# Calculation of U and J using Cococcioni’s approach¶

## How to determine U for DFT+U in ABINIT ? Cococcioni’s approach.¶

This lesson aims to show how you can determine U for further DFT+U
calculations consistently and in a fast an easy way. You will learn to prepare
the input files for the determination and to use the main parameters implemented for this aim.

It is supposed that you already know how to run ABINIT in the PAW mode (lesson PAW1).
Obviously, you should also read the lesson DFT+U, and likely the lesson PAW2,
to generate PAW atomic data.

This lesson should take about ½ hour.

## 1 Summary of linear response method to determine U¶

The linear response method has been introduced by several authors [1-6]. It is based on the fact that U corresponds to the energy to localize an additional electron on the same site: U=E[n+1]+E[n-1]-2E[n] [4]. This can be reformulated as the response to an infinitesimal change of of occupation of the orbital by the electrons dn. Then U is the second derivative of the energy with respect to the occupation U=d^2E/ d^2n. The first method fixed the occupation by cutting the hopping terms of localized orbitals. Later propositions constrained the occupation through Lagrange multipliers [3,5]. The Lagrange multiplier \alpha corresponds to a local potential that has to be applied to augment or decrease the occupation by ±1 electron. Note that the occupation need not to vary by 1 electron, but the occupation shift can be infinitesimal.

It is recommended to read the following papers to understand the basic concepts of the linear response calculations to calculate U:

[1] “A LDA+U study of selected iron compounds “, M. Cococcioni, Ph.D. thesis,
International School for Advanced Studies (SISSA), Trieste (2002)

[2] “Linear response approach to the calculation of the effective interaction
parameters in the LDA + U method”, M. Cococcioni and S. de Gironcoli, Physical
Review B 71, 035105 (2005)

Some further reading:

[3] “Ground States of Constrained Systems: Application to Cerium Impurities”,
P. H. Dederichs, S. Blugel, R. Zeller, and H. Akai, Phys. Rev. Lett. 53, 2512 (1984)

[4] “Calculation of Coulomb-interaction parameters for La2CuO4 using a
constrained-density-functional approach”, M. S. Hybertsen, M. Schluter, and N.
E. Christensen, Phys. Rev. B 39, 9028 (1989)

[5] “Density-functional calculation of effective Coulomb interactions in
metals”, V. I. Anisimov and O. Gunnarsson, Phys. Rev. B42, 7570 (1991)

[6] “Reformulation of the LDA+U method for a local-orbital basis”, W. E.
Pickett, S. C. Erwin, and E. C. Ethridge, Phys. Rev. B58, 1201 (1998)

The implementation of the determination of U in ABINIT is described in the following paper, soon to appear:

[7] “Consistent determination of U in the PAW approximation”, D. Adams, B. Amadon, S. Biermann, unpublished (2010)

## 2 Determine U in ABINIT¶

*Before continuing, you might consider to work in a different subdirectory as
for the other lessons. Why not “Work_udet”?*

Important

In what follows, the name of files are mentioned as if you were in this subdirectory.

All the input files can be found in the ~abinit/tests/tutorial/Input directory
You can compare your results with reference output files located in
_~abinit/tests/tutorial/Refs directories (for the present tutorial they are named tudet*.out).

The input file tudet_1.in is an example of a file to prepare a wave function for further processing. You might use the file tudet_1.files as a “files” file, and get the corresponding output file ../Refs/tudet_1.out).

Copy the files tudet_1.in and tudet_1.files in your work directory, and run ABINIT:

abinit < tudet_1.files > tudet_1.log

In the meantime, you can read the input file and see that this is a usual DFT+U calculation, with U=0.

tudet_1.in tudet_1.out tudet_1i tudet_1o tudet_1x ../../../Psps_for_tests/26fe.paw

################################################################# # Automatic test for ABINIT: # # Prelimirary step for test v5#39 (macro_uj) # # and v5#40 (testirdden) # # # # Fe bcc 2 atomic supercell - ferromag.- PAW DJA 2010 & MT 2009 # ################################################################# #Unit cell acell 3*5.42 chkprim 0 # 0: do not check if uc primitive rprim 1.00 0.00 0.00 0.00 1.00 0.00 0.00 0.00 1.00 #Spin polarization nsppol 2 #1 unpolarized / 2 polarized spinat 0 0 2.843062 0 0 2.843062 #Definition of the atom types ntypat 1 znucl 26 #Definition of the atoms natom 2 typat 2*1 # atomic types xred 0.0 0.0 0.0 0.5 0.5 0.5 ecut 8 # Energy cutoff pawecutdg 20 # pawecutdg > 2*ecut nband 25 # Fe_2 minband=17 #Definition of the SCF procedure nstep 15 # max number SCF cycles tolvrs 10d-12 #Definition of the k-point grid kptopt 1 # 1: automatic generation of k points ngkpt 3 3 3 # n x n x n nshiftk 1 shiftk 0.5 0.5 0.5 #Smearing occopt 4 tsmear 0.05 eV #DFT+U usepawu 1 # 1 at lim dble cnt / 2 rnd m fld dle cnt lpawu 2 # ang moments corrrected #Save disk space & Miscelaneous prteig 0 prtden 1 # This is the default value ## After modifying the following section, one might need to regenerate the pickle database with runtests.py -r #%%<BEGIN TEST_INFO> #%% [setup] #%% executable = abinit #%% test_chain = tudet_1.in, tudet_2.in, tudet_3.in #%% [files] #%% files_to_test = #%% tudet_1.out, tolnlines= 2, tolabs= 7.000e-09, tolrel= 2.000e-10 #%% psp_files = 26fe.paw #%% [paral_info] #%% max_nprocs = 1 #%% [extra_info] #%% authors = M. Torrent #%% keywords = PAW, LDAU #%% description = #%% Fe bcc 2 atomic supercell - ferromag.- PAW DJA 2010 & MT 2009 #%% Prelimirary step for test v5#39 (macro_uj) #%% and v5#40 (testirdden) #%%<END TEST_INFO>

This setting allows us to read the occupations of
the Fe 3d orbitals (lpawu 2). The cell contains 2 atoms. This is the
minimum to get reasonable response matrices. We converge the electronic
structure to a high accuracy (tolvrs 10d-12), which usually allows to
determine occupations with a precision of 10d-10. The ecut is chosen very low,
in order to accelerate calculations.

We do not suppress the writing of the _WFK file, because this is the input for
the calculations of U.

Once this calculation has finished, run the second one:

Copy the files tudet_2.in and tudet_2.files in your work directory, and run ABINIT:

abinit < tudet_2.files > tudet_1.log

tudet_2.in tudet_2.out tudet_1o tudet_2o tudet_2x ../../../Psps_for_tests/26fe.paw

################################################################### ## Automatic test for ABINIT: ## ## determine U from change of occupation on atoms upon potential ## ## shift on atom 1 ## ## Fe bcc structure - ferromagnetic PAW DJA 2010 ## ################################################################### # 2 atomic supercell acell 3*5.42 chkprim 0 # 0: do not check if uc primitive rprim 1.00 0.00 0.00 0.00 1.00 0.00 0.00 0.00 1.00 nsppol 2 #Definition of the atom types ntypat 1 znucl 26 #Definition of the atoms natom 2 typat 2*1 # atomic types xred 0.0 0.0 0.0 0.5 0.5 0.5 ecut 8 # Energy cutoff pawecutdg 40 # pawecutdg > 2*ecut nband 25 # Fe_2 minband=17 #Definition of the k-point grid kptopt 1 # 1: automatic generation of k points ngkpt 3 3 3 # n x n x n nshiftk 1 shiftk 0.5 0.5 0.5 #Smearing occopt 4 tsmear 0.05 eV #DFT+U usepawu 1 # 1 at lim dble cnt / 2 rnd m fld dle cnt lpawu 2 # ang moments corrrected nsym 48 # nsym&symrel: break cubic symmetry of crystal: allow # individual ionicoccupations symrel 1 0 0 0 1 0 0 0 1 -1 0 0 0 -1 0 0 0 -1 -1 0 0 0 1 0 0 0 -1 1 0 0 0 -1 0 0 0 1 -1 0 0 0 -1 0 0 0 1 1 0 0 0 1 0 0 0 -1 1 0 0 0 -1 0 0 0 -1 -1 0 0 0 1 0 0 0 1 0 1 0 1 0 0 0 0 1 0 -1 0 -1 0 0 0 0 -1 0 -1 0 1 0 0 0 0 -1 0 1 0 -1 0 0 0 0 1 0 -1 0 -1 0 0 0 0 1 0 1 0 1 0 0 0 0 -1 0 1 0 -1 0 0 0 0 -1 0 -1 0 1 0 0 0 0 1 0 0 1 1 0 0 0 1 0 0 0 -1 -1 0 0 0 -1 0 0 0 -1 1 0 0 0 -1 0 0 0 1 -1 0 0 0 1 0 0 0 -1 -1 0 0 0 1 0 0 0 1 1 0 0 0 -1 0 0 0 1 -1 0 0 0 -1 0 0 0 -1 1 0 0 0 1 0 1 0 0 0 0 1 0 1 0 -1 0 0 0 0 -1 0 -1 0 -1 0 0 0 0 1 0 -1 0 1 0 0 0 0 -1 0 1 0 -1 0 0 0 0 -1 0 1 0 1 0 0 0 0 1 0 -1 0 1 0 0 0 0 -1 0 -1 0 -1 0 0 0 0 1 0 1 0 0 1 0 0 0 1 1 0 0 0 -1 0 0 0 -1 -1 0 0 0 -1 0 0 0 1 -1 0 0 0 1 0 0 0 -1 1 0 0 0 -1 0 0 0 -1 1 0 0 0 1 0 0 0 1 -1 0 0 0 1 0 0 0 -1 -1 0 0 0 -1 0 0 0 1 1 0 0 0 0 1 0 1 0 1 0 0 0 0 -1 0 -1 0 -1 0 0 0 0 -1 0 1 0 -1 0 0 0 0 1 0 -1 0 1 0 0 0 0 -1 0 -1 0 1 0 0 0 0 1 0 1 0 -1 0 0 0 0 1 0 -1 0 -1 0 0 0 0 -1 0 1 0 1 0 0 pawujat 1 # default, the atom on which U is determined pawujv 0.1 eV # default, size of the potential shift macro_uj 1 # activate determination of U pawujrad 2.66866 # optional, radius ASA-sphere to which U should be extrapolated #Only to accelerate test irdwfk 1 # default for macro_uj = 1 # nline 2 # nnsclo 2 tolvrs 10d-9 # default for macro_uj = 1 #Save disk space prteig 0 prtwf 0 prtden 0 ## After modifying the following section, one might need to regenerate the pickle database with runtests.py -r #%%<BEGIN TEST_INFO> #%% [setup] #%% executable = abinit #%% test_chain = tudet_1.in, tudet_2.in, tudet_3.in #%% input_prefix = tudet_1o #%% [files] #%% files_to_test = #%% tudet_2.out, tolnlines= 4, tolabs= 8.000e-03, tolrel= 6.500e-01, fld_options = -easy #%% psp_files = 26fe.paw #%% [paral_info] #%% max_nprocs = 1 #%% [extra_info] #%% authors = D.J. Adams #%% keywords = PAW, LDAU #%% description = #%% Fe bcc structure - ferromagnetic PAW #%% determine U from change of occupation on atoms upon potential #%% shift on atom 1 #%%<END TEST_INFO>

As you can see from the tudet_2.files file, this run uses the tudet_1o_WFK as an input. In the tudet_2.in all the symmetry relations are specified explicitly. In the tudet_2.log you can verify that none of the symmetries connects atoms 1 with atom 2:

symatm: atom number 1 is reached starting at atom 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 symatm: atom number 2 is reached starting at atom 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2

This is important. Otherwise the occupation numbers have no freedom to evolve separately on the atoms surrounding the atom on which you apply the perturbation.

You can generate these symmetries, in a separate run, where you specify the atom where the perturbation is done as a different species. From the output you read the number of symmetries (nsym), the symmetry relations (symrel) and the non-symmorphic vectors (tnons). This is already done here and inserted in the tudet_2.in file. Note that you can alternatively just break all the symmetries (nsym=1), or break specific symmetries by displacing the impurity atom in the preliminary run. However, for the determination of U, the positions should be the ideal positions and only the symmetry should be reduced.

For the rest, usually it is enough to set macro_uj 1 to run the calculation of U. Note also, that the irdwfk 1 and the tolvrs 1d-8 need not be set explicitly, because they are the defaults with macro_uj 1.

Once the calculation tudet_2 is converged, you can have look at the output. You can see, that the atomic shift (atvshift) is automatically set:

atvshift 0.00367 0.00367 0.00367 0.00367 0.00367 0.00367 0.00367 0.00367 0.00367 0.00367 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000

This means, that all the 10 3d spin-spin orbitals on the first Fe atom where shifted by 0.1 eV (=0.00367 Ha). On the second atom no shift was applied. Self-consistency was reached twice: Once for a positive shift, once for the negative shift:

grep SCF tudet_2.out

The lines starting with URES

URES ii nat r_max U(J)[eV] U_ASA[eV] U_inf[eV] URES 1 2 4.69390 4.74555 3.67983 3.20150 URES 2 16 9.38770 8.77694 6.80588 5.92122 URES 3 54 14.08160 9.17082 7.11130 6.18694 URES 4 128 18.77540 9.25647 7.17772 6.24472 URES 5 250 23.46930 9.28509 7.19991 6.26403

contain U for different supercells. The column “nat” indicates how many atoms were involved in the supercell, r_max indicates the maximal distance of the impurity atoms in that supercell. The column U indicates the actual U you calculated and should use in your further calculations. U_ASA is an estimate of U for more extended projectors and U_\inf is the estimate for a projector extended even further.

Although it is enough to set macro_uj 1, you can further tune your runs.
As a standard, the potential shift to the 1^{st} atom treated in DFT+U, with a
potential shift of 0.1 eV. If you wish to determine U on the second atom you
put pawujat 2. To change the size of the potential shift use e.g.
pawujv 0.05 eV. Our tests show that 0.1 eV is the optimal value, but the
linear response is linear in a wide range (1-0.001 eV).

## 3 The ujdet utility¶

In general the calculation of U with abinit as described above is sufficient. For some post-treatment that goes beyond the standard applications, a separate executable ujdet was created. The output of abinit is formatted so that you can easily “cut” the part with the ujdet input variables : you can generate the standard input file for the ujdet utility by typing:

sed -n "/MARK/,/MARK/p" tudet_2.out > ujdet.in

Note that the input for the ujdet utility is always called ujdet.in

It contains the potential shifts applied vsh (there are 4 shifts: vsh1, vsh3 for non-selfconsistent calculations that allows to extract the contribution to U originating from a non-interacting electron gas, and vsh2, vsh4 for positive and negative potential shift). The same applies for the occupations occ[1-4].

We now calculate U for an even larger supercell: Uncomment the line scdim in ujdet.in and add

scdim 6 6 6

to specify a 6 6 6 supercell or

scdim 700 0 0

to specify the maximum total number of atoms in the supercell. Then, run ujdet:

rm ujdet.[ol]* ; ujdet > ujdet.log grep URES ujdet.out URES ii nat r_max U(J)[eV] U_ASA[eV] U_inf[eV] URES 1 2 4.69390 4.74555 3.67983 3.20150 URES 2 16 9.38770 8.77694 6.80588 5.92122 URES 3 54 14.08160 9.17082 7.11130 6.18694 URES 4 128 18.77540 9.25647 7.17772 6.24472 URES 5 250 23.46930 9.28509 7.19991 6.26403 URES 6 432 28.16310 9.29738 7.20944 6.27232

As you can see, U has now been extrapolated to a supercell containing 432 atoms.

The value of U depends strongly on the extension of the projectors used in the calculation. If you want to use U in LMTO-ASA calculations you can use the keyword pawujrad in the ujdet.in file to get grips of the U you want to use there. Just uncomment the line and add the ASA-radius of the specific atom e.g.

pawujrad 2.5

Running

rm ujdet.[ol]* ; ujdet > ujdet.log

gives now higher values in the column U_ASA than in the runs before (8.07 eV compared to 7.21 eV): For more localized projectors the U value has to be bigger.