$cm^{-1}$. The acoustic modes tends (nearly-)linearly to zero, while the optic modes are close to their values at Gamma : 344.3 $cm^{-1}$ and 379.6 $cm^{-1}$.

---

This ABINIT tutorial is now finished. You are advised to read now the second tutorial on DFPT