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..^[1]

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.