# Tutorial on DFT+U¶

## The projected density of states of NiO.¶

You will learn what is a DFT+U calculation and what are the main input variables controlling this type of calculation.

It is supposed that you already know how to do PAW calculations using ABINIT. Please follow the two tutorials on PAW in ABINIT (PAW1, PAW2), if this is not the case.

This tutorial should take about 1 hour to complete.

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.

## 0 Short summary of the DFT+U method¶

The standard Local Density Approximation (LDA), where the exchange and correlation energy is fit to homogeneous electron gas results, is a functional that works well for a vast number of compounds. But, for some crystals, the interactions between electrons are so important that they cannot be represented by the LDA alone. Generally, these highly correlated materials contain rare-earth metals or transition metals, which have partially filled d or f bands and thus localized electrons.

The LDA tends to delocalize electrons over the crystal, and each electron feels an average of the Coulombic potential. For highly correlated materials, the large Coulombic repulsion between localized electrons might not be well represented by a functional such as the LDA. A way to avoid this problem is to add a Hubbard-like, localised term, to the LDA density functional. This approach is known as DFT+U (formerly referred to as LDA+U). In the actual implementation, we separate localized d or f electrons, on which the Hubbard term will act, from the delocalized ones (s and p electrons). The latter are correctly described by the usual LDA calculation. In order to avoid the double counting of the correlation part for localized electrons (already included in the LDA, although in an average manner), another term - called the double-counting correction - is subtracted from the Hamiltonian.

In Abinit, two double-counting corrections are currently implemented:

-The Full localized limit (FLL) [Liechtenstein1995] (usepawu=1)

-The Around Mean Field (AMF) [Czyzyk1994] (usepawu=2)

For some systems, the result might depend on the choice of the double-counting method. However, the two methods generally give similar results.

## 1 Ground state calculation of NiO using LDA¶

Before continuing, you might consider to work in a different subdirectory as for the other tutorials. Why not Work_dftu? In what follows, the names of files will be mentioned as if you were in this subdirectory.

Copy the file tdftu_1.abi from $ABI_TESTS/tutorial/Input to your Work_dftu directory with: cd$ABI_TESTS/tutorial/Input
mkdir Work_dftu
cd Work_dftu
cp ../tdftu_1.abi .


Now run the code as usual. The job should take less than 20 seconds on a laptop. It calculates the LDA ground state of the NiO crystal. A low cutoff and a small number of k-points are used in order to speed up the calculation. During this time you can take a look at the input file.

The NiO crystallizes in the rocksalt structure, with one Ni and one O atom in the primitive cell (the crystallographic primitive cell). However, NiO is known to exhibit an antiferromagnetic ordering at low temperature (along the <111> direction). From the electronic point of view, the true unit cell has two Ni and two O atoms: the local magnetic moment around the first Ni atom will have a sign opposite to the one of the other Ni atom.

You should take some time to examine the values used for the input variables xred, rprim (note the last line!), typat, spinat, nsppol, and nspden, that define this antiferromagnetic ordering along the <111> direction (of a conventional cubic cell).

If you take a look at the output file (tdftu_1.out), you can see the integrated total density in the PAW spheres (see the PAW1 and PAW2 tutorials on PAW formalism). This value roughly estimates the magnetic moment of NiO:

 Integrated electronic and magnetization densities in atomic spheres:
---------------------------------------------------------------------
Radius=ratsph(iatom), smearing ratsm=  0.0000. Diff(up-dn)=approximate z local magnetic moment.
Atom    Radius    up_density   dn_density  Total(up+dn)  Diff(up-dn)
1   1.81432     8.564383     7.188016     15.752398     1.376367
2   1.81432     7.188016     8.564383     15.752398    -1.376367
3   1.41465     2.260902     2.260902      4.521804    -0.000000
4   1.41465     2.260902     2.260902      4.521804     0.000000


The atoms in the output file, are listed as in the typat variable (the first two are nickel atoms and the last two are oxygen atoms). The results indicate that spins are located in each nickel atom of the doubled primitive cell. Fortunately, the LDA succeeds to give an antiferromagnetic ground state for the NiO. But the result does not agree with the experimental data.

The magnetic moment (the difference between up and down spin on the nickel atom) range around 1.6-1.9 according to experiments ([Cheetham1983],[Neubeck1999],[Sawatzky1984], [Hufner1984]) Also, as the Fermi level is at 0.33748 Ha (see the tdftu_1.abo file), one can see (on the tdftu_1.o_EIG file that contains eigenvalues for the three k-point of this calculation) that the band gap obtained between the last (24th) occupied band (0.31537 Ha, at k point 3) and the first (25th) unoccupied band (0.35671 Ha, at kpoint 3) is approximately 1.1 eV which is lower than the measured value of 4.0-4.3 eV (This value could be modified using well-converged parameters but would still be much lower than what is expected). A easier and graphical way to evaluate the gap would be to plot the density of states (see last section of this tutorial).

Making abstraction of the effect of insufficiently convergence parameters, the reason for the discrepancy between the DFT-LDA data and the experiments is first the fact the DFT is a theory for the ground state and second, the lack of correlation of the LDA. Alone, the homogeneous electron gas cannot correctly represent the interactions among $d$ electrons of the Ni atom. That is why we want to improve our functional, and be able to manage the strong correlation in NiO.

## 2 DFT+U with the FLL double-counting¶

As seen previously, the LDA does not gives good results for the magnetization and band gap compared to experiments. At this stage, we will try to improve the correspondence between calculation and experimental data. First, we will use the DFT(LDA)+U with the Full localized limit (FLL) double-counting method.

FLL and AMF double-counting expressions are given in the papers listed above, and use the adequate number of electrons for each spin. For the Hubbard term, the rotationally invariant interaction is used.

Note

It is important to notice that in order to use DFT+U in Abinit, you must employ PAW pseudopotentials.

You should run abinit with the tdftu_2.abi input file. This calculation takes also less than 20 seconds on a laptop. During the calculation, you can take a look at the input file.

Some variable describing the DFT+U parameters have been added to the previous file. All other parameters were kept constant from the preceding calculation. First, you must set the variable usepawu to one (for the FLL method) and two (for the AMF method) in order to enable the DFT+U calculation. Then, with lpawu you give for each atomic species (znucl) the values of angular momentum (l) for which the DFT+U correction will be applied. The choices are 1 for p-orbitals, 2 for d-orbitals and 3 for f-orbitals. You cannot treat s orbitals with DFT+U in the present version of ABINIT. Also, if you do not want to apply DFT+U correction on a species, you can set the variable to -1. For the case of NiO, we put lpawu to 2 for Ni and -1 for O.

Note

The current implementation applies DFT+U correction only inside atomic sphere. To check if this approximation is realistic, relaunch the calculation with pawprtvol equal to three. Then search for ph0phiint in the log file:

pawpuxinit: icount, ph0phiint(icount)= 1  0.90467


This line indicates that the norm of atomic wavefunctions inside atomic sphere is 0.90, rather close to one. In the case of nickel, the approximation is thus realistic. The case where the norm is too small (close to 0.5) is discussed in [Geneste2017].

Finally, as described in the article cited above for FLL and AMF, we must define the screened Coulomb interaction between electrons that are treated in DFT+U, with the help of the variable upawu and the screened exchange interaction, with jpawu. Note that you can choose the energy unit by indicating at the end of the line the unit abbreviation (e.g. eV or Ha). For NiO, we will use variables that are generally accepted for this type of compound:

upawu  8.0 0.0 eV
jpawu  0.8 0.0 eV


You can take a look at the result of the calculation. The magnetic moment is now:

 Integrated electronic and magnetization densities in atomic spheres:
---------------------------------------------------------------------
Radius=ratsph(iatom), smearing ratsm=  0.0000. Diff(up-dn)=approximate z local magnetic moment.
Atom    Radius    up_density   dn_density  Total(up+dn)  Diff(up-dn)
1   1.81432     8.749919     6.987384     15.737302     1.762535
2   1.81432     6.987384     8.749919     15.737302    -1.762535
3   1.41465     2.290397     2.290397      4.580793    -0.000000
4   1.41465     2.290397     2.290397      4.580793    -0.000000


NiO is found antiferromagnetic, with a moment that is in reasonable agreement with experimental results. Moreover, the system is a large gap insulator with about 5.3 eV band gap (the 24th band at k point 3 has an eigenenergy of 0.26699 Ha, much lower than the eigenenergy of the 25th band at k point 1, namely 0.46243 Ha, see the tdftu_2.o_EIG file). This number is very approximative, since the very rough sampling of k points is not really appropriate to evaluate a band gap, still one obtains the right physics.

A word of caution is in order here. It is NOT the case that one obtain systematically a good result with the DFT+U method at the first trial. Indeed, due to the nature of the modification of the energy functional, the landscape of this energy functional might present numerous local minima (see for examples [Jomard2008] or [Dorado2009]).

Unlike DFT+U, for the simple LDA (without U), in the non-spin-polarized case, there is usually only one minimum, that is the global minimum. So, if it converges, the self-consistency algorithm always find the same solution, namely, the global minimum. This is already not true in the case of spin- polarized calculations (where there might be several stable solutions of the SCF cycles, like ferromagnetic and ferromagnetic), but usually, there are not many local minima, and the use of the spinat input variables allows one to adequately select the global physical characteristics of the sought solution.

By contrast, with the U, the spinat input variable is too primitive, and one needs to be able to initialize a spin-density matrix on each atomic site where a U is present, in order to guide the SCF algorithm.

The fact that spinat works for NiO comes from the relative simplicity of this system.

## 3 Initialization of the density matrix¶

You should begin by running the tdftu_3.abi file before continuing.

In order to help the DFT+U find the ground state, you can define the initial density matrix for correlated orbitals with dmatpawu. For $d$ orbitals, this variable must contains $5\times5$ square matrices. There should be one square matrix per nsppol and atom. So in our case, there are 2 square matrices. Also, to enable this feature, usedmatpu must be set to a non-zero value (default is 0). When positive, the density matrix is kept to the dmatpawu value for the usedmatpu value steps. For our calculation(tdftu_3.abi) , usedmatpu is 5, thus the spin-density matrix is kept constant for 5 SCF steps. Let’s examinates the input dmatpawu

To understand the density matrix used in the variable dmatpawu in this input file, have a look to the section on this variable dmatpawu. This section show the order to orbitals in the density matrix. With the help of this section, one can understand that the density matrix corresponds to all orbitals filled except $e_g$ orbitals for one spin.

In the log file (not the usual output file), you will find for each step, the calculated density matrix, followed by the imposed density matrix. After the first 5 SCF steps, the initial density matrix is no longer imposed. Here is a section of the log file, in which the imposed occupation matrices are echoed:

-------------------------------------------------------------------------

Occupation matrix for correlated orbitals is kept constant
and equal to dmatpawu from input file !
----------------------------------------------------------

== Atom   1 == Imposed occupation matrix for spin 1 ==
0.90036    0.00000   -0.00003    0.00000    0.00000
0.00000    0.90036   -0.00001    0.00000    0.00002
-0.00003   -0.00001    0.91309   -0.00001    0.00000
0.00000    0.00000   -0.00001    0.90036   -0.00002
0.00000    0.00002    0.00000   -0.00002    0.91309

== Atom   1 == Imposed occupation matrix for spin 2 ==
0.89677   -0.00001    0.00011   -0.00001    0.00000
-0.00001    0.89677    0.00006    0.00001   -0.00010
0.00011    0.00006    0.11580    0.00006    0.00000
-0.00001    0.00001    0.00006    0.89677    0.00010
0.00000   -0.00010    0.00000    0.00010    0.11580

== Atom   2 == Imposed occupation matrix for spin 1 ==
0.89677   -0.00001    0.00011   -0.00001    0.00000
-0.00001    0.89677    0.00006    0.00001   -0.00010
0.00011    0.00006    0.11580    0.00006    0.00000
-0.00001    0.00001    0.00006    0.89677    0.00010
0.00000   -0.00010    0.00000    0.00010    0.11580

== Atom   2 == Imposed occupation matrix for spin 2 ==
0.90036    0.00000   -0.00003    0.00000    0.00000
0.00000    0.90036   -0.00001    0.00000    0.00002
-0.00003   -0.00001    0.91309   -0.00001    0.00000
0.00000    0.00000   -0.00001    0.90036   -0.00002
0.00000    0.00002    0.00000   -0.00002    0.91309


Generally, the DFT+U functional meets the problem of multiple local minima, much more than the usual LDA or GGA functionals. One often gets trapped in a local minimum. Trying different starting points might be important…

## 4 AMF double-counting method¶

Now we will use the other implementation for the double-counting term in DFT+U (in Abinit), known as AMF. As the FLL method, this method uses the number of electrons for each spin independently and the complete interactions $U(m_1,m_2,m_3,m_4)$ and $J(m_1,m_2,m_3,m_4)$.

As in the preceding run, we will start with a fixed density matrix for d orbitals. You might now start your calculation, with the tdftu_4.abi, or skip the calculation, and rely on the reference file provided in the \$ABI_TESTS/tutorial/Refs directory. Examine the tdftu_4.abi file.

The only difference in the input file compared to tdftu_3.abi is the value of usepawu = 2. We obtain a band gap of 4.75 eV. The value of the band gap with AMF and FLL is different. However, we have to remember that these results are not well converged. By contrast, the magnetization,

  Integrated electronic and magnetization densities in atomic spheres:
---------------------------------------------------------------------
Radius=ratsph(iatom), smearing ratsm=  0.0000. Diff(up-dn)=approximate z local magnetic moment.
Atom    Radius    up_density   dn_density  Total(up+dn)  Diff(up-dn)
1   1.81432     8.675718     6.993823     15.669541     1.681895
2   1.81432     6.993823     8.675718     15.669541    -1.681895
3   1.41465     2.288681     2.288681      4.577361    -0.000000
4   1.41465     2.288681     2.288681      4.577361     0.000000


is very similar to the DFT+U FLL. For other systems, the difference can be more important. FLL is designed to work well for systems in which occupations of orbitals are 0 or 1 for each spin. The AMF should be used when orbital occupations are near the average occupancies.

## 5 Projected density of states in DFT+U¶

Using prtdos 3, you can now compute the projected d and f density of states. For more information about projected density of states, for more details see the PAW1 tutorial.