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XG, RC

Second (basic) tutorial

The H₂ molecule, with convergence studies.

This tutorial aims at showing how to get converged values for the following physical properties:

the bond length
the atomisation energy

You will learn about the numerical quality of the calculations, then make convergence studies with respect to the number of planewaves and the size of the supercell, and finally consider the effect of the XC functional. The problems related to the use of different pseudopotential are not examined. You will also finish to read the help:abinit.

This tutorial should take about 1 hour.

[TUTORIAL_README]

Summary of the previous tutorial

We studied the H$_2$ molecule in a big box. We used 10 Ha as cut-off energy, a 10x10x10 Bohr$^3$ supercell, the local-density approximation (as well as the local-spin-density approximation) in the Perdew-Wang parametrization (ixc = -1012) and a pseudopotential from the pseudodojo http://www.pseudo-dojo.org/.

At this stage, we compared our results:

bond length: 1.486 Bohr
atomisation energy at that bond length: 0.1704 Ha = 4.635 eV

with the experimental data (as well as theoretical data using a much more accurate technique than DFT)

bond length: 1.401 Bohr
atomisation energy: 4.747 eV

The bond length is rather bad (about 6% off), and the atomisation energy is a bit too low, 2.5% off.

2 The convergence in ecut (I)

2.1.a Computing the bond length and corresponding atomisation energy in one run.

Before beginning, you might consider to work in a different subdirectory as for tutorial 1. Why not Work2?

Because we will compute many times the bond length and atomisation energy, it is worth to make a single input file that will do all the associated operations. You should try to use 2 datasets (try to combine $ABI_TESTS/tutorial/Input/tbase1_3.abi with tbase1_5.abi). Do not try to have the same position of the H atom as one of the H$_2$ atoms in the optimized geometry.

cd $ABI_TESTS/tutorial/Input
mkdir Work2
cd Work2
cp ../tbase2_1.abi .

The input file tbase2_1.abi is an example of file that will do the job,

{% dialog tests/tutorial/Input/tbase2_1.abi %}

while tbase2_1.abo is an example of output file:

{% dialog tests/tutorial/Refs/tbase2_1.abo %}

Execute the code with:

abinit tbase2_1.abi > log &

The run should take less than one minute.

You should obtain the values:

       etotal1    -1.1182883137E+00
       etotal2    -4.7393103688E-01

and

        xcart1    -7.4307169181E-01  0.0000000000E+00  0.0000000000E+00
                   7.4307169181E-01  0.0000000000E+00  0.0000000000E+00

These are similar to those determined in tutorial 1, although they have been obtained in one run. You can also check that the residual forces are lower than 5.0d-4. Convergence issues are discussed in help:abinit#numerical-quality of the abinit help file, on numerical quality. You should read it. By the way, you have read many parts of the abinit help file! You are missing the sections (or part of) help:abinit#2, help:abinit#pseudopotential-files, help:abinit#numerical-quality.

You are also missing the description of many input variables. We suggest that you finish reading entirely the abinit help file now, while the knowledge of the input variables will come in the long run.

2.1.b Many convergence parameters have already been identified. We will focus only on ecut and acell. This is because

the convergence of the SCF cycle and geometry determination are well under control thanks to toldfe, toldff and tolmxf (this might not be the case for other physical properties)
there is no k point convergence study to be done for an isolated system in a big box: no additional information is gained by adding a k-point beyond one
the boxcut value (see boxcutmin) is automatically chosen larger than 2 by ABINIT, see the determination of the input variable ngfft by preprocessing
we are using ionmov = 2 for the determination of the geometry.

3 The convergence in ecut (II)

For the check of convergence with respect to ecut, you have the choice between doing different runs of the tbase2_1.abi file with different values of ecut, or doing a double loop of datasets, as proposed in $ABI_TESTS/tutorial/Input/tbase2_2.abi. The values of ecut have been chosen between 10 Ha and 35 Ha, by step of 5 Ha. If you want to make a double loop, you might benefit of reading again the help:abinit#loop of the abinit_help file.

2.2.a You have likely seen a big increase of the CPU time needed to do the calculation. You should also look at the increase of the memory needed to do the calculation (go back to the beginning of the output file). The output data are as follows:

       etotal11   -1.1182883137E+00
       etotal12   -4.7393103688E-01
       etotal21   -1.1325211211E+00
       etotal22   -4.7860857539E-01
       etotal31   -1.1374371581E+00
       etotal32   -4.8027186429E-01
       etotal41   -1.1389569555E+00
       etotal42   -4.8083335144E-01
       etotal51   -1.1394234882E+00
       etotal52   -4.8101478048E-01
       etotal61   -1.1395511942E+00
       etotal62   -4.8107063412E-01

        xcart11   -7.4307169181E-01  0.0000000000E+00  0.0000000000E+00
                   7.4307169181E-01  0.0000000000E+00  0.0000000000E+00
        xcart12    0.0000000000E+00  0.0000000000E+00  0.0000000000E+00
        xcart21   -7.3687974546E-01  0.0000000000E+00  0.0000000000E+00
                   7.3687974546E-01  0.0000000000E+00  0.0000000000E+00
        xcart22    0.0000000000E+00  0.0000000000E+00  0.0000000000E+00
        xcart31   -7.3014665027E-01  0.0000000000E+00  0.0000000000E+00
                   7.3014665027E-01  0.0000000000E+00  0.0000000000E+00
        xcart32    0.0000000000E+00  0.0000000000E+00  0.0000000000E+00
        xcart41   -7.2642579309E-01  0.0000000000E+00  0.0000000000E+00
                   7.2642579309E-01  0.0000000000E+00  0.0000000000E+00
        xcart42    0.0000000000E+00  0.0000000000E+00  0.0000000000E+00
        xcart51   -7.2563260546E-01  0.0000000000E+00  0.0000000000E+00
                   7.2563260546E-01  0.0000000000E+00  0.0000000000E+00
        xcart52    0.0000000000E+00  0.0000000000E+00  0.0000000000E+00
        xcart61   -7.2554339763E-01  0.0000000000E+00  0.0000000000E+00
                   7.2554339763E-01  0.0000000000E+00  0.0000000000E+00
        xcart62    0.0000000000E+00  0.0000000000E+00  0.0000000000E+00

The corresponding atomisation energies and interatomic distances are:

ecut (Ha)	atomisation energy (Ha)	interatomic distance (Bohr)
10	.1704	1.486
15	.1753	1.474
20	.1769	1.460
25	.1773	1.453
30	.1774	1.451
35	.1774	1.451

In order to obtain 0.2% relative accuracy on the bond length or atomisation energy, one should use a kinetic cut-off energy of 25 Ha. We will keep in mind this value for the final run.

4 The convergence in acell

The same technique as for ecut should be now used for the convergence in acell. We will explore acell starting from 8 8 8 to 18 18 18, by step of 2 2 2. We keep ecut 10 for this study. Indeed, it is a rather general rule that there is little cross-influence between the convergence of ecut and the convergence of acell.

The file $ABI_TESTS/tutorial/Input/tbase2_3.abi can be used as an example.

{% dialog tests/tutorial/Input/tbase2_3.abi %}

The output results in $ABI_TESTS/tutorial/Refs/tbase2_3.abo are as follows:

       etotal11   -1.1305202335E+00
       etotal12   -4.8429570903E-01
       etotal21   -1.1182883137E+00
       etotal22   -4.7393103688E-01
       etotal31   -1.1165450484E+00
       etotal32   -4.7158917506E-01
       etotal41   -1.1165327748E+00
       etotal42   -4.7136118536E-01
       etotal51   -1.1167740301E+00
       etotal52   -4.7128698890E-01
       etotal61   -1.1168374331E+00
       etotal62   -4.7129589330E-01

        xcart11   -7.6471149217E-01  0.0000000000E+00  0.0000000000E+00
                   7.6471149217E-01  0.0000000000E+00  0.0000000000E+00
        xcart12    0.0000000000E+00  0.0000000000E+00  0.0000000000E+00
        xcart21   -7.4307169181E-01  0.0000000000E+00  0.0000000000E+00
                   7.4307169181E-01  0.0000000000E+00  0.0000000000E+00
        xcart22    0.0000000000E+00  0.0000000000E+00  0.0000000000E+00
        xcart31   -7.3778405090E-01  0.0000000000E+00  0.0000000000E+00
                   7.3778405090E-01  0.0000000000E+00  0.0000000000E+00
        xcart32    0.0000000000E+00  0.0000000000E+00  0.0000000000E+00
        xcart41   -7.3794243127E-01  0.0000000000E+00  0.0000000000E+00
                   7.3794243127E-01  0.0000000000E+00  0.0000000000E+00
        xcart42    0.0000000000E+00  0.0000000000E+00  0.0000000000E+00
        xcart51   -7.3742475720E-01  0.0000000000E+00  0.0000000000E+00
                   7.3742475720E-01  0.0000000000E+00  0.0000000000E+00
        xcart52    0.0000000000E+00  0.0000000000E+00  0.0000000000E+00
        xcart61   -7.3733248368E-01  0.0000000000E+00  0.0000000000E+00
                   7.3733248368E-01  0.0000000000E+00  0.0000000000E+00
        xcart62    0.0000000000E+00  0.0000000000E+00  0.0000000000E+00

The corresponding atomisation energies and interatomic distances are:

acell (Bohr)	atomisation energy (Ha)	interatomic distance (Bohr)
8	.1619	1.529
10	.1704	1.486
12	.1734	1.476
14	.1738	1.478
16	.1742	1.475
18	.1742	1.475

In order to reach 0.2% convergence on the atomisation energy and interatomic distance one needs acell 16 16 16. We will use acell 16 16 16 for the final run.

For most solids the size of the unit cell will be smaller than that. We are treating a lot of vacuum in this supercell ! So, the H$_2$ study, with this pseudopotential, turns out to be not really easy. Of course, the number of states to be treated is minimal! This allows to have reasonable CPU time still.

5 The final calculation in Local (Spin) Density Approximation

We now use the correct values of both ecut and acell. Well, you should modify the tbase2_3.abi file to make a calculation with acell 16 16 16 and ecut 25. You can still use the double loop feature with udtset 1 2 (which reduces to a single loop), to minimize the modifications to the file.

The file $ABI_TESTS/tutorial/Input/tbase2_4.abi can be taken as an example of input file:

{% dialog tests/tutorial/Input/tbase2_4.abi %}

while $ABI_TESTS/tutorial/Refs/tbase2_4.abo is as an example of output file:

{% dialog tests/tutorial/Refs/tbase2_4.abo %}

Since we are doing the calculation at a single (ecut, acell) pair, the total CPU time is not as much as for the previous determinations of optimal values through series calculations. However, the memory needs have still increased a bit.

The output data are:

       etotal11   -1.1369766875E+00
       etotal12   -4.7827555035E-01

        xcart11   -7.2259811794E-01  0.0000000000E+00  0.0000000000E+00
                   7.2259811794E-01  0.0000000000E+00  0.0000000000E+00
        xcart12    0.0000000000E+00  0.0000000000E+00  0.0000000000E+00

The corresponding atomisation energy is 0.1804 Ha = 4.910 eV
The interatomic distance is 1.445 Bohr.
These are our final data for the local (spin) density approximation.

Without our choice of pseudopotential, the value of ixc was -1012, corresponding to the Perdew-Wang cite:Perdew1992a parameterization of the LDA XC functional. It is in principle the same as using ixc = 1. Other expressions for the local (spin) density approximation ixc=[2, 3 ... 7] are possible. The values 1, 2, 3 and 7 should give about the same results, since they all start from the XC energy of the homogeneous electron gas, as determined by Quantum Monte Carlo calculations. Other possibilities (ixc = 4, 5, 6) are older local density functionals, that could not rely on these data.

6 The use of the Generalized Gradient Approximation

We will use the Perdew-Burke-Ernzerhof functional proposed in cite:Perdew1996

For GGA, we use another pseudopotential than for LDA. In principle we should redo the ecut convergence test, possibly coming to the conclusion that another value of ecut should be used. However, for the special case of Hydrogen, and in general pseudopotentials with a very small core (including only the 1s orbital), pseudopotentials issued from the LDA and from the GGA are very similar. So, we will not redo an ecut convergence test.

!!! important

*ecut* is often characteristic of the pseudopotentials that are used in a calculation.

Independently of the pseudopotential, an acell convergence test should not be done again, since the vacuum is treated similarly in LDA or GGA.

So, our final values within GGA will be easily obtained by changing the pseudopotential with respect to the one used in tbase2_4.abi.

{% dialog tests/tutorial/Input/tbase2_5.abi %}

       etotal11   -1.1658082573E+00
       etotal12   -4.9940910146E-01

        xcart11   -7.0870975055E-01 -2.3009273766E-32 -6.6702161522E-32
                   7.0870975055E-01  2.3009273766E-32  6.6702161522E-32
        xcart12    0.0000000000E+00  0.0000000000E+00  0.0000000000E+00

The corresponding atomisation energy is 0.1670 Ha = 4.544 eV
The interatomic distance is 1.417 Bohr.
These are our final data for the generalized gradient approximation.

Once more, here are the experimental data:

bond length: 1.401 Bohr
atomisation energy: 4.747 eV

In GGA, we are within 2% of the experimental bond length, but 5% of the experimental atomisation energy. In LDA, we were within 3% of the experimental bond length, and about 3.5% of the experimental atomisation energy.