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XG, JWZ

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.

WARNING : This tutorial has been updated by removing the use of the ANADDB input variable brav in trf2_4.abi, trf2_5.abi, and trf2_7.abi, with the accompanying modifications in the text of the tutorial. Also, due to the change of default values for rfdir and rfatpol in ABINITv9.8 with respect to the prior versions of ABINIT, the example input files from this tutorial will not work with such prior versions of ABINIT. Please, use ABINITv9.8 or more recent versions of ABINIT, or adjust the values of rfdir and rfatpol. See point A.3 of the v9.8 release notes.

[TUTORIAL_README]

0 Generation of necessary input wavefunctions

Before beginning, you might consider working in a different subdirectory as for the other tutorials, in order to help keep track of all the files. Why not create Work_rf2 in $ABI_TESTS/tutorespfn/Input?

As the DFPT equations at higher order always start from input wavefunctions obtained at lower order, we begin by generating the ground state wavefunctions and DDK wavefunctions for our system, AlAs. The ground state wavefunctions are needed for the phonon and electric field perturbations, and the electric field perturbation needs the DDK wavefunctions as well, as explained in the first tutorial on DFPT.

Copy 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

Execute this file with

abinit trf2_1.abi >& log &

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

{% dialog tests/tutorespfn/Input/trf2_1.abi %}

This input file consists of two datasets--the first generates the ground state wavefunctions, with tight convergence: these are output in the file trf2_1o_DS1_WFK. The second dataset uses these wavefunctions to generate the DDK wavefunctions, which are output in the files trf2_1o_DS2_1WF7, trf2_1o_DS2_1WF8, and trf2_1o_DS2_1WF9. Depending on the details of your build, these files might be appended with .nc, if they are produced in netcdf format.

A note on the naming convention--the DDK files here are named 1WF7, 1WF8, and 1WF9. The "1" signifies a first-order wavefunction. Then, ABINIT numbers each perturbation, such that the phonons are numbered 1...3N, for an N atom system. Here N = 2 for AlAs. The DDK perturbations are numbered 3N+1, 3N+2, and 3N+3, for the three cartesian directions, giving in this case 7, 8, and 9.

1 Generation of a derivative database

With the necessary input wavefunctions generated, we are ready to compute the derivative database.

Copy the file trf2_2.abi from $ABI_TESTS/tutorespfn/Input* into your working directory:

cp ../trf2_2.abi .

Before you run it, you will have to name the input wavefunctions that you prepared in step 0 appropriately, so ABINIT can find them.

Examine the input file you are about to run:

{% dialog tests/tutorespfn/Input/trf2_2.abi %}

Notice the lines ndtset 8, getddk 98, and getwfk 99. Since we will be looping over multiple datasets (8, in fact), in order to compute multiple perturbations in a single file, we use multi dataset mode. But this requires that any file read in be given dataset numbers in the same format. We avoid low numbers because those are the new datasets we are about to run; it is thus convenient to relabel the ground state and DDK wavefunctions with the dataset labels 99 and 98. Thus you can rename with

mv trf2_1o_DS1_WFK trf2_2o_DS99_WFK;
mv trf2_1o_DS2_1WF7 trf2_2o_DS98_1WF7;
mv trf2_1o_DS2_1WF8 trf2_2o_DS98_1WF8;
mv trf2_1o_DS2_1WF9 trf2_2o_DS98_1WF9

Append each new file with ".nc" if the original files were ".nc" files.

Now you can execute the file with

abinit trf2_2.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 literature relative to the first tutorial on DFPT. You might find additional material, related to the present section, in cite:Gonze1997a -especially section IX-, cite:Lee1995 and cite:Baroni2001. If you haven't read parts of these references, we strongly advise you to 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.

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. The input file trf2_2.abi consists of a loop over a small set of phonons, and all other phonons will be determined later by interpolation. Moreover, the electric field perturbation is also a long wavelength, hence Gamma phonon perturbation, and can be computed at the same time. Thus our first dataset in trf2_2.abi is the Gamma phonon and the electric field both, and the other datasets are the additional nonzero q point phonons we wish to consider explicitly.

The values of the phonon q wavevectors are not determined arbitrarily. 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 need 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 conveniently, and automatically, in the input file, using qptopt, ngqpt, nshiftq, and shiftq, quite analogously to what can be done with the k point grid. Look again at the file trf2_2.abi and note that ngqpt 4 4 4 is used, mirroring ngkpt, but with nshiftq 1 and shiftq 0 0 0 to produce an unshifted set, which will include the Gamma point.

!!! tip Never forget to use nqpt 1, whenever qpt is being input, either directly or implicitly. If nqpt is absent in the input file, it defaults to zero which leads to qpt being ignored.

This approach is conveniently combined with ndtset and iqpt to step automatically through the generated q points, as is done in our file with ndtset 8 and iqpt: 1 iqpt+ 1. This raises the question, though, of how one would know in advance that there will be 8 q points in the current set. For this one can use abitk, a convenient tool built along with ABINIT and used to examine output files in netcdf format. Run it on our original ground state file as:

abitk ibz trf2_1o_DS1_GSR.nc --ngkpt 4 4 4 --shiftk 0 0 0

and you should obtain a list of the k points (equivalently q points) generated with a 4 4 4 unshifted mesh, for the current input structure. There are 8 of them, and the set includes the Gamma point, as desired.

At this point, it might be worth examining in some detail one of the Derivative Databases that has been created by the trf2_2 run. We suppose that the file trf2_2o_DS1_DDB has already been created. It corresponds to the first dataset, namely the response to q = 0 and electric field. Open this file, and read the help:respfn#ddb of the respfn help file. Examine the trf2_2o_DS1_DDB file carefully.

Seven other similar files will be generated by the trf2_2 run, containing the same header, but a different 2DTE block. It will be the job of the MRGDDB utility (described in the next section) to gather all these files and merge them into a single DDB file.

2 Manipulation of the derivative databases (the MRGDDB utility)

The use of the MRGDDB utility is described in its help:mrgddb, which you should read carefully now.

Use MRGDDB to create the merged DDB from the eight DDB's corresponding to the datasets of the trf2_2 job, which contain 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.

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

{% dialog tests/tutorespfn/Input/trf2_3.abi %}

You can copy it in the Work_rf2 directory, noting that if your DDB files were produced in netcdf format, you will need to include properly the .nc suffixes in the file names listed in trf2_3.abi. Then run the merge as follows:

mrgddb < trf2_3.abi

Note the chevron in the call: MRGDDB reads from standard input by default, so you can make it read instead from a file using the < operator.

3 Analysis of the derivative databases

An introduction to the use of the ANADDB utility is described in its help:anaddb. 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 the commonly used 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 anaddb:ifcflag and anaddb:thmflag.

4 The computation of interatomic force constants

You can copy the file trf2_4.abi from $ABI_TESTS/tutorespfn/Input to the Work_rf2 directory. Examine the file trf2_4.abi. Note that anaddb:ifcflag is activated. It also specifies the input ddb file, and output file naming (anaddb:ddb_filepath, anaddb:output_file, and anaddb:outdata_prefix).

{% dialog tests/tutorespfn/Input/trf2_4.abi %}

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 are not exactly the same as the names used to generate the k and q 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.abi > trf2_4.abi.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.

{% dialog tests/tutorespfn/Refs/trf2_4.abo %}

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 anaddb: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 file trf2_5.abi from $ABI_TESTS/tutorespfn/Input to the Work_rf2 directory. Then open trf2_5.abi.

{% dialog tests/tutorespfn/Input/trf2_5.abi %}

Note that anaddb: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 anaddb:dipdip only.

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

  • anaddb:eivec: flag to turn on the analysis of phonon eigenvectors
  • anaddb:nqpath: number of q-points defining the band structure path
  • anaddb:qpath: list of q-points defining the band structure path
  • anaddb:ndivsm: number of divisions in the shortest band structure path segment
  • anaddb:nph2l: number of q-directions for LO-TO correction
  • anaddb:qph2l: list of q-directions for LO-TO correction

Now, you should issue:

anaddb trf2_5.abi > trf2_5.log

It will last only a few seconds. The variables anaddb:nqpath, anaddb:qpath, and anaddb:ndivsm have analogous counterparts in electronic band structure calculations, see the 3rd basic tutorial.

The file trf2_5.abo contains the list of eigenvalues, for all the needed q-wavevectors. You can open 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.

{% dialog tests/tutorespfn/Refs/trf2_5.abo %}

Please, open also the other output file, named trf2_5o_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, trf2_6.files and trf2_6.in_freq to the Work_rf2 directory. Note that band2eps uses the old format using the files file. Then, issue

band2eps < trf2_6.files > trf2_6.log

{% dialog tests/tutorespfn/Input/trf2_6.files tests/tutorespfn/Input/trf2_6.abi %}

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 visualize 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_5o_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 cite: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 cite: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 commands! First, copy the ddb file to a file with extension DDB, then, execute abiview.py:

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 cite:Petretto2018a at https://materialsproject.org/materials/mp-2172/

!!! tip

For further information on the AbiPy API, please consult the |DdbFileNb| .
To learn how to automate DFPT calculations with Python, see
[this jupyter notebook](https://nbviewer.jupyter.org/github/abinit/abitutorials/blob/master/abitutorials/dfpt/lesson_dfpt.ipynb).

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 cite:Lee1995.

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

{% dialog tests/tutorespfn/Input/trf2_7.abi %}

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 test:v5_22

!!! important

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