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, and then walk into the famous trap of this system, the case where the run stays metallic even with U on, and learn to diagnose it. Knowing this trap is what lets you read FeO+U results in the literature critically.

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 trap section)

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 the U proper to 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: measured

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 the trap below
FeO spin-resolved DOS: GGA vs GGA+U
Measured 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: the measured face of the "U alone does not make it an insulator" trap.

The trap: why it stays metallic with U on (with measurements)

The DOS shows U doing real work (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 (the mailing-list case cited in Chapter 13).

The standard 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
/

We report what actually happened when we 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 simply 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 precisely this situation). A living example of "converged does not mean correct".

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. You will feel in your hands that "U comes packaged 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