Calculate U for LSDA+U

Task

In this exercise you will calculate the U parameter for the LSDA+U treatment of Ni d-electrons in NiO using the linear response ansatz of Cococcioni et al..

POSCAR

For this calculation we will use a 2×2×2 supercell of AFM-II NiO:

AFM  NiO
   4.03500000 
 2.0000000000   1.0000000000   1.0000000000 
 1.0000000000   2.0000000000   1.0000000000 
 1.0000000000   1.0000000000   2.0000000000 
  1 15   16 
Direct
 0.0000000000   0.0000000000   0.0000000000 
 0.2500000000   0.2500000000   0.2500000000 
 0.0000000000   0.0000000000   0.5000000000 
 0.2500000000   0.2500000000   0.7500000000 
 0.0000000000   0.5000000000   0.0000000000 
 0.2500000000   0.7500000000   0.2500000000 
 0.0000000000   0.5000000000   0.5000000000 
 0.2500000000   0.7500000000   0.7500000000 
 0.5000000000   0.0000000000   0.0000000000 
 0.7500000000   0.2500000000   0.2500000000 
 0.5000000000   0.0000000000   0.5000000000 
 0.7500000000   0.2500000000   0.7500000000 
 0.5000000000   0.5000000000   0.0000000000 
 0.7500000000   0.7500000000   0.2500000000 
 0.5000000000   0.5000000000   0.5000000000 
 0.7500000000   0.7500000000   0.7500000000 
 0.1250000000   0.1250000000   0.1250000000 
 0.3750000000   0.3750000000   0.3750000000 
 0.1250000000   0.1250000000   0.6250000000 
 0.3750000000   0.3750000000   0.8750000000 
 0.1250000000   0.6250000000   0.1250000000 
 0.3750000000   0.8750000000   0.3750000000 
 0.1250000000   0.6250000000   0.6250000000 
 0.3750000000   0.8750000000   0.8750000000 
 0.6250000000   0.1250000000   0.1250000000 
 0.8750000000   0.3750000000   0.3750000000 
 0.6250000000   0.1250000000   0.6250000000 
 0.8750000000   0.3750000000   0.8750000000 
 0.6250000000   0.6250000000   0.1250000000 
 0.8750000000   0.8750000000   0.3750000000 
 0.6250000000   0.6250000000   0.6250000000 
 0.8750000000   0.8750000000   0.8750000000

Atoms 1-16 are Ni and atoms 17-32 are O.

Note that the Ni atoms are split into two groups: atom 1, and atom 2-15. This trick breaks the symmetry of the Ni sub-lattice and allows us to treat atom 1 differently from atom 2-15. Our POTCAR file has to reflect the fact that we now formally have 3 "species" (2 ×Ni + 1×O), i.e., we concatenate two Ni POTCAR files and one O POTCAR file:

cat Ni/POTCAR Ni/POTCAR O/POTCAR > POTCAR

To check whether you have a suitable POTCAR type:

grep TITEL POTCAR

This should yield something like:

   TITEL  = PAW Ni 02Aug2007
   TITEL  = PAW Ni 02Aug2007
   TITEL  = PAW O 22Mar2012

i.e., two Ni entries followed by one O entry.

KPOINTS

Gamma only
 0
Monkhorst
 1 1 1 
 0 0 0

The DFT groudstate

We will calculate the DFT groundstate of our NiO system with the following INCAR:

SYSTEM       = NiO AFM 

PREC         = A

EDIFF        = 1E-6

ISMEAR       = 0
SIGMA        = 0.2

ISPIN        = 2
MAGMOM       = 1.0 -1.0  1.0 -1.0  1.0 \
              -1.0  1.0 -1.0  1.0 -1.0 \
               1.0 -1.0  1.0 -1.0  1.0 \
              -1.0  1.0 -1.0  1.0 -1.0 \
            16*0.0

LORBIT       = 11

LMAXMIX      = 4 

Instrumental here is that we correctly specify the initial magnetic moments (by means of MAGMOM-tag). The setting above is consistent with the AFM-II magnetic structure.

Secondly we set LORBIT=11: at the end of the OUTCAR file VASP will write the number of (d-) electrons per site. This information we will need to compute the U-parameter.

Last but not least, we set LMAXMIX=4: this is needed to be able to perform non-selfconsistent (ICHARG=11) LSDA+U calculations (LDAUTYPE=3) in the following.

Non-selfconsistent response

ICHARG       = 11

LDAU         = .TRUE.
LDAUTYPE     =  3
LDAUL        =  2 -1 -1
LDAUU        =  $v 0.00 0.00
LDAUJ        =  $v 0.00 0.00

Selfconsistent response

$U = \chi^{-1}-\chi_0^{-1} \approx \left(\frac{\partial N^{\rm SCF}_{I}}{\partial V_{I}}\right)^{-1} - \left(\frac{\partial N^{\rm NSCF}_{I}}{\partial V_{I}}\right)^{-1} = \frac{1}{0.131333}-\frac{1}{0.492333} = 5.58 \; eV$

References

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