EXAMPLE · E11

FeO with DFT+U

Turn on U with the HUBBARD card, measure the Hubbard splitting, and diagnose the famous metallic trap.

Goal

Add three HUBBARD lines to the input of E10 and turn on DFT+U. Measure what U does to the Fe-3d manifold, then look at a known pitfall of this system: the run stays metallic even with U on. The example shows how to diagnose it, which also helps when reading FeO+U results in the literature.

New cards and variables

Item Role
HUBBARD (ortho-atomic) The v7.1+ card with its projector (Chapter 13)
U Fe1-3d 4.6 U (eV) per label and manifold
starting_ns_eigenvalue Steering the d occupations toward a chosen minimum (see the section on the metallic solution)

Input file

Download feo_u.scf.in · scan_U.sh

The input is E10 (with prefix = 'FeO_U' so the E10 files survive) plus this card at the very end:

! The v7.1+ Hubbard card. (ortho-atomic) picks the projector; U values are
! in eV, addressed per label and manifold. The Fe1/Fe2 label split from E10
! is what lets the two sublattices carry independent ns matrices.
! U = 4.6 eV is a conventional literature value; E12 computes 5.22 eV for
! this exact setup from first principles.
HUBBARD (ortho-atomic)
U Fe1-3d 4.6
U Fe2-3d 4.6

Computing U for your own system is E12.

Run

mpirun -np 8 pw.x -nk 4 -in feo_u.scf.in > feo_u.scf.out

What to check

Item GGA (E10) GGA+U (this example)
Total energy −741.81592 Ry −741.52737 Ry (do not compare directly: different functionals)
total / absolute magnetization 0.00 / 7.17 μB −0.00 / 7.50 μB
Fe local moments ±3.31 μB ±3.46 μB (U strengthens the d localization)
Electronic structure Metal Still (semi)metallic: see below
FeO spin-resolved DOS: GGA vs GGA+U
Spin-resolved DOS of FeO (QE 7.5). With U on (right), the occupied and unoccupied d manifolds split apart and Hubbard gaps open above and below. But in the ideal cubic cell a narrow band of minority-spin t2g character stays pinned at the Fermi level, so U alone does not make it an insulator.

Why it stays metallic with U on

U does change the electronic structure (larger moments, wide Hubbard splitting). The problem is that the single minority-spin electron of Fe²⁺ must choose among three t2g orbitals that are exactly degenerate in the ideal cubic cell. Unable to pick one, it spreads across all three and forms a narrow metallic band. Since QE 7.1 the starting d occupations are read from the pseudopotential (previously hardcoded), so the same input can land in different metallic solutions on different versions, and neither is the correct ground state (see Chapter 13).

The usual prescription is to steer the occupations:

&SYSTEM
  ...
  starting_ns_eigenvalue(5,2,1) = 1.d0   ! Fe1: fill the top minority eigenvalue
  starting_ns_eigenvalue(5,1,2) = 1.d0   ! Fe2: mirrored spin channel
/

Here is what happened when I tried:

  1. With the seed above, the SCF returned to the same metallic solution (energies agree to 4×10⁻⁷ Ry).
  2. A second attempt that pinned the full minority occupation pattern to [1,0,0,0,0], with mixing_beta lowered to 0.1 and degauss to 0.005, oscillated near 10⁻⁴ Ry for 98 iterations without converging.

In this geometry (ideal rocksalt) and setup, the metallic basin is very deep. Real FeO distorts rhombohedrally along [111] below its Néel temperature, and that distortion lifts the t2g degeneracy so that orbital order and the insulating state settle in together. To reach the full insulating solution you would combine (1) the experimentally distorted structure, (2) a broader search over starting_ns_eigenvalue patterns, and (3) the orbital-resolved DFT+U of QE 7.5 (different U for t2g and eg, introduced for this situation). The SCF converged here, but the result is not the correct ground state.

Exercises

  1. Use scan_U.sh to scan U = 0, 2, 4, 6, 8 eV and tabulate the Fe moment and the Hubbard splitting in the DOS.
  2. Switch the projector between atomic and ortho-atomic at fixed U and compare. The value of U only means something together with its projector.
  3. Add a ~3% rhombohedral distortion along [111] to CELL_PARAMETERS and rerun with the occupation seed. Does a gap open now?
  4. Try DFT+U+J0 by adding J0 Fe1-3d 0.8.
Common mistakes

Concluding "U is on and the SCF converged, so it must be an insulator now". Always check whether the Fermi energy is disappeared in favor of highest occupied, lowest unoccupied level, and inspect the Tr[ns(na)] and ns eigenvalue blocks (verbosity='high') for a physical occupation pattern. And never compare GGA and GGA+U total energies directly; they belong to different functionals.

13 DFT+U and the HUBBARD card · 14 Computing U with hp.x · 12 Spin polarization and magnetism