THEOS

DFT+Hubbard (i.e. DFT+U and its generalizations DFT+U+V, DFT+U+J, ...) is a popular and powerful tool to model the structural, electronic, magnetic and vibrational properties of compounds with localised d or f electrons. The functional extension from DFT to DFT+U is simple and has low computational costs (compared with more computationally expensive methods, such as hybrid functionals); in addition, it is often accurate (we actually hope it could be more accurate than hybrids, if applied properly, since it's aware of its surroundings, when applied using the techniques below). You can find some of the key pioneering references here: V.I. Anisimov et al., Phys. Rev. B 44, 943 (1991), A.I. Liechtenstein et al., Phys. Rev. B 52, R5467(R) (1995), S. L. Dudarev et al., Phys. Rev. B 57, 1505 (1998). The key ingredients for a proper DFT+Hubbard study are the Hubbard parameters, such as the on-site U, or, in the cases discussed below, also the inter-site V. Understanding what are the right values for U and V is key to the correct use of this computational technology - in the following we address this issue in more detail.

DFT+Hubbard: correlations or self-interactions? DFT+U has historically been aimed at describing electronic correlation in solids; in fact, it was inspired by the Hubbard model of electrons on a lattice. But how does this approach actually address correlations when applied not to a lattice, but to the inhomogeneous interacting electron gas of a solid, as described by density-functional theory? A first key step, in our opinion, has been made by showing that the DFT+U functional aims to correct the curvature of the energy with respect to the occupancy of the Hubbard manifold that is exhibited by standard DFT functionals M. Cococcioni and S. de Gironcoli, Phys. Rev. B 71, 035105 (2005)). Mind you, there is no theorem that justifies that this is an exact property to satisfy, but one could argue that in the dissociation of a very simple system (the hydrogen molecular ion H2+) the correct left-right degeneracy is obtained only if linearity in the energy exists for the addition/removal of an arbitrary fraction of an electron to the left/right atom. This consideration can be heuristically extended to a localised Hubbard manifold (i.e. the set of 3d or 4 f electrons) in a solid, with DFT+U aiming to linearise the energy when a fraction of electron is being added or removed from the manifold - justifying the linearisation argument above. And of course H2+ is a very well defined case of a system suffering from one-electron self-interaction, so what the Hubbard term is actually doing is fixing self-interaction, not correlation! This novel point was made by taking the case of molecules (rather than solids - if the system contains only one transition-metal atom, there is no correlation as intended from the Hubbard model) and showing that DFT+U improves very significantly their description (H.J. Kulik et al., Phys. Rev. Lett. 97, 103001 (2006)). Many papers on molecules followed, reinforcing the case that, when embedded into a real DFT calculation as opposed to a lattice model, the Hubbard correction does not deal at all with with correlations, but with self-interactions (this point can be made more clear by showing that the U controls the charge transfer from the Hubbard manifold to the environment - in a lattice model there is no other place to go than another site, but in a real DFT calculations the electron that exits the Hubbard manifold will go and sit on the ligands, or wander around, but will not go to another site/Hubbard manifold).

DFT+Hubbard: energies or band gaps? DFT is a theory of total energies. The auxiliary system of non-interacting Kohn-Sham electrons that are used to solve the Kohn-Sham equations is able to capture reasonably well the true quantum kinetic energy, and thus the ground state energy of the interacting system, but the Kohn-Sham levels have no theoretical foundation to be interpreted as band structures (the HOMO does, but just because it determines the asymptotic decay of the charge density, that exact DFT must describe exactly). For this reason, asking DFT+U or its extensions to represent spectral properties - such as the band gap, or any other charged excitation - is, well, incorrect.

What is the Hubbard V? DFT+U is now widely used, but not everybody is aware of DFT+U+V, which turns out to be a valuable and important extension of DFT+U (V.L. Campo Jr and M. Cococcioni, J. Phys.: Condens. Matter 22, 055602 (2010)). V, like U, is a Hubbard parameter, but contrary to U it describes inter-site electronic interactions and not on-site ones (e.g. how a 3d Hubbard manifold talks to the 2p orbitals of oxygen ligands). The idea of including V in the DFT+Hubbard formulation comes from the "extended Hubbard model", where both on-site U and inter-site V parameters are used. So, DFT+U+V is a natural extension of DFT+U. Is it important to include V in calculations? Yes, when covalent interactions between the Hubbard manifold and its surrounding are strong in the material of interest. For example, the inclusion of inter-site V turns out to be very important to compute accurately the structure of transition-metal di-oxides (H. J. Kulik and N. Marzari, J. Chem. Phys. 134, 094103 (2011)), the voltages of Li-ion batteries (M. Cococcioni and N. Marzari, Phys. Rev. Materials 3, 033801 (2019)), or the formation energies of oxygen vacancies in transitional-metal oxides (C. Ricca, I. Timrov, M. Cococcioni, N. Marzari, and U. Aschauer, Phys. Rev. Research 2, 023313 (2020)).

Which value of Hubbard parameters to choose? Very often, empirical Hubbard parameters are used. For example, it is common practice to choose a value of U able to reproduce some properties of interest (sometimes even the experimental band gap, that is not really correct from a conceptual point of view - especially because DFT is not a theory of band gaps, but also because the frontier states might be originating from atomic orbitals that are not involved in the U term). Also, if there are no experimental data available for a material of interest, U is often taken from studies on similar materials, but doing so is not justified: see the next point below regarding the transferability of Hubbard parameters). In addition, using an empirical U makes DFT+U a semi-empirical, rather than a first-principles approach, because U is adjusted/tuned by hand. We believe that the correct approach is to compute U (and V) from first principles is as a linear response property, and this was the key suggestion of (M. Cococcioni and S. de Gironcoli, Phys. Rev. B 71, 035105 (2005)). Its practical implementation remained somewhat cumbersome for years, due to the need to calculate linear-response properties by finite differences in large supercells. It has now been recast in reciprocal space (i.e. using primitive cells and sampling monochromatic perturbations, rather than using supercells) thanks to the use of density-functional perturbation theory (DFPT) (I. Timrov, N. Marzari, and M. Cococcioni, Phys. Rev. B 98, 085127 (2018)). This approach to compute Hubbard U and V based on DFPT is implemented in Quantum ESPRESSO in the HP code, where "HP" stands for "Hubbard Parameters" (I. Timrov, N. Marzari, and M. Cococcioni, Phys. Rev. B 98, 085127 (2018)). (Note: currently the HP code allows to compute Hubbard parameters only for open-shell systems; for closed-shell systems Hubbard parameters are unphysically large K. Yu and E.A. Carter, J. Chem. Phys. 140, 121105 (2014)).

Are Hubbard parameters universal/transferable? Mostly, not. When Hubbard parameters (U and V) are computed from first principles (e.g. using linear response theory - see above), their value depends on many physical or technical details: first and foremost, the actual size/shape of the Hubbard manifold (i.e. which localised functions are used for the Hubbard projections: atomic orbitals, orthogonalised atomic orbitals, Wannier functions, ...); these will change depending on which pseudopotentials are used, and which was the oxidation state of the original all-electron atomic calculation (see a discussion of this in the Appendix of H.J. Kulik and N. Marzari, J. Chem. Phys. 129, 134314 (2008)). Also, which exchange-correlation functional (LDA, PBE, PBEsol,...) has been used will matter, and whether U and V are computed as a one-shot response property, or iterated self-consistently (see e.g. M. Cococcioni and N. Marzari, Phys. Rev. Materials 3, 033801 (2019)). Therefore, Hubbard U and V computed under one set of technical assumptions (e.g. using PBEsol, ortho-atomic orbitals, ultrasoft pseudopotentials from the GBRV library) cannot be used for a different DFT+U+V (or DFT+U) calculation (e.g. using LDA, atomic orbitals, norm-conserving pseudopotentials from the Pseudo Dojo library). Consistency between how Hubbard parameters are determined, and how they are used, is crucial. Moreover, the Hubbard parameters for the same element/pseudopotential but in different environments can vary quite significantly (see the U for Co-3d when used in LiCoO2 and in CoO, whose values differ significantly, see e.g. A. Floris, I. Timrov, B. Himmetoglu, N. Marzari, S. de Gironcoli, and M. Cococcioni, Phys. Rev. B 101, 064305 (2020)). Hopefully, the addition of the V term, or using orthogonalised atomic projectors, ameliorates transferability.

Atomic or ortho-atomic orbitals? As discussed above, the values of U and V depend strongly on the technical details of calculations. Concerning pseudopotentials, we recommend to use those suggested in the SSSP library. Regarding the exchange-correlation functional, for solids PBEsol is probably the best straightforward choice, as PBE ia for molecules (SCAN these days is becoming quite popular - but there are still no SCAN pseudo potentials available, AFAWK). Last, for the Hubbard manifold, should one use atomic or ortho-atomic orbitals (here we refer to two options implemented in Quantum ESPRESSO)? Ortho-atomic orbitals seem to provide more accurate results - maybe because if atomic (i.e. non-orthogonal) orbitals are used, the Hubbard correction can be added twice in the regions where orbitals overlap, or, when using the U alone, the orthogonalisation procedure makes the system more "aware" of the presence of the ligands (but, at the end, we think the most general approach is to use both U and V in a ortho-atomic formulation).