[QE-users] Advice on HP for a PBA material

[Eric Macke]

Dear Krystian,

Calculating PBAs with "standard" DFT+U formulations is a quite tricky 
thing to do for fundamental reasons. While the method works well for 
many other battery materials, it tends to fail for PBAs, and 
LR-cDFT/DFPT indeed often yield inexplicably large Hubbard parameters. 
So what is happening?

The LS sites (the transition metals sitting at the C-terminations) of 
PBAs are subject to an extremely strong (~octahedral) ligand field, and 
carbon itself is an element with a much lower electronegativity than 
oxygen, which most of the DFT+U community is used to seeing as a ligand. 
This results in a very appreciable covalent character of the 
metal-ligand bond (Fe--C in your case); so far so good. But what does 
this mean for DFT+U and for the calculation of U parameters from 
perturbative methods?

1. While you didn't mention it, I strongly assume that you used atomic 
or ortho-atomic Hubbard projectors for your calculations. Now lets look 
at the eigenvalue spectrum of the d-orbital occupation matrix in 
Mn[Fe(CN)6], as outputted by QE:

      SPIN  1
      eigenvalues:
        0.471  0.511  0.928  0.928  0.957
      SPIN  2
      eigenvalues:
        0.039  0.407  0.459  0.905  0.905

As expected for a LS-Fe(III) species, we see that there are 5 almost 
full d-orbitals, namely those with occupation eigenvalues >0.9. We also 
see that in the spin-down manifold, there is an eigenvalue that is very 
close to 0, and this is the unoccupied "t2g-like" orbital (careful that 
the point symmetry of Fe in Mn[Fe(CN)6] is not really O_h, but we'll 
approximate it as such). But then what are the other 4 orbitals with 
eigenvalues ~0.5? Well, these should be the e_g orbitals, but as you can 
see, they are half-occupied even though fundamental chemistry tells us 
they should be empty. What's going on here is that the atomic-like 
projectors we're using are simply very very bad approximations for a 
state that is so hybridized. The e_g orbitals of Fe are forming sigma 
bonds with the CN ligand MOs, and this hybridization leads to an MO with 
a shape vastly different from that of a d-orbital in a free atom 
calculation (for which the projector was generated!). So it is the 
mismatch between and projector and the state we seek to represent that 
explains these deceptively "fractional" occupation values.

2. This has direct implications for SCF calculations with a finite U 
value. If you apply, say, U(Fe-3d)=6eV to these states using the Dudarev 
functional, then every single one of these orbitals will give rise to a 
punitive Hubbard energy of U/2*(0.5*(1-0.5) = 0.75eV, which is a huge 
amount of energy. Consequently, the system will try to avoid these 
fractional occupations at all cost, either by unphysically switching to 
the HS state or by unphysically extending to bond lengths far beyond 
their experimental values if you allow for relaxations (see Mariano et 
al., J. Chem. Theory Comput. 2020, 16, 11, 6755–6762 for a detailed 
analysis performed on Fe(II) hexacomplexes including Fe[CNH]6).

3. The problem is aggravated when you start computing the U values from 
first principles using DFPT or the equivalent LR-cDFT approach 
(Cococcioni & de Gironcoli, Phys. Rev. B *71*, 035105). When you 
calculate the response of a manifold (such as Fe-3d) by perturbing it to 
obtain U, you essentially calculate how much the non-perturbed states 
are able to screen this perturbation. If your perturbation is 
well-screened, you'll get low U parameters, and vice versa. Perturbing 
the Fe-3d shell in Mn[Fe(CN)6], you'd expect that the screening should 
be mainly due to the C and N 2p orbitals. However, and here is the 
problem, if you 3d projector already includes a large chunk of the C-2p 
(and eventually even N-2p) orbitals, then these orbitals are INSIDE the 
perturbed space, and therefore cannot screen the perturbation. Thus, you 
obtain huge Hubbard U parameters that are only partially screened when 
they should be fully screened.

So what can you do? One option is to change your Hubbard projector basis 
to one that offers a better (more localized!) representation of your 
hybridized states, such as Wannier functions. This way you'd get rid of 
these deceptive fractional occupations and also find lower U values. 
However, the drawback is that the Wannierization can be quite cumbersome 
and then you cannot compute forces and stresses anymore.

An easier ad-hoc fix is orbital-resolved DFT+U, which has been published 
by us in J. Chem. Theory Comput. 2024, 20, 11, 4824–4843. Here the idea 
is to circumvent the problem of non-ideal projector orbitals by simply 
excluding the hardly projectable eigenstates from the Hubbard manifold. 
This leads to orbital-resolved Hubbard parameters, such as 
U(Fe-t2g)=3.5eV instead of U(Fe-3d)=8.0eV. The difficulty here is to 
find a suitable Hubbard manifold, for which you need to invoke group 
theory and also analyze the electronic structure (see Warda et al., J. 
Chem. Theory Comput. 2026, 22, 2, 1016–1029 for an example).

In any case, I would advise against using "standard" DFT+U calculating 
PBAs, especially not in combination with U parameters obtained from 
shell-averaged DFPT. For simple tests, it might be best to simply use an 
empirical value of 2eV>U>5eV for most d-shells, for anything more 
serious I'd use OR-DFT+U or DFT+U with Wannier projectors.

All the best for you and your work,

Eric

-- 
Eric Macke

PhD Student
Hybrid Materials Interfaces Group

University of Bremen
Faculty of Production Engineering
TAB-Building, Room 3.29
Am Fallturm 1
28359 Bremen, DE
http://www.hmi.uni-bremen.de/

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