CHAPTER 14 · ADVANCED · MAGNETISM

Computing U with hp.x

Computing U from first principles with hp.x: the linear-response workflow and its convergence parameters.

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Instead of copying a U value from the literature, you can compute the U appropriate to your own system by linear response (DFPT). The tool is hp.x (Timrov, Cococcioni, Marzari, Comput. Phys. Commun. 2022, 279, 108455), and the result is far easier to defend methodologically.

The principle in one line

Hubbard U is defined by how strongly the system pushes back when the occupation of the localized manifold is perturbed. hp.x computes that response matrix χ with density-functional perturbation theory and returns the screened interaction

$$U = (\chi_0^{-1} - \chi^{-1})$$

with no empirical input.

Workflow

pw.x (scf with a tiny U, conv_thr 1e-12) → hp.x → prefix.Hubbard_parameters.dat

Step 1. Use the input of Example E11 with the U set to a negligible value. The HUBBARD card must be present so hp.x knows which atoms and manifolds to perturb.

HUBBARD (ortho-atomic)
U Fe1-3d 1.0d-8
U Fe2-3d 1.0d-8

Set conv_thr in &ELECTRONS to about 1.0d-12; linear response is sensitive to the quality of the ground-state density.

Step 2. The hp.x input (&INPUTHP):

&INPUTHP
  prefix       = 'FeO'
  outdir       = './tmp/'
  nq1 = 2, nq2 = 2, nq3 = 2
  conv_thr_chi = 1.0d-6
  iverbosity   = 2
/

pw.x -in feo_hp_scf.in > feo_hp_scf.out
hp.x -in feo.hp.in     > feo.hp.out
cat FeO.Hubbard_parameters.dat

The result file lists U per atom. The measured numbers are in Example E12.

conv_thr_chi is the convergence threshold for the response function χ. In systems whose GGA ground state is metallic (FeO is one), the numerical noise floor of χ sits around 10⁻⁷, so 1.0d-8 may never be reached; the measured evidence is in Example E12.

Cautions

  • Convergence testing over the nq grid is mandatory. A U from 1×1×1 cannot be trusted, and the k-grid of the underlying scf must be converged along with it.
  • hp.x is expensive. The number of perturbations scales with the number of inequivalent Hubbard atoms, and each perturbation runs a linear-response cycle per q-point. -nk pool parallelism applies, and q-points can be split across jobs with start_q/last_q.
  • Feeding the computed U back into the HUBBARD card and repeating scf → hp.x gives a self-consistent U; one or two rounds usually stabilize it.
  • If you split labels (Fe1/Fe2), hp.x recognizes symmetry-equivalent atoms and skips redundant perturbations.
Common mistakes

Transplanting an hp.x U into a run with a different projector or pseudopotential. U is meaningful only within the conditions it was computed for (projector, pseudopotential, magnetic order). Also, a loose conv_thr on the preceding scf injects noise into the response matrix and swings U by whole eV: keep the 1.0d-12.