quantum-espresso/PW/examples/example16

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Introduction:
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In this example, we compute the total energy of the low-spin 
transition-metal compound LiCoO2 using orbital-resolved DFT+U.
While most previous implementations of the DFT+U method treat 
nl-subshells (e.g., the d-shell of Co) as a single Hubbard manifold,
the orbital-resolved method allows to assign on-site Hubbard U
parameters on the level of (spin)-orbitals.
This is particularly useful for shells whose (spin-)orbitals are strongly
degenerate in terms of energy and where the "shell-averaged" approximation
of current implementations collapses. See Macke et al., 
arXiv:2312.13580 (2023) for further details.

In the present example, the Co atoms are octahedrally coordinated
by O atoms (O_h point group symmetry), which entails the formation of two
distinct manifolds in the d-shell of Co: the doubly degenerate e_g states
on the one hand and the triply degenerate t_2g ones on the other hand.
Formally, the low-lying t_2g manifold of Co is fully occupied and -- given 
the large number of localized electrons -- can be expected to suffer from
electron self-interaction. Therefore, this manifold should benefit from 
Hubbard U corrections.
In contrast, the e_g manifold is high in energy and formally unoccupied.
However, it participates in the formation of strong sigma-bonds with 
the neighboring O atoms. Since this interaction is not at all "on-site"
but rather "inter-site" (see example13), we should
not treat the e_g manifold with Hubbard U corrections.

In the following, we explain how this can be done using a "two-step"
procedure whose first step is to investigate the order of the eigenstates,
which are then targeted by orbital-resolved U corrections in the second run.

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Explanation of the input:
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The first run is just a plain PBE calculation for LiCoO2 in the primitive
unit cell except that we assign a tiny value of the conventional Hubbard U
to Co in order to force the code to write out the occupation matrices for
the Co-3d shell (but without affecting the PBE result).

Looking at the occupation eigenvalues printed in the output we can clearly
observe the aforementioned degeneracy in the eigenvalues of the Co-3d occupation matrix.
While the first two eigenvalues are ~0.42 (this is the e_g manifold), the last
three eigenvalues are >0.95 and must correspond to the occupied t_2g manifold.

Based on this observation, we can now set up the second input file:
 - We change the value of the Hubbard parameter to the finite value of 5.0eV.
   (NOTE: this is an example value, in practice Hubbard parameters should be
    calculated from first principles, e.g., using LR-cDFT)
 - In the same line, we add the indices "3 4 5" behind the Hubbard U parameter 
   to tell the code that we wish to treat ONLY the 3rd, 4th and 5th eigenstate
   of the Co-3d manifold (== the t_2g orbitals) with Hubbard U corrections
 - Finally, we change the value of the keyword "startingpot" to 'file', to
   ensure that orbital-resolved DFT+U will target the correct eigenstates
      - N.B.: Restarting from a converged charge density is not mandatory, but
	      strongly recommended unless you already know the order of the
              eigenstates after a few self-consistent iterations (see example 16).

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Analysis of the output:
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Focussing on the results, we can see that the orbital-resolved PBE+U solution
is slightly higher in energy than plain PBE, as expected. However, the 
difference between the two is ~0.02Ry (>0.3eV), which is far smaller than 
the difference between the PBE solution and the PBE+U+V solution (example13), 
despite the use of the same Hubbard U parameter!

This shows that the orbital-resolved approach can be useful to
avoid applying spurious (over-)corrections to delocalized states such as e_g.
Moreover, looking at the energy of the highest occupied molecular orbital (HOMO)
and at that of the lowest unoccupied one (LUMO), we can see that the U correction
strongly affects the former, lowering it by about 1.55eV, while having 
no impact on the latter. Therefore, the band gap expands due to the 
Hubbard U-induced downshift of the HOMO.