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Second tutorial on DFPT:

Phonon band structures, thermodynamical properties.

In this tutorial you will learn how to post-process the raw data of the Abinit DFPT calculations to get the following physical properties of periodic solids:

  • Interatomic forces constants
  • Phonon band structures
  • Thermodynamical properties

Visualisation tools are NOT covered in this tutorial. Powerful visualisation procedures have been developed in the Abipy context, relying on matplotlib. See the README of Abipy and the Abipy tutorials.

This tutorial should take about 1 hour.

Note

Supposing you made your own installation of ABINIT, the input files to run the examples are in the ~abinit/tests/ directory where ~abinit is the absolute path of the abinit top-level directory. If you have NOT made your own install, ask your system administrator where to find the package, especially the executable and test files.

In case you work on your own PC or workstation, to make things easier, we suggest you define some handy environment variables by executing the following lines in the terminal:

export ABI_HOME=Replace_with_absolute_path_to_abinit_top_level_dir # Change this line
export PATH=$ABI_HOME/src/98_main/:$PATH      # Do not change this line: path to executable
export ABI_TESTS=$ABI_HOME/tests/             # Do not change this line: path to tests dir
export ABI_PSPDIR=$ABI_TESTS/Psps_for_tests/  # Do not change this line: path to pseudos dir

Examples in this tutorial use these shell variables: copy and paste the code snippets into the terminal (remember to set ABI_HOME first!) or, alternatively, source the set_abienv.sh script located in the ~abinit directory:

source ~abinit/set_abienv.sh

The ‘export PATH’ line adds the directory containing the executables to your PATH so that you can invoke the code by simply typing abinit in the terminal instead of providing the absolute path.

To execute the tutorials, create a working directory (Work*) and copy there the input files of the lesson.

Most of the tutorials do not rely on parallelism (except specific tutorials on parallelism). However you can run most of the tutorial examples in parallel with MPI, see the topic on parallelism.

1 Generation of a derivative database

Before beginning, you might consider to work in a different subdirectory as for the other tutorials. Why not create Work_rf2 in $ABI_TESTS/tutorespfn/Input?

Then copy the file trf2_1.abi from $ABI_TESTS/tutorespfn/Input* to Work_rf2:

cd $ABI_TESTS/tutorespfn/Input
mkdir Work_rf2
cd Work_rf2
cp ../trf2_1.abi .

This tutorial starts by the generation of a database, that might be quite time-consuming. We suggest you to start immediately this computation with

abinit trf2_1.abi >& log &

It takes about 1-2 minutes to be completed on a PC 2.8 GHz.

In order to do interatomic force constant (IFC) calculations, and to compute associated phonon band structure and thermodynamical properties, you should first have some theoretical background. Let us assume that you have read the litterature relative to the first tutorial on DFPT. You might find additional material, related to the present section, in [Gonze1997a] -especially section IX-, [Lee1995] and [Baroni2001]. If you haven’t read parts of these references, we strongly advise you take the time to read them now.

In short, the idea is that, in order to compute properties for which the phonon frequencies are needed in the full Brillouin zone, one can use an elaborate Fourier interpolation, so that only few dynamical matrices need to be computed directly. Others will be computed by interpolation. A schematic representation of the different steps required to compute the dynamical matrix in the IBZ and post-process the results with anaddb is given below.

Let us have a look at the input file trf2_1.abi.

The calculation is done for AlAs, the same crystalline material as for the first tutorial on DFPT. Many input parameters are also quite similar, both at the level of the description of the unit cell and for the choice of cut-off energy and k point grid.

Still, this input file is rather complex: in one single run, one produces the Derivative Databases (DDBs) needed for the rest of this tutorial. So, it starts with a ground-state calculation (dataset 1), followed by the computation of the response to the d/dk perturbation (dataset 2), and the response to electric fields, and phonons at Gamma (dataset 3). Datasets 4 to 10 generate the dynamical matrices at 7 q wavevectors, other than Gamma. At present (v8.6), one can only compute one q point per dataset, that is why so many datasets are needed.

Also, the values of these q wavevectors are not determined automatically. They must correspond to the q wavevectors needed by the ANADDB utility (see later), that is, they should form a reduced set of symmetry-inequivalent wavevectors, corresponding to a regularly spaced grid. In principle, they might not include the Gamma point, but it is recommended to have it in the set, in order for the Fourier interpolation not to introduce errors at that important point.

Tip

In order to minimize the number of preliminary non-self-consistent calculations, it is advised to take a q point mesh that is adjusted to the k point mesh used for the electronic structure: all q wavevectors should connect two k point wavevectors from this grid.

Such a set of q wavevectors can be generated straightforwardly by running a GS calculation with kptopt = 1, nshiftk = 1, shiftk = 0 0 0 (to include gamma) and taking the output kpt set file as this qpt set. One might set nstep = 1 and nline = 1, so only one iteration runs, or even nstep = 0 and prtvol = -1, so no real DFT calculation is done.

The input file $ABI_TESTS/tutorespfn/Input/trf2_2.abi is precisely an input file that can be used to generate such a set of k points. Copy it in the present Work_rf2 directly, as well as the accompanying trf2_2.files. Examine these files, then run this calculation (it is very rapid - it won’t hurt the trf2_1 job). The following k point set is obtained:

       kpt    0.00000000E+00  0.00000000E+00  0.00000000E+00
              2.50000000E-01  0.00000000E+00  0.00000000E+00
              5.00000000E-01  0.00000000E+00  0.00000000E+00
              2.50000000E-01  2.50000000E-01  0.00000000E+00
              5.00000000E-01  2.50000000E-01  0.00000000E+00
             -2.50000000E-01  2.50000000E-01  0.00000000E+00
              5.00000000E-01  5.00000000E-01  0.00000000E+00
             -2.50000000E-01  5.00000000E-01  2.50000000E-01

It is, as promised, the same as the q point set in the trf2_1.abi file.

Now, it might be worth to examine in some detail one of the Derivative Database that has been created by the trf2_1 run. We suppose that the file trf2_1o_DS3_DDB has already been created. It corresponds to the third dataset, namely the response to q = 0 and electric field. Open this file, and read the 6.5 section of the respfn help file. Examine the trf2_1o_DS3_DDB file carefully.

Seven other similar files will be generated by the trf2_1 run, containing the same header, but a different 2DTE block. It will be the duty of the MRGDDB utility, next section, to gather all these information and merge them into a single DDB file.

Now, there might be two possibilities: either the trf2_1 run is finished, and you can continue the tutorial with the section 2 about the MRGDDB utility, or the run is not finished. In the latter case, instead of waiting for trf2_1 to be finished, we suggest you to pursue with section 3. You will use as DDB file the one that can be found in $ABI_TESTS/tutorespfn/Refs, with the name trf2_3.ddb.abo, instead of the one that would result from the section 2. Copy this file to the present directory, then go to section section 3 of this tutorial. You might come back to section 2 afterwards.

2 Manipulation of the derivative databases (the MRGDDB utility)

The use of the MRGDDB utility is described in its help file. Please, read it carefully now.

Use MRGDDB to create the merge DDB from the eight DDB’s corresponding to datasets 3 to 10 of the trf2_1 job, containing the dynamical matrices for the 8 q points, as well as the response to the electric field (dielectric tensor and Born effective charges). Name the new DDB trf2_3.ddb.abo.

Note

Including also the DDB from dataset 1 won’t hurt (it contains the forces and stresses), but is not needed for the computation of phonon band structure, interatomic force constants, and thermodynamical properties.

File $ABI_TESTS/tutorespfn/Input/trf2_3.abi is an example of input file for MRGDDB.

You can copy it in the Work_rf2 directory, and run the merge as follows:

mrgddb < trf2_3.abi

Note the chevron in the call.

3 Analysis of the derivative databases

An introduction to the use of the ANADDB utility is described in its help file. Please, read it carefully.

This ANADDB utility is able to perform many different tasks, each governed by a selected set of input variables, with also some input variables common to many of the different tasks. The list of tasks to be done in one run is governed by different flags. Here is the list of flags:

Please, take some time to read the description of each of these flags. Note that some of these flags might be required to allow to run another task. In this tutorial, we will focus on the flags ifcflag and thmflag.

4 The computation of interatomic force constants

You can copy the files trf2_4.abi and trf2_4.files from $ABI_TESTS/tutorespfn/Input to the Work_rf2 directory. Open the file trf2_4.abi. Note that anaddb use the old format to start calculations which means it needs a files file. Also note that ifcflag is activated.

Related input variables can be split in three groups. The first group of variables define the grid of q wavevectors:

Unfortunately, the names of input variables and their meaning is not exactly the same as the names used to generate the k points in ABINIT. This is a shame, a remnant of history. Please read carefully the documentation that describes these input variables.

The second group of variables allows to impose the acoustic sum rule on the dynamical matrices and the charge neutrality on Born effective charges before proceeding with the analysis:

Please, read carefully the explanation for these input variables.

Finally, a third group of variables is related specifically to the analysis of the IFC:

Here also, spend some time to read the associated documentation.

Now, you should issue:

anaddb < trf2_4.files > trf2_4.log

It will last only a few seconds.

The file trf2_4.abo contains the list of interatomic force constants, as well as some analysis.

Open this file and find the following paragraph:

 Analysis of interatomic force constants

 Are given : column(1-3), the total force constant
       then  column(4-6), the Ewald part
       then  column(7-9), the short-range part
 Column 1, 4 and 7 are related to the displacement
       of the generic atom along x,
 column 2, 5 and 8 are related to the displacement
       of the generic atom along y,
 column 3, 6 and 9 are related to the displacement
       of the generic atom along z.

The interatomic force constants are output for the nuclei specified by the input variable atifc. Here, only atom 1 is considered. The IFCs with respect to the other nuclei is given, by order of increasing distance. For each pair of nuclei involving atom 1, there is first the output of the IFCs in cartesian coordinates, as well as their decomposition into an Ewald and a short-range part, then, the analysis with respect to a local system of coordinate. The latter is chosen such that it diagonalizes the IFC tensor, in case of the self-force constant, and in the other cases, the first vector is the vector joining the two nuclei, in order to decompose the IFC into a longitudinal and a transverse component.

5 Computation of phonon band structures with efficient interpolation

You can copy the files trf2_5.abi and trf2_5.files from $ABI_TESTS/tutorespfn/Input to the Work_rf2 directory. Then open trf2_5.abi.

Note that ifcflag is again activated. Indeed, in order to compute a phonon band structure using the Fourier interpolation, the IFCs are required. This is why the two first groups of variables, needed to generate the IFCs are still defined. The third group of variables is now restricted to dipdip only.

Then, come the input variables needed to define the list of q wavevectors in the band structure:

  • eivec: flag to turn on the analysis of phonon eigenvectors
  • nph1l: number of q-points for phonon interpolation
  • qph1l: list of q-points for phonon interpolation
  • nph2l: number of q-directions for LO-TO correction
  • qph2l: list of q-directions for LO-TO correction

Now, you should issue:

anaddb < trf2_5.files > trf2_5.log

It will last only a few seconds.

The file trf2_5.abo contains the list of eigenvalues, for all the needed q-wavevectors. You can iopen it, and have a look at the different sections of the file. Note that the interatomic force constants are computed (they are needed for the Fourier interpolation), but not printed.

Please, open also the other output file, named trf2_5_B2EPS.freq. It contains the frequencies, in a format suitable for graphical output, using the program band2eps (the latter should be more documented, and will not be described in the present tutorial).

You can copy the files trf2_6.abi and trf2_6.files to the Work_rf2 directory. Note that, like anaddb, band2eps use the old format using the files file. Then, issue

band2eps < trf2_6.files > trf2_6.log

The file trf2_6.abo.eps has been produced. It is an .eps file (eps stand for Encapsulated PostScript). You can use the program ghostview to vizualize it. The command to issue will depend on the way you have configured your machine, but the following might perhaps do the work:

gv trf2_6.abo.eps

You should see a nice phonon band structure for AlAs. Well, not so nice, after all, because there are two strange dips for the highest phonon band, at the Gamma point. This is due to the lack of LO-TO splitting for the ANADDB treatment of the first list of vector. The correct phonon band structure is:

You can correct the LO-TO splitting by the following little hack.

Open the file trf2_5_B2EPS.freq, and note that the value of the frequency, in the sixth column, has a discontinuity exactly for the Gamma point (the three first columns give the k point coordinates), that is, at lines 1 and 31:

 0.0000000000E+00  0.0000000000E+00  0.0000000000E+00  0.1568561346E-02  0.1568561346E-02  0.1568561346E-02

Replace these values (sixth column, line 1 and 31) by the correct value, including the LO-TO splitting, that you can find in the file trf2_5.abo, at the end, second list of vector. That is, the lines 1 and 31 should now read:

 0.000000E+00  0.000000E+00  0.000000E+00  1.568561E-03  1.568561E-03  1.730570E-03

Now, run band2eps again. Your phonon band structure should be perfect!

It can be compared with the AlAs phonon band structure published in [Giannozzi1991].

Of course, one should make a convergence study, on the k and q point grids (separately!), as well as on the energy cut-off, and also test LDA and GGA… But this is left to the user! You can have a look at the paper [Petretto2018] for a careful analysis of phonon dispersion convergence with Abinit.

Plotting phonon bands with AbiPy

If AbiPy is installed on your machine, you can use the abiopen script with the --expose option to visualize the phonon band structure stored in the PHBST.nc file produced by anaddb. For instance:

abiopen.py trf2_5_PHBST.nc --expose --seaborn=talk

produces the following plot without LO-TO splitting:

(left) Phonon bands without LO-TO splitting (right) Plot with band connection estimated from the overlap of the eigenvectors at adjacent q-points

Alternatively, we can start from the DDB file and use the abiview script. In this case, AbiPy will generate the anaddb input file with all the variables required to handle the plotting of the LO-TO splitting, invoke anaddb for us and finally plot the results. All of this with just two lines:

# Copy the tutorial output file to have the correct file extension (DDB)
# otherwise abiview does not know how to handle our file.
cp trf2_3.ddb.abo trf2_3_DDB

abiview.py ddb trf2_3_DDB -sns=talk

We can also compare our results with the phonon band structure available on the materials project .

First of all, let’s find the materials project identifier associated to this particular phase of AlAs. Of course, one could use the materials project web interface but we can also do it from the shell by just passing our Abinit input file to the abistruct script:

abistruct.py mp_match trf2_1.abi

# Found 1 structures in Materials Project database (use `verbose` to get further info)

######################### abivars input for mp-2172 #########################
# Full Formula (Al1 As1)
# Reduced Formula: AlAs
# abc   :   4.054377   4.054377   4.054377
# angles:  60.000000  60.000000  60.000000
#
# Spglib space group info (magnetic symmetries are not taken into account).
# Spacegroup: F-43m (216), Hall: F -4 2 3, Abinit spg_number: None
# Crystal_system: cubic, Lattice_type: cubic, Point_group: -43m
#
#   Idx  Symbol    Reduced_Coords              Wyck      EqIdx
# -----  --------  --------------------------  ------  -------
#     0  Al        +0.00000 +0.00000 +0.00000  a             0
#     1  As        +0.25000 +0.25000 +0.25000  d             1

 natom 2
 ntypat 2
 typat 1 2
 znucl 13 33
 xred
    0.0000000000    0.0000000000    0.0000000000
    0.2500000000    0.2500000000    0.2500000000
 acell    1.0    1.0    1.0
 rprim
    6.6351943530    0.0000000000    3.8308312587
    2.2117314510    6.2557212277    3.8308312587
    0.0000000000    0.0000000000    7.6616624984

AbiPy found one entry in the MP database that matches the structure given in our input file and has generated the corresponding input file. Now we know that this phase of AlAs corresponds to mp-2172 and we can look at the phonon band structure computed by [Petretto2018a] at https://materialsproject.org/materials/mp-2172/

Tip

For further information on the AbiPy API, please consult the DdbFile notebook . To learn how to automate DFPT calculations with Python, see this jupyter notebook.

6 Thermodynamical properties

We will give only a very short example of the use of ANADDB to compute thermodynamical properties. This is because this part of ANADDB is likely the farthest from a clean, stable, usage. By exploring the input variables, the user should be able to produce figures and data like the ones for SiO2 quartz and stishovite, published in [Lee1995].

You can copy the files trf2_7.abi from $ABI_TESTS/tutorespfn/Input to Work_rf2 and have a look at them. The same DDB as for trf2_4 and trf2_5 is used, namely trf2_3.ddb.abo.

The following additional input variables are present:

Examine the input file, the input variables, then run anaddb as usual. Then, open the output file. You should be able to find the crucial section:

 # At  T     F(J/mol-c)     E(J/mol-c)     S(J/(mol-c.K)) C(J/(mol-c.K)) Omega_mean(cm-1)
 # (A mol-c is the abbreviation of a mole-cell, that is, the
 #  number of Avogadro times the atoms in a unit cell)
  2.000E+01  8.1406018E+03  8.1484316E+03  3.9149240E-01  1.4057917E+00  7.2615609E+01
  4.000E+01  8.1084535E+03  8.2384509E+03  3.2499352E+00  7.8730812E+00  9.4376064E+01
  6.000E+01  8.0007856E+03  8.4587550E+03  7.6328229E+00  1.3972961E+01  1.1313071E+02
  8.000E+01  7.8007781E+03  8.7924243E+03  1.2395577E+01  1.9312368E+01  1.3603493E+02
  1.000E+02  7.5044659E+03  9.2281056E+03  1.7236397E+01  2.4166544E+01  1.5713727E+02
  1.200E+02  7.1116611E+03  9.7549590E+03  2.2027483E+01  2.8405448E+01  1.7346361E+02
  1.400E+02  6.6242892E+03  1.0359674E+04  2.6681323E+01  3.1951237E+01  1.8536427E+02
  1.600E+02  6.0456925E+03  1.1028647E+04  3.1143464E+01  3.4844495E+01  1.9397760E+02
  1.800E+02  5.3800094E+03  1.1749746E+04  3.5387425E+01  3.7181668E+01  2.0029176E+02
  2.000E+02  4.6317003E+03  1.2512909E+04  3.9406045E+01  3.9067753E+01  2.0500949E+02

There, one finds, the phonon free energy, the phonon internal energy, the phonon entropy and the phonon heat capacity. The atomic temperature factors can also be computed. An example is presented in v5[22]

Important

Do not forget that we are working in the harmonic approximation; beyond some temperature, anharmonic effects will have a sizeable contributions.