CHAPTER 13 · ADVANCED · MAGNETISM

DFT+U and the HUBBARD card

Self-interaction error and DFT+U, the v7.1+ HUBBARD card syntax, and why (and how) a gap opens in FeO.

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Why it is needed

GGA mishandles the self-interaction error of localized 3d electrons. The spurious interaction of an electron with itself delocalizes orbitals that should be localized, and in systems like FeO the result is a computed metal where experiment sees an insulator (gap ≈ 2.4 eV). You can watch this failure happen in Example E10.

DFT+U adds a Hubbard correction on a chosen orbital manifold (Fe-3d, say) and repairs the error at nearly zero extra cost.

The new syntax: the HUBBARD card (v7.1+)

U parameters go in a HUBBARD card at the end of the input, not in &SYSTEM.

HUBBARD (ortho-atomic)
U Fe1-3d 4.6
U Fe2-3d 4.6

The grammar:

HUBBARD (<projector>)
<parameter> <label>-<manifold> <value(eV)>
Field Options Notes
Projector atomic, ortho-atomic, norm-atomic, wf, pseudo ortho-atomic recommended. With atomic, the correction double-counts in orbital-overlap regions
Parameter U, J0, J, B, E2, E3, V, alpha
Manifold 3d, 2p, 4f, ... Up to 3 channels per atomic type

Inter-site interactions (DFT+U+V) also take neighbor indices (numbered by order in ATOMIC_POSITIONS):

HUBBARD (ortho-atomic)
U Fe1-3d 4.6
U Fe2-3d 4.6
V Fe1-3d O-2p  1 3  0.8

The authoritative document is Doc/Hubbard_input.pdf (online).

Do not use the old syntax

The lda_plus_u = .true. / Hubbard_U(1) = 4.6 / U_projection_type style was retired in v7.1. Much of the internet still shows it, and pasting it gets you input that is silently ignored or fails. The only DFT+U variables left in &SYSTEM are starting_ns_eigenvalue and Hubbard_occ.

Measured: what U does in FeO, and what it cannot do alone

Example E11 is the same FeO AFM-II cell with just the three HUBBARD lines added.

FeO DOS: GGA (metallic) vs GGA+U (Hubbard splitting)
Measured spin-resolved DOS of FeO (QE 7.5, PBE, AFM-II). Left, GGA: Fe-3d states sit at the Fermi level and the system is metallic (experiment: an insulator). Right, GGA+U (4.6 eV, ortho-atomic): the Hubbard splitting opens gaps above and below, but in the ideal cubic cell a narrow band derived from the minority-spin t2g states survives at the Fermi level. That last trap is treated below and in E11.

If it is still metallic with U on: the occupation-pattern trap

It is common to switch U on and still converge to a metal. The culprit is the occupation pattern of the d orbitals. Since QE 7.1 the initial d occupations are read from the pseudopotential (they used to be hardcoded), so the same input can converge to different metallic solutions on different versions. Both are wrong ground states (this is the mailing-list case the community knows well).

The prescription is to steer the occupations explicitly with starting_ns_eigenvalue:

&SYSTEM
  ...
  starting_ns_eigenvalue(5, 2, 1) = 0.0d0   ! (orbital index, spin, atom type)
/

The constraint holds only for the first few SCF iterations and is then released; think of it as a device that pushes the run into the basin of the correct minimum. After convergence, inspect the Tr[ns(na)] values and the ns eigenvalue blocks in the output (verbosity='high' required) to confirm the occupation pattern is physical. This is the canonical example of "converged does not mean correct".

U comes as a package with its projector

The same U = 4.6 eV gives different results under atomic and ortho-atomic projectors. A U value has meaning only together with its projector (and pseudopotential), so when borrowing U from a paper, check the projector conventions, and when publishing, record your own. The way to avoid borrowing altogether is to compute U for your own system: Chapter 14, hp.x.

QE 7.5 adds orbital-resolved DFT+U (Macke & Timrov, JCTC 2024), which can assign different U values to t2g and eg within one 3d manifold. It was designed for octahedral transition-metal oxides, exactly the FeO situation.