# Lesson on elastic properties¶

## Elastic and piezoelectric properties.¶

This lesson shows how to calculate physical properties related to strain, for an insulator and a metal:

• the rigid-atom elastic tensor
• the rigid-atom piezoelectric tensor (insulators only)
• the internal strain tensor
• the atomic relaxation corrections to the elastic and piezoelectric tensor

You should complete lessons RF1 and RF2 to introduce the density-functional perturbation theory (DFPT) features of ABINIT before starting this lesson. You will learn to use additional DFPT features of ABINIT, and to use relevant parts of the associated codes Mrgddb and Anaddb.

This lesson should take about two hours.

## 1 The ground-state geometry of (hypothetical) wurtzite AlAs¶

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

You should copy the files ~abinit/tests/tutorespfn/Input/telast_1.files and telast_1.in into Work_elast. You may wish to start the calculation (less than one minute on a standard 3GHz machine) before you read the following. You should open your input file telast_1.in with an editor and examine it as you read this discussion.

The hypothetical wurtzite structure for AlAs retains the tetrahedral coordination of the atoms of the actual zincblende structure of AlAs, but has a hexagonal lattice. It was chosen for this lesson because the atomic positions are not completely determined by symmetry. Both the atomic positions and the lattice constants should be optimized before beginning DFPT calculations, especially those related to strain properties. While GS structural optimization was treated in lessons 1-3, we are introducing a few new features here, and you should look at the following new input variables which will be discussed below:

There are two datasets specified in telast_1.in. First, let us examine the common input data. We specify a starting guess for acell, and give an accurate decimal specification for rprim. The definition of the atom types and atoms follows lesson RF1. The reduced atomic positions xred are a starting approximation, and will be replaced by our converged results in the remaining input files, as will acell.

We will work with a fixed plane wave cutoff ecut (=6 Ha), but introduce ecutsm (0.5 Ha)as in lesson 3 to smear the cutoff, which produces smoothly varying stresses as the lattice parameters are optimized. We will keep the same value of ecutsm for the DFPT calculations as well, since changing it from the optimization run value could reintroduce non-zero forces and stresses. For the k-point grid, we must explicitly specify shiftk since the default value results in a grid shifted so as to break hexagonal symmetry. The RF strain calculations check this, and will exit with an error message if the grid does not have the proper symmetry. The self-consistency procedures follow lesson RF1.

Dataset 1 optimizes the atomic positions keeping the lattice parameters fixed, setting ionmov=2 as in lesson 1. The optimization steps proceed until the maximum force component on any atom is less than tolmxf. It is always advised to relax the forces before beginning the lattice parameter optimization. Dataset 2 optimizes the lattice parameters with optcell=2 as in lesson 3. However, lesson 3 treated cubic Si, and the atom positions in reduced coordinates remained fixed. In the present, more general case, the reduced atomic coordinates must be reoptimized as the lattice parameters are optimized. Note that it is necessary to include getxred = -1 so that the second dataset is initialized with the relaxed coordinates. Coordinate and lattice parameter optimizations actually take place simultaneously, with the computed stresses at each step acting as forces on the lattice parameters. We have introduced strfact which scales the stresses so that they may be compared with the same tolmxf convergence test that is applied to the forces. The default value of 100 is probably a good choice for many systems, but you should be aware of what is happening.

From the hexagonal symmetry, we know that the positions of the atoms in the a-b basal plane are fixed. However, a uniform translation along the c axis of all the atoms leaves the structure invariant. Only the relative displacement of the Al and As planes along the c axis is physically relevant. We will fix the Al positions to be at reduced c-axis coordinates 0 and ½ (these are related by symmetry) by introducing natfix and iatfix to constrain the structural optimization. This is really just for cosmetic purposes, since letting them all slide an arbitrary amount (as they otherwise would) won’t change any results. However, you probably wouldn’t want to publish the results that way, so we may as well develop good habits.

Now we shall examine the results of the structural optimization run. As always, we should first examine the log file to make sure the run has terminated cleanly. There are a number of warnings, but none of them are apparently serious. Next, let us edit the output file, telast_1.out.

The first thing to look for is to see whether Abinit recognized the symmetry of the system. In setting up a new data file, it’s easy to make mistakes, so this is a valuable check. We see

DATASET    1 : space group P6_3 m c (#186); Bravais hP (primitive hexag.)


which is correct. Next, we confirm that the structural optimization converged. The following lines from dataset 1 and dataset2 tell us that things are OK:

At Broyd/MD step   4, gradients are converged :
max grad (force/stress) = 1.0670E-08 < tolmxf= 1.0000E-06 ha/bohr (free atoms)

At Broyd/MD step  11, gradients are converged :
max grad (force/stress) = 7.8147E-08 < tolmxf= 1.0000E-06 ha/bohr (free atoms)


We can also confirm that the stresses are relaxed:

Cartesian components of stress tensor (hartree/bohr^3)
sigma(1 1)= -3.76811543E-10  sigma(3 2)=  0.00000000E+00
sigma(2 2)= -3.76811542E-10  sigma(3 1)=  0.00000000E+00
sigma(3 3)=  7.81471561E-10  sigma(2 1)=  0.00000000E+00


Now would be a good time to copy telast_2.in and telast_2.files into your working directory, since we will use the present output to start the next run. Locate the optimized lattice parameters and reduced atomic coordinates near the end of telast_1.out:

     acell2   7.5389647789E+00  7.5389647789E+00  1.2277795511E+01 Bohr

xred2   3.3333333333E-01  6.6666666667E-01  0.0000000000E+00
6.6666666667E-01  3.3333333333E-01  5.0000000000E-01
3.3333333333E-01  6.6666666667E-01  3.7608587611E-01
6.6666666667E-01  3.3333333333E-01  8.7608587611E-01


With your editor, copy and paste these into telast_2.in at the indicated places in the “Common input data” area. Be sure to change acell2 and xred2 to acell and xred since these common values will apply to all datasets in the next set of calculations.

## 2 Response-function calculations of several second derivatives of the total energy¶

We will now compute second derivatives of the total energy (2DTE’s) with respect to all the perturbations we need to compute elastic and piezoelectric properties. You may want to review the first paragraphs of the respfn help file which you studied in lesson RF1. We will introduce only one new input variable for the strain perturbation,

The treatment of strain as a perturbation has some subtle aspects. It would be a good idea to read Metric tensor formulation of strain in density-functional perturbation theory, by D. R. Hamann, Xifan Wu, Karin M. Rabe, and David Vanderbilt [Hamann2005] especially Sec. II and Sec. IV. We will do all the RF calculations you learned in lesson RF1 together with strain, so you should review the variables

It would be a good idea to copy telast_2.files into Work_elast and start the calculation while you read (less than 2 minutes on a standard 3GHz machine). Look at telast_2.in in your editor to follow the discussion, and double check that you have copied acell and xred as discussed in the last section.

This has been set up as a self-contained calculation with three datasets. The first is simply a GS run to obtain the GS wave functions we will need for the DFPT calculations. We have removed the convergence test from the common input data to remind ourselves that different tests are needed for different datasets. We set a tight limit on the convergence of the self-consistent potential with tolvrs. Since we have specified nband=8, all the bands are occupied and the potential test also assures us that all the wave functions are well converged. This issue will come up again in the section on metals. We could have used the output wave functions telast_1o_DS2_WFK as input for our RF calculations and skipped dataset 1, but redoing the GS calculation takes relatively little time for this simple system .

Dataset 2 involves the calculation of the derivatives of the wave functions with respect to the Brillouin-zone wave vector, the so-called ddk wave functions. Recall that these are auxiliary quantities needed to compute the response to the electric field perturbation and introduced in lesson RF1. It would be a good idea to review the relevant parts of section 1 of the respfn_help file.

Examining this section of telast_2.in, note that electric field as well as strain are uniform perturbations, only are defined for qpt= 0 0 0. rfelfd= 2 specifies that we want the ddk calculation to be performed, which requires iscf= -3. The ddk wave functions will be used to calculate both the piezoelectric tensor and the Born effective charges, and in general we need them for k derivatives in all three (reduced) directions, rfdir= 1 1 1. Since there is no potential self-consistency in the ddk calculations, we must specify convergence in terms of the wave function residuals using tolwfr.

Finally, dataset 3 performs the actual calculations of the needed 2DTE’s for the elastic and piezoelectric tensors. Setting rfphon=1 turns on the atomic displacement perturbation, which we need for all atoms (rfatpol= 1 4) and all directions (rfdir= 1 1 1). Abinit will calculate first-order wave functions for each atom and direction in turn, and use those to calculate 2DTE’s with respect to all pairs of atomic displacements and with respect to one atomic displacement and one component of electric field. These quantities, the interatomic force constants (at gamma) and the Born effective charges will be used later to compute the atomic relaxation contribution to the elastic and piezoelectric tensor.

First-order wave functions for the strain perturbation are computed next. Setting rfstrs = 3 specifies that we want both uniaxial and shear strains to be treated, and rfdir= 1 1 1 cycles through strains xx, yy, and zz for uniaxial and yz, xz, and xy for shear. We note that while other perturbations in Abinit are treated in reduced coordinates, strain is better dealt with in Cartesian coordinates for reasons discussed in the reference cited above. These wave functions are used to compute three types of 2DTE’s. Derivatives with respect to two strain components give us the so-called rigid-ion elastic tensor. Derivatives with respect to one strain and one electric field component give us the rigid-ion piezoelectric tensor. Finally, derivatives with respect to one strain and one atomic displacement yield the internal- strain force-response tensor, an intermediate quantity that will be necessary to compute the atomic relaxation corrections to the rigid-ion quantities. As in lesson RF1, we specify convergence in terms of the residual of the potential (here the first-order potential) using tolvrs .

Your run should have completed by now. Abinit should have created quite a few files.

• telast_2.log (log file)
• telast_2.out (main output file)
• telast_2o_DS1_DDB (first derivatives of the energy from GS calculation)
• telast_2o_DS3_DDB (second derivatives from the RF calculation)
• telast_2o_DS1_WFK (GS wave functions)
• telast_2o_DS2_1WF* (ddk wave functions)
• telast_2o_DS3_1WF* (RF first-order wave functions from various perturbations)

The log and out files are diagnostics and readable output information for a wide variety of properties. The derivative database DDB files are ascii and readable, but primarily for subsequent analysis by anaddb which we will undertake in the next section. Finally, the various wave function binary files are primarily of use for subsequent calculations, where they could cut the number of needed iterations in, for example, convergence testing. We take note of a few conventions in the file names. The root output file name telast_2o is from the 4th line of the “files” file. The dataset producing the file is next. Finally, the first-order wave function 1WF files have a final “pertcase” number described in section 1 of the respfn_help file. While telast_2.in specifies all atomic displacements, only the symmetry- inequivalent perturbations are treated, so the “pertcase” list is incomplete. All cases specified in the input data are treated for the strain perturbation.

First, take a look at the end of the telast_2.log file to make sure the run has completed without error. You might wish to take a look at the WARNING’s, but they all appear to be harmless. Next, edit your telast_2.out file. Searching backwards for ETOT you will find

     iter   2DEtotal(Ha)       deltaE(Ha) residm    vres2
-ETOT  1   2.3955208361366     -6.519E+00 6.313E-01 4.126E+02
ETOT  2   1.3040286462220     -1.091E+00 4.926E-04 4.735E+00
ETOT  3   1.2898966738702     -1.413E-02 1.857E-05 3.504E-01
ETOT  4   1.2891923712805     -7.043E-04 2.937E-07 8.931E-03
ETOT  5   1.2891781500347     -1.422E-05 7.582E-09 5.989E-05
ETOT  6   1.2891780804502     -6.958E-08 4.440E-11 7.674E-07
ETOT  7   1.2891780792714     -1.179E-09 1.236E-12 2.972E-08
ETOT  8   1.2891780791847     -8.667E-11 2.674E-14 7.671E-10
ETOT  9   1.2891780791827     -2.006E-12 9.086E-16 4.128E-12

At SCF step    9       vres2   =  4.13E-12 < tolvrs=  1.00E-10 =>converged.


Abinit is solving a set of Schroedinger-like equations for the first-order wave functions, and these functions minimize a variational expression for the 2DTE (Technically, they are called self-consistent Sternheimer equations) The energy convergence looks similar to that of GS calculations. The fact that vres2, the residual of the self-consistent first-order potential, has reached tolvrs well within nstep (40) iterations indicates that the 2DTE calculation for this perturbation (xy strain) has converged. It would pay to examine a few more cases for different perturbations (unless you have looked through all the warnings in the log).

Another convergence item to examine in your .out file is

 Seventeen components of 2nd-order total energy (hartree) are
1,2,3: 0th-order hamiltonian combined with 1st-order wavefunctions
kin0=   9.10477366E+00 eigvalue=   3.11026184E-01  local=  -3.66858410E+00
4,5,6,7: 1st-order hamiltonian combined with 1st and 0th-order wfs
loc psp =  -8.91644855E+00  Hartree=   4.33575581E+00     xc=  -6.58530138E-01
kin1=  -8.62111363E+00
8,9,10: eventually, occupation + non-local contributions
edocc=   0.00000000E+00     enl0=   6.43290228E-01   enl1=  -1.55388963E-01
1-10 gives the relaxation energy (to be shifted if some occ is /=2.0)
erelax=  -7.62521951E+00
11,12,13 Non-relaxation  contributions : frozen-wavefunctions and Ewald
fr.hart=  -1.18530360E-01   fr.kin=   5.20015318E+00 fr.loc=   4.18792202E-01
14,15,16 Non-relaxation  contributions : frozen-wavefunctions and Ewald
fr.nonl=   2.94970622E-01    fr.xc=   9.41457939E-02  Ewald=   3.02486615E+00
17 Non-relaxation  contributions : pseudopotential core energy
pspcore=   0.00000000E+00
Resulting in :
2DEtotal=    0.1289178079E+01 Ha. Also 2DEtotal=    0.350803195765E+02 eV
(2DErelax=   -7.6252195073E+00 Ha. 2DEnonrelax=    8.9143975865E+00 Ha)
(  non-var. 2DEtotal :    1.2891781360E+00 Ha)


This detailed breakdown of the contributions to 2DTE is probably of limited interest, but you should compare “2DEtotal” and “non-var. 2DEtotal” from the last three lines. While the first-order wave function for the present perturbation minimizes a variational expression for the second derivative with respect to this perturbation as we just saw, the various 2DTE given as elastic tensors, etc. in the output and in the DDB file are all computed using non-variational expressions. Using the non-variational expressions, mixed second derivatives with respect to the present perturbation and all other perturbations of interest can be computed directly from the present first- order wave functions. The disadvantage is that the non-variational result has errors which are linearly proportional to convergence errors in the GS and first-order wave functions. Since errors in the variational 2DEtotal are second-order in wave-function convergence errors, comparing this to the non-variational result for the diagonal second derivative will give an idea of the accuracy of the latter and perhaps indicate the need for tighter convergence tolerances for both the GS and RF wave functions.
This is discussed in X. Gonze and C. Lee, Phys. Rev. B 55, 10355 (1997) , Sec. II.
For an atomic-displacement perturbation, the corresponding breakdown of the 2DTE is headed “Thirteen components.”

Now let us take a look at the results we want, the various 2DTE’s. They begin

 ==> Compute Derivative Database <==

2nd-order matrix (non-cartesian coordinates, masses not included,
asr not included )
cartesian coordinates for strain terms (1/ucvol factor
for elastic tensor components not included)
j1       j2             matrix element
dir pert dir pert     real part     imaginary part

1    1   1    1         5.4508668454         0.0000000000
1    1   2    1        -2.7254334227         0.0000000000
1    1   3    1         0.0000000000         0.0000000000
.....


These are the “raw” 2DTE’s, in reduced coordinates for atom-displacement and electric-field perturbations, but Cartesian coordinates for strain perturbations. This same results with the same organization appear in the file telast_2_DS3_DDB which will be used later as input for automated analysis and converted to more useful notation and units by anaddb. A breakout of various types of 2DTE’s follows (all converted to Cartesian coordinates and in atomic units):

  Dynamical matrix, in cartesian coordinates,
if specified in the inputs, asr has been imposed
j1       j2                   matrix element
dir pert dir pert          real part        imaginary part

1    1   1    1         0.0959051967         0.0000000000
1    1   2    1         0.0000000000         0.0000000000
1    1   3    1         0.0000000000         0.0000000000
.....


This contains the interatomic force constant data that will be used later to include atomic relaxation effects. “asr” refers to the acoustic sum rule, which basically is a way of making sure that forces sum to zero when an atom is displaced.

  Effective charges, in cartesian coordinates,
(from phonon response)
if specified in the inputs, asr has been imposed
j1       j2                   matrix element
dir pert dir pert          real part        imaginary part

1    6   1    1         1.8290469443         0.0000000000
2    6   1    1         0.0000000000         0.0000000000
3    6   1    1         0.0000000000         0.0000000000
.....


The Born effective charges will be used to find the atomic relaxation contributions of the piezoelectric tensor.

  Rigid-atom elastic tensor , in cartesian coordinates,
j1       j2                   matrix element
dir pert dir pert          real part        imaginary part

1    7   1    7         0.0056418385         0.0000000000
1    7   2    7         0.0013753710         0.0000000000
1    7   3    7         0.0007168444         0.0000000000
.....


The rigid-atom elastic tensor is the 2DTE with respect to a pair of strains. We recall that “pert” = natom+3 and natom+4 for unaxial and shear strains, respectively.

  Internal strain coupling parameters, in cartesian coordinates,
zero average net force deriv. has been imposed
j1       j2                   matrix element
dir pert dir pert          real part        imaginary part

1    1   1    7         0.1249319229         0.0000000000
1    1   2    7        -0.1249319273         0.0000000000
1    1   3    7         0.0000000000         0.0000000000
.....


These 2DTE’s with respect to one strain and one atomic displacement are needed for atomic relaxation corrections to both the elastic tensor and piezoelectric tensor. While this set of parameters is of limited direct interest, it should be examined in cases when you think that high symmetry may eliminate the need for these corrections. You are probably wrong, and any non-zero term indicates a correction.

  Rigid-atom proper piezoelectric tensor, in cartesian coordinates,
j1       j2                   matrix element
dir pert dir pert          real part        imaginary part

1    1   1    7         0.1249319273         0.0000000000

1    1   2    7        -0.1249319211         0.0000000000

1    1   3    7         0.0000000000         0.0000000000


Finally, we have the piezoelectric tensor, the 2DTE with respect to one strain and one uniform electric field component. (Yes, there are non-zero elements.)

## 3 ANADDB calculation of atom-relaxation effects¶

In this section, we will run the program anaddb, which analyzes DDB files generated in prior RF calculations. You should copy telast_3.in and telast_3.files in your Work_elast directory. You should now go to the anaddb help file introduction. The bulk of the material in this help file is contained in the description of the variables. You should read the descriptions of

For the theory underlying the incorporation of atom-relaxation corrections, it is recommended you see X. Wu, D. Vanderbilt, and D. R. Hamann [Wu2005].

Anaddb can do lots of other things, such as calculate the frequency-dependent dielectric tensor, interpolate the phonon spectrum to make nice phonon dispersion plots, calculate Raman spectra, etc., but we are focusing on the minimum needed for the elastic and piezoelectric constants at zero electric field.

We also mention that mrgddb is another utility program that can be used to combine DDB files generated in several different datasets or in different runs into a single DDB file that can be analyzed by anaddb. One particular usage would be to combine the DDB file produced by the GS run, which contains first-derivative information such as stresses and forces with the RF DDB. It is anticipated that anaddb in a future release will implement the finite-stress corrections to the elastic tensor discussed in notes by A. R. Oganov .

Now would be a good time to edit telast_3.in and observe that it is very simple, consisting of nothing more than the four variables listed above set to appropriate values.

The telast_3.files file is used with anaddb in the same manner as the abinit .files you are by now used to. The first two lines specify the .in and .out files, the third line specifies the DDB file, and the last two lines are dummy names which would be used in connection with other capabilities of anaddb. Now you should run the calculation, which is done in the same way as you are now used to for abinit:

../../anaddb <telast_3.files >&telast_3.log


This calculation should only take a few seconds. You should edit the log file, go to the end, and make sure the calculation terminated without error. Next, examine telast_3.out. After some header information, we come to tables giving the “force-response” and “displacement-response” internal strain tensors. These represent, respectively, the force on each atom and the displacement of each atom in response to a unit strain of the specified type. These numbers are of limited interest to us, but represent important intermediate quantities in the treatment of atomic relaxation (see the X. Wu paper cited above).

Next, we come to the elastic tensor output:

 Elastic Tensor(clamped ion)(unit:10^2GP):

1.6598859   0.4046480   0.2109029   0.0000000   0.0000000   0.0000000
0.4046480   1.6598859   0.2109029   0.0000000  -0.0000000   0.0000000
0.2109030   0.2109030   1.8258575   0.0000000  -0.0000000   0.0000000
-0.0000000  -0.0000000  -0.0000000   0.4081820  -0.0000000   0.0000000
0.0000000   0.0000000   0.0000000  -0.0000000   0.4081820  -0.0000000
-0.0000000  -0.0000000  -0.0000000   0.0000000  -0.0000000   0.6276190

Elastic Tensor(relaxed ion)(unit:10^2GP):
(at fixed electric field boundary condition)

1.3526217   0.5445039   0.3805291  -0.0000000   0.0000000   0.0000000
0.5445039   1.3526217   0.3805292   0.0000000  -0.0000000   0.0000000
0.3805292   0.3805293   1.4821104   0.0000000  -0.0000000   0.0000000
-0.0000000   0.0000000   0.0000000   0.3055071  -0.0000000   0.0000000
0.0000000  -0.0000000  -0.0000000  -0.0000000   0.3055071   0.0000000
-0.0000000   0.0000000  -0.0000000   0.0000000   0.0000000   0.4040588


While not labeled, the rows and columns 1-6 here represent xx, yy, zz, yz, xz, xy strains and stresses in the conventional Voigt notation.
The clamped-ion results were calculated in the telast_2 RF run, and are simply converted to standard GPa units by anaddb (the terms “clamped ion,” “clamped atom,” and “rigid atom” used in various places are interchangeable, similarly for “relaxed.”)
The relaxed-ion result was calculated by anaddb by combining 2DTE’s for internal strain and interatomic force constants which are stored in the input DDB file. Comparing the clamped and relaxed results, we see that all the diagonal elastic constants have decreased in value.
This is plausible, since allowing the internal degrees of freedom to relax should make a material less stiff. These tensors should be symmetric, and certain tensor elements should be zero or identical by symmetry.
It’s a good idea to check these properties against a standard text such as J. F. Nye, Physical Properties of Crystals (Oxford U. P., Oxford 1985). Departures from expected symmetries (there are a few in the last decimal place here) are due to either convergence errors or, if large, incorrectly specified geometry (however, see the final comments on symmetry below).

Next in telast_3.out we find the piezoelectric tensor results:

 Proper piezoelectric constants(clamped ion)(unit:c/m^2)

0.00000000      0.00000000      0.38490079
0.00000000      0.00000000      0.38490075
0.00000000      0.00000000     -0.73943025
0.00000000      0.43548797      0.00000000
0.43548796      0.00000000     -0.00000000<
0.00000000      0.00000000      0.00000001

Proper piezoelectric constants(relaxed ion)(unit:c/m^2)

0.00000000     -0.00000000     -0.01187149
-0.00000000      0.00000000     -0.01187169
-0.00000000      0.00000000      0.06462779
-0.00000000     -0.04828847     -0.00000000
-0.04828832     -0.00000000      0.00000000


The 3 columns here represent x, y, and z electric polarization, and the 6 rows the Voigt strains. The clamped-ion result was calculated in the telast_2 RF run, and is simply scaled to conventional units by anaddb. The ion relaxation contributions are based on 2DTE’s for internal strain, interatomic force constants, and Born effective charges, and typically constitute much larger corrections to the piezoelectric tensor than to the elastic tensor. Once again, symmetries should be checked. (The slight discrepancies seen here can be removed by setting tolvrs3=1.0d-18 in telast_2.in.) One should be aware that the piezoelectric tensor is identically zero in any material which has a center of symmetry.

Since we are dealing with a hypothetical material, there is no experimental data with which to compare our results. In the next section, we will calculate a few of these numbers by a finite-difference method to gain confidence in the RF approach.

## 4 Finite-difference calculation of elastic and piezoelectric constants¶

You should copy telast_4.in and telast_4.files into your Work_elast directory.

Editing telast_4.in, you will see that it has four datasets, the first two with the c-axis contracted 0.01% and the second two with it expanded 0.01%, which we specified by changing the third row of rprim. The common data is essentially the same as telast_2.in, and the relaxed acell values and xred from telast_1.out have already been included. Datasets 1 and 3 do the self-consistent convergence of the GS wave functions for the strained lattices and compute the stress. Datasets 2 and 4 introduce a new variable.

Electric polarization in solids is a subtle topic which has only recently been rigorously resolved. It is now understood to be a bulk property, and to be quantitatively described by a Berry phase formulation introduced by R. D. King-Smith and D. Vanderbilt, Phys. Ref. B 47, 1651(1993) . It can be calculated in a GS calculation by integrating the gradient with respect to k of the GS wave functions over the Brillouin zone. In GS calculations, the gradients are approximated by finite-difference expressions constructed from neighboring points in the k mesh. These are closely related to the ddk wave functions used in RF calculations in section 2 and introduced in lesson RF1, section 5. We will use berryopt = -1, which utilizes an improved coding of the calculation, and must specify rfdir = 1 1 1 so that the Cartesian components of the polarization are computed.

Now, run the telast_4 calculation, which should only take a minute or two, and edit telast_4.out. To calculate the elastic constants, we need to find the stresses sigma(1 1) and sigma(3 3). We see that each of the four datasets have stress results, but that there are slight differences between those from, for example dataset 1 and dataset 2, which should be identical. Despite our tight limit, this is still a convergence issue. Look at the following convergence results,

Dataset 1:
At SCF step   13       vres2   =  6.24E-21 < tolvrs=  1.00E-18 =>converged.

Dataset 2:
At SCF step    1       vres2   =  5.08E-21 < tolvrs=  1.00E-18 =>converged.


Since dataset 2 has better convergence, we will use this and the dataset 4 results, choosing those in GPa units,

- sigma(1 1)= -2.11918835E-03  sigma(3 2)=  0.00000000E+00
- sigma(3 3)= -1.82392050E-02  sigma(2 1)=  0.00000000E+00

- sigma(1 1)=  2.09886408E-03  sigma(3 2)=  0.00000000E+00
- sigma(3 3)=  1.82778679E-02  sigma(2 1)=  0.00000000E+00


Let us now compute the numerical derivative of sigma(3 3)and compare to our RF result. Recalling that our dimensionless strains were ±0.0001, we find 182.5853 GPa. This compares very well with the value 182.58575 GPa, the 3,3 element of the Rigid-ion elastic tensor we found from our anaddb calculation in section 3. (Recall that our strain and stress were both 3 3 or z z or Voigt 3.) Similarly, the numerical derivative of sigma(1 1)is 21.09026 GPa, compared to 21.09029 GPa, the 3,1 elastic-tensor element.

The good agreement we found from this simple numerical differentiation required that we had accurately relaxed the lattice so that the stress of the unstrained structure was very small. Similar numerical-derivative comparisons for systems with finite stress are more complicated, as discussed in notes by A. R. Oganov. Numerical- derivative comparisons for the relaxed-ion results are extremely challenging since they require relaxing atomic forces to exceedingly small limits.

Now let us examine the electric polarizations found in datasets 2 and 4, focusing on the C/m^2 results,

           Polarization    -1.578272218E-11 C/m^2
Polarization     1.578207434E-11 C/m^2
Polarization    -2.979936062E-01 C/m^2

Polarization    -1.577757536E-11 C/m^2
Polarization     1.577753205E-11 C/m^2
Polarization    -2.981427239E-01 C/m^2


While not labeled as such, these are the Cartesian x, y, and z components, respectively, and the x and y components are zero within numerical accuracy as they must be from symmetry. Numerical differentiation of the z component yields -0.745589 C/m^2. This is to be compared with the z,3 element of our rigid-ion piezoelectric tensor from section 3, -0.73943025 C/m^2, and the two results do not compare as well as we might wish.

What is wrong? There are two possibilities. The first is that the RF calculation produces the proper piezoelectric tensor, while numerical differentiation of the polarization produces the improper piezoelectric tensor. This is a subtle point, for which you are referred to D. Vanderbilt, J. Phys. Chem. Solids 61, 147 (2000). The improper-to-proper transformation only effects certain tensor elements, however, and for our particular combination of crystal symmetry and choice of strain there is no correction. The second possibility is the subject of the next section.

## 5 Alternative DFPT calculation of some piezoelectric constants¶

Our GS calculation of the polarization in section 4 used, in effect, a finite- difference approximation to ddk wave functions, while our RF calculations in section 2 used analytic results based on the RF approach. Since the k grid determined by ngkpt = 4 4 4 and nshiftk = 1 is rather coarse, this is a probable source of discrepancy. Since this issue was noted previously in connection with the calculation of Born effective charges by Na Sai, K. M. Rabe, and D. Vanderbilt, Phys. Rev. B 66, 104108 (2002) , Abinit has incorporated the ability to use finite-difference ddk wave functions from GS calculations in RF calculations of electric-field-related 2DTE’s. Copy telast_5.in and telast_5.files into Work_elast, and edit telast_5.in.

You should compare this with our previous RF data, telast_2.in, and note that dataset1 and the Common data (after entering relaxed structural results) are essentially identical. Dataset 2 has been replaced by a non-self-consistent GS calculation with berryopt = -2 specified to perform the finite-difference ddk wave function calculation. (The finite-difference first-order wave functions are implicit but not actually calculated in the GS polarization calculation.) We have restricted rfdir to 0 0 1 since we are only interested in the 3,3 piezoelectric constant. Now compare dataset 3 with that in telast_2.in. Can you figure out what we have dropped and why? Run the telast_5 calculation, which will only take about a minute with our simplifications.

Now edit telast_5.out, looking for the piezoelectric tensor,

  Rigid-atom proper piezoelectric tensor, in cartesian coordinates,
j1       j2                   matrix element
dir pert dir pert          real part        imaginary part

3    6   3    7        -0.0130314050         0.0000000000


We immediately see a problem – this output, like most of the .out file, is in atomic units, while we computed our numerical derivative in conventional C/m^2 units. While you might think to simply run anaddb to do the conversion as before, its present version is not happy with such an incomplete DDB file as telast_5 has generated and will not produce the desired result. While it should be left as an exercise to the student to dig the conversion factor out of the literature, or better yet out of the source code, we will cheat and tell you that 1a.u.=57.2147606 C/m^2 Thus the new RF result for the 3,3 rigid- ion piezoelectric constant is -0.7455887 C/m^2 compared to the result found in section 4 by a completely-GS finite difference calculation, -0.745589 C/m^2. The agreement is now excellent!

The fully RF calculation in section 2 in fact will converge much more rapidly with k sample than the partial-finite-difference method introduced here. Is it worthwhile to have learned how to do this? We believe that is always pays to have alternative ways to test results, and besides, this didn’t take much time. (Have you found the conversion factor on your own yet?)

## 6 Response-function calculation of the elastic constants of Al metal¶

For metals, the existence of partially occupied bands is a complicating feature for RF as well as GS calculations.
Now would be a good time to review lesson 4 which dealt in detail with the interplay between k-sample convergence and Fermi-surface broadening, especially section 3. You should copy telast_6.in and telast_6.files into Work_elast, and begin your run while you read on, since it involves a convergence study with multiple datasets and may take about two minutes.

While the run is in progress, edit telast_6.in. As in t43.in, we will set udtset to specify a double loop. In the present case, however, the outer loop will be over 3 successively larger meshes of k points, while the inner loop will be successively

1. GS self-consistent runs with optimization of acell.
2. GS density-generating run for the next step.
3. Non-self-consistent GS run to converge unoccupied or slightly-occupied bands.
4. RF run for symmetry-inequivalent elastic constants.

In Section 1, we did a separate GS structural optimization run and transferred the results by hand to RF run section 2. Because we are doing a convergence test here, we have combined these steps, and use getcell to transfer the optimized coordinates from the first dataset of the inner loop forward to the rest. If we were doing a more complicated structure with internal coordinates that were also optimized, we would need to use both this and getxred to transfer these, as in telast_1.in.

The specific data for inner-loop dataset 1 is very similar to that for telast_1.in.
Inner-loop dataset 2 is a bit of a hack. We need the density for inner-loop dataset 3, and while we could set prtden = 1 in dataset 1, this would produce a separate density file for every step in the structural optimization, and it isn’t clear how to automatically pick out the last one. So, dataset 2 picks up the wave functions from dataset 1 (only one file of these is produced, for the optimized structure), does one more iteration with fixed geometry, and writes a density file.

Inner-loop dataset 3 is a non-self-consistent run whose purpose is to ensure that all the wave functions specified by nband are well converged. For metals, we have to specify enough bands to make sure that the Fermi surface is properly calculated. Bands above the Fermi level which have small occupancy or near-zero occupancy if their energies exceed the Fermi energy by more than a few times tsmear, will have very little effect on the self-consistent potential, so the tolvrs test in dataset 1 doesn’t ensure their convergence. Using tolwfr in inner-loop dataset 3 does. Partially- occupied or unoccupied bands up to nband play a different role in constructing the first-order wave functions than do the many unoccupied bands beyond nband which aren’t explicitly treated in Abinit, as discussed in S. de Gironcoli, Phys. Rev. B 51, 6773 (1995). By setting nband exactly equal to the number of occupied bands for RF calculations for semiconductors and insulators, we avoid having to deal with the issue of converging unoccupied bands. Could we avoid the extra steps by simply using tolwfr instead of tolvrs in dataset 1? Perhaps, but experience has shown that this does not necessarily lead to as well-converged a potential, and it is not recommended. These same considerations apply to phonon calculations for metals, or in particular to qpt= 0 0 0 phonon calculations for the interatomic force constants needed to find atom-relaxation contributions to the elastic constants for non-trivial structures as in section 2 and section 3.

The data specific to the elastic-tensor RF calculation in inner-loop dataset 4 should by now be familiar. We take advantage of the fact that for cubic symmetry the only symmetry-inequivalent elastic constants are C 11, C 12 , and C 44 . Abinit, unfortunately, does not do this analysis automatically, so we specify rfdir=1 0 0 to avoid duplicate calculations. (Note that if atom relaxation is to be taken into account for a more complex structure, the full set of directions must be used.)

When the telast_6 calculations finish, first look at telast_6.log as usual to make sure they have run to completion without error. Next, it would be a good idea to look at the band occupancies occ?? (where ?? is a dual-loop dataset index) reported at the end (following END DATASET(S)). The highest band, the fourth in this case, should have zero or very small occupation, or you need to increase nband or decrease tsmear . Now, use your newly perfected knowledge of the Abinit perturbation indexing conventions to scan through telast_6.out and find C 11 , C12 , and C 44 for each of the three k -sample choices, which will be under the ” Rigid-atom elastic tensor” heading. Also find the lattice constants for each case, whose convergence you studied in lesson 4. You should be able to cut-and-paste these into a table like the following,

            C_11           C_12           C_44           acell
ngkpt=3*6   0.0037773556   0.0022583541   0.0013453692   7.5710952266
ngkpt=3*8   0.0042004431   0.0020423388   0.0013076763   7.5693986665
ngkpt=3*10  0.0042034396   0.0020343437   0.0012956768   7.5694820855


We can immediately see that the lattice constant converges considerably more rapidly with k sample than the elastic constants. For ngkpt =3*6, acell is converged to 0.02%, while the C’s have 5-10% errors. For ngkpt =3*8, the C’s are converged to better than 1%, much better for the largest, C11, which should be acceptable.

As in lesson 4, the ngkpt convergence is controlled by tsmear. The smaller the broadening, the denser the k sample that is needed to get a smooth variation of occupancy, and presumably stress, with strain. While we will not explore tsmear convergence in this lesson, you may wish to do so on your own. We believe that the value tsmear = 0.02 in telast_6.in gives results within 1% of the fully-converged small-broadening limit.

We find thatoccopt =3, standard Fermi-Dirac broadening , gives much better convergence of the C’s than “cold smearing.” Changing occopt to 4 in telast_6.in, the option used in lesson 4, the C’s show no sign of convergence. At ngkpt=3*16, errors are still ~5%. The reasons that this supposedly superior smoothing function performs so poorly in this context is a future research topic. The main thing to be learned is that checking convergence with respect to all relevant parameters is always the user’s responsibility. Simple systems that include the main physical features of a complex system of interest will usually suffice for this testing. Don’t get caught publishing a result that another researcher refutes on convergence grounds, and don’t blame such a mistake on Abinit!

Now we make a comparison with experiment. Converting the C’s to standard units (Ha/Bohr^3 = 2.94210119E+04 GPa) and using zero-temperature extrapolated experimental results from P. M. Sutton, Phys. Rev. 91, 816 (1953), we find

                     C_11(GPa)  C_12(GPa)  C_44(GPa)
Calculated        123.7      59.9       38.1
Experiment (T=0)  123.0      70.8       30.9


Is this good agreement? There isn’t much literature on DFT calculations of full sets of elastic constants. Many calculations of the bulk modulus (K=(C11+2C 12 )/3 in the cubic case) typically are within 10% of experiment for the LDA. Running telast_6 with ixc=11, the Perdew-Burke-Enzerhof GGA, increases the calculated C’s by 1-2%, and wouldn’t be expected to make a large difference for a nearly-free-electron metal.

### Comment on symmetry¶

It is important to bear in mind that the way a tensor like the elastic tensor appears is a function of the frame used. Thus for the aluminum fcc case considered above, the nonzero elements are C 11, C 12, and C 44, provided that the crystal axes are aligned with the laboratory frame. For an arbitrary alignment of the crystal axes, many more C ij elements will be non-zero, and this can be confusing.

It’s easy to see why this happens if you imagine actually measuring the elastic tensor elements. If you start with the conventional cubic cell, and apply pressure to one face, you can measure C 11. But if you turn the cell to some random angle, you’ll measure a response that is a mixture of C 11 and C 12.

Within ABINIT, if the aluminum fcc cell is described using angdeg and acell, then an axis of the primitive cell will be aligned along the laboratory z axis but this will not lead to a (conventional) cell alignment with the laboratory frame. The resulting elastic tensor will be correct but will appear to be more complicated than in the illustration above. It can be rotated back to a simple frame by hand (bearing in mind that all four indices of the fourth-rank elastic tensor have to be rotated!) but it’s easier to start with a more conventional alignment of the unit cell.

If you use a standard text like Bradley and Cracknell, The Mathematical Theory of Symmetry in Solids, Oxford you can find the standard primitive cell descriptions for the Bravais lattice types and these are aligned as much as possible with a standard laboratory frame.