Subspace representations in ab initio methods for strongly correlated systems

We present a generalized definition of subspace occupancy matrices in ab initio methods for strongly correlated materials, such as DFT$+U$ (density functional theory $+$ Hubbard $U$) and DFT$+$DMFT (dynamical mean-field theory), which is appropriate to the case of nonorthogonal projector functions. By enforcing the tensorial consistency of all matrix operations, we are led to a subspace-projection operator for which the occupancy matrix is tensorial and accumulates only contributions which are local to the correlated subspace at hand. For DFT$+U$, in particular, the resulting contributions to the potential and ionic forces are automatically Hermitian, without resort to symmetrization, and localized to their corresponding correlated subspace. The tensorial invariance of the occupancies, energies, and ionic forces is preserved. We illustrate the effect of this formalism in a DFT$+U$ study using self-consistently determined projectors.

Figure 1

Isosurfaces of the set of nonorthogonal generalized Wannier functions (NGWFs) on a nickel atom in NiO. The NGWFs are those computed at projector self-consistency in the tensorial representation at LDA$+U=6$ eV. Those in the left column (predominantly $3d-t_{2g}$ character) and the top and bottom NGWFs in the middle column (predominantly $3d-e_g$ character) are those used as Hubbard projectors, while the remaining NGWFs (pseudized $4s$-like in the center and pseudized $4p$-like in the right column) lie outside the correlated subspace on that atom. The isosurface is set to half of the maximum for the $4s$ and $4p$-like NGWFs and ${10}^{-3}$ times the maximum for the $3d$-like NGWFs.

Figure 2

The total occupancy of a correlated subspace in NiO (left axis) and the maximum element on the diagonal of the occupancy matrix (right axis) as functions of the interaction $U$. Values are computed with orthonormal hydrogenic Hubbard projectors and self-consistent NGWF projectors in both the dual and tensorial representations.

Figure 3

The projection of the magnetic dipole moment onto the DFT$+U$ correlated subspace on nickel atoms in NiO, computed as a function of the interaction $U$. Values are computed as in Fig. 2.

Figure 4

Top: Density of Kohn–Sham states per spin per atom of NiO at LDA$+U=6$ eV, together with its projection onto the union of correlated subspaces using hydrogenic Hubbard projectors and NGWF projectors in the tensorial and dual representations. Bottom: The decomposition, in the NGWF-tensorial representation, of the density of states for a given spin channel into its contributions from NGWFs on nickel atoms with magnetization aligned (majority) and antialigned (minority) spins, the correlated subspace projections of each, and the contribution due to NGWFs on oxygen atoms.

Figure 5

The Hubbard $U$ dependence of the Kohn–Sham band gap of NiO at LDA$+U$.

Figure 6

Spin-density isosurfaces at 5% of maximum in Cu(ii)Pc${}_2$ at projector self-consistent GGA$+U=6$ eV within the NGWF-tensorial subspace representation.

Figure 7

The average magnitude of the projection of the magnetic dipole onto the correlated subspaces of Cu(ii)Pc${}_2$, plotted as a function of $U$ for various definitions of the subspace projection.

Figure 8

The HOMO-LUMO energy gap (top) and the energy levels adjacent to the Fermi energy (bottom) of Cu(ii)Pc${}_2$, plotted as a function of $U$. In the bottom panel, solid lines show energy levels of states of predominantly Cu-centered $b_{1g}$ character, and to which the DFT$+U$ correction strongly applies, while dashed lines show energy levels of states of more delocalized nature.