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

This lesson should take about 2 hours.

This file is distributed under the terms of the GNU General Public License, see ~abinit/COPYING or http://www.gnu.org/copyleft/gpl.txt .

For the initials of contributors, see ~abinit/doc/developers/contributors.txt .

- 1 The ground-state geometry of AlAs.
- 2 Frozen-phonon calculation of a second derivative of the total energy.
- 3 Response-function calculation of a second derivative of the total energy.
- 4 Response-function calculation of the dynamical matrix at Gamma.
- 5 Response-function calculation of the effect of an homogeneous electric field.
- 6 Response-function calculation of phonon frequencies at non-zero q.

*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 ? *

NOTE : 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-response ones), but ~abinit/tests/tutorespfn . This will be the case for all the response-function based part of the tutorial.

The file ~abinit/tests/tutorespfn/Input/trf1_x.files lists the file names and root names. You can copy it in the Work_rf1 directory (and change it, as usual). 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 is the first time that you encounter this situation in the tutorials.

You can also copy the file ~abinit/tests/tutorespfn/Input/trf1_1.in in Work_rf1. This is your input file. You should edit it, read it carefully, and, because of the use of two types of atoms, have a look at the following input variables :

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 response-function 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 problems with some tutorial examples was mentioned briefly in a side note to the answer to question 1 of lesson 1. 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 convergence 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), and obtain the following value for the energy, in the final echo section :

etotal -9.7626837450E+00However, we will rely later 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= -9.76268374500289E+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 response functions 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.
**So, in the corresponding "files" file for all the following runs, at third line,
pay attention TO KEEP "trf1_1i", even if you change
the root name for output files (fourth line) to "trf1_2o" or "trf1_3o", as well as
the first, second and fifth lines of this file.**

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 response function 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 response-function 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. 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.

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 0.005007930232 0.002526304574 0.002526304574 2 -0.005007868293 -0.002526274885 -0.002526274885 ... >>>>>>>>> Etotal= -9.76268124105780E+00The change of reduced coordinate 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 (-9.76268374500102E+00 hartree) of the previous run, and the perturbed value (-9.76268124105590E+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 5.00791024 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 0.005007930232 (hartree). At first order, this quantity is the product of the 2DTE by the reduced coordinate difference. The estimate of the 2DTE is thus 5.007930232 hartree. The agreement with the other estimate is rather good (2.10^-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 0.010016299779 0.005097509981 0.005097509981 2 -0.010016176675 -0.005097455174 -0.005097455174 ...From these data, taking into account that the perturbation was twice stronger, the same procedure as above leads to the values 5.00800270 hartree (from finite difference of energy) and 5.009149889 hartree (from finite difference of forces, the value 0.010016299779 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 0.0001104 Ha, which divided by three, and subtracted from 5.00789230 hartree, gives 5.0078555 hartree. The same higher-order procedure for force estimates gives 5.0078552 hartree. So, the agreement between total-energy estimate and force estimate of the 2DTE can be observed up to the 7th digit, inclusive.

Before comparing this result with the 2DTE directly computed from the response-function 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, for example for the doubled displacement :

rms dE/dt= 7.1249E-03; max dE/dt= 1.0016E-02; dE/dt below (all hartree) 1 0.010016299779 0.005097509981 0.005097509981 2 -0.010016176675 -0.005097455174 -0.005097455174show a small, but non-negligible difference between the two atoms. 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

cartesian forces (hartree/bohr) at end: 1 -0.00000421123276 -0.00047199792687 -0.00047199792687 2 0.00000421123276 0.00047199792687 0.00047199792687This 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) tend 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 to reduce this effect.

We now compute the second derivative of the total energy with respect to the same atomic displacement, using the response-function 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 edit 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.

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

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 four different files :

- trf1_3.log (the log file)
- trf1_3.out (the output file)
- trf1_3o_1WF1 (the 1st-order wavefunction file)
- trf1_3o_DDB (the derivative database)

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. 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 6.5156051312863 -1.464E+01 1.146E-02 1.947E+02 ETOT 2 5.0216331638978 -1.494E+00 9.267E-04 2.027E+00 ETOT 3 5.0082678671217 -1.337E-02 4.772E-06 7.929E-02 ETOT 4 5.0078677958105 -4.001E-04 1.980E-07 2.712E-03 ETOT 5 5.0078558860285 -1.191E-05 6.103E-09 5.074E-05 ETOT 6 5.0078557520180 -1.340E-07 1.258E-10 9.613E-06 ETOT 7 5.0078557017091 -5.031E-08 2.768E-11 3.841E-07 ETOT 8 5.0078557001188 -1.590E-09 2.983E-12 6.089E-09From these data, you can see that the 2DTE determined by the response-function technique is in excellent agreement with the higher-order finite-difference formula for the 2DTE, determined in the previous section : 5.0078555 hartree from the energy differences, and 5.0078552 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 6.4 of the respfn_help file.

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

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.

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 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. 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 : 2.586632E-06 2.590723E-06 2.614440E-06 1.568560E-03 1.568560E-03 1.568560E-03 Phonon frequencies in cm-1 : - 5.677000E-01 5.685980E-01 5.738033E-01 3.442590E+02 3.442590E+02 - 3.442590E+02

*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 mode, 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, by P. Giannozzi, S. de Gironcoli, P. Pavone and S. Baroni, Phys. Rev. B 43, 7231 (1991) , 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.

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 P. Giannozzi, S. de Gironcoli, P. Pavone and S. Baroni, Phys. Rev. B 43, 7231 (1991) . The reader will find in X. Gonze, Phys. Rev. B 55, 10337 (1997) and X. Gonze and C. Lee, Phys. Rev. B 55, 10355 (1997) more detailed information of this perturbation, closely related to the ABINIT implementation. There is also an extensive discussion of the Born effective charges by Ph. Ghosez, J.-P. Michenaud and X. Gonze, Phys. Rev. B 58, 6224 (1998).

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. 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 1The 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= 1.0028582985E+00that 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= 4Much 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 9.7606048428 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 9.7606048428 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 9.7606048428 0.0000000000It is diagonal and isotropic, and corresponds to a dielectric constant of 9.7606048428 .

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 X. Gonze, Phys. Rev. B55, 10355 (1997) :

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 : 2.586632E-06 2.590723E-06 2.614440E-06 1.568560E-03 1.568560E-03 1.568560E-03 Phonon frequencies in cm-1 : - 5.677000E-01 5.685980E-01 5.738033E-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 : 2.590670E-06 2.590723E-06 4.101029E-06 1.568560E-03 1.568560E-03 1.729575E-03 Phonon frequencies in cm-1 : - 5.685864E-01 5.685980E-01 9.000719E-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 : 2.586632E-06 2.614440E-06 4.088526E-06 1.568560E-03 1.568560E-03 1.729575E-03 Phonon frequencies in cm-1 : - 5.677000E-01 5.738033E-01 8.973277E-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 : 2.590723E-06 2.610339E-06 4.088540E-06 1.568560E-03 1.568560E-03 1.729575E-03 Phonon frequencies in cm-1 : - 5.685980E-01 5.729031E-01 8.973308E-01 3.442590E+02 3.442590E+02 - 3.795979E+02The 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 1cm^-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 the Gianozzi et al paper mentioned above). 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 ( X. Gonze, Ph. Ghosez and R. Godby, Phys. Rev. Lett. (1995)). 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.

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 response-function 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 X. Gonze, Phys. Rev. B55, 10337 (1997).

You can copy the file ~abinit/tests/tutorespfn/Input/trf1_6.in in Work_rf1. This is your input file. You should edit it. 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 95cm^-1, 216cm^-1, 337cm^-1 and 393cm^-1, while we get 92.5cm^-1, 204.6cm^-1, 313.9cm^-1 and 375.9cm^-1. With ecut=6 hartree, we get 89.7cm^-1, 212.3cm^-1, 328.5cm^-1 and 385.8cm^-1.

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

At q=(0.1 0 0), we get 31.6cm^-1, 63.6cm^-1, 342.0cm^-1 and 379.7cm^-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 response functions