Density functional versus spin-density functional and the choice of correlated subspace in multivariable effective action theories of electronic structure

Modern extensions of density functional theory such as the density functional theory plus U and the density functional theory plus dynamical mean field theory require choices, including selection of variable (charge vs spin density) for the density functional and specification of the correlated subspace. This paper examines these issues in the context of the “plus U” extensions of density functional theory, in which additional correlations on specified correlated orbitals are treated using a Hartree-Fock approximation. Differences between using charge-only or spin-density-dependent exchange-correlation functionals and between Wannier and projector-based definitions of the correlated orbitals are considered on the formal level and in the context of the structural energetics of the rare-earth nickelates. It is demonstrated that theories based on spin-dependent exchange-correlation functionals can lead to large and in some cases unphysical effective on-site exchange couplings. Wannier and projector-based definitions of the correlated orbitals lead to similar behavior near ambient pressure, but substantial differences are observed at large pressures. Implications for other beyond density functional methods such as the combination of density functional and dynamical mean field theory are discussed.

Figure 1

Average Ni-O bond length difference $\delta a$ as a function of volume computed using the GGA functional with DFT+U (top panel) and SDFT+U (lower panel) for ${\mathrm{LuNiO}}_3$ (red square), ${\mathrm{NdNiO}}_3$ (blue diamond), and ${\mathrm{LaNiO}}_3$ (green circle). $U=5$ eV and $J=1$ eV are used. The same orthonormalized projector method is used to construct the correlated subspace in all cases. $V_0$ is the zero-pressure volume computed for the given compound by the given method.

Figure 2

Bond disproportionation $\delta a$ plotted against relative volume change $(V-V_0)/V_0$ computed for ${\mathrm{LaNiO}}_3$ for different Hund's coupling $J$ at $U=5$ eV using both (a) GGA and (b) LDA functionals and different methods indicated in the legends (DFT+U vs SDFT+U). The same orthonormalized projector method is used to construct the correlated subspace in all cases. $V_0$ is the zero-pressure volume computed using the given method and $J=1$ eV. The change of $V_0$ at different $J$ is negligible for SDFT+U and rather notable for DFT+U (see Fig. 5) but it does not affect any results discussed here.

Figure 3

The trace of the site-resolved occupancy matrix of the $d$ electrons, $N_d$, per Ni atom computed for ${\mathrm{LaNiO}}_3$ as a function of volume using GGA+U with $J=1$ eV (pentagon dots), $J=2$ eV (square dots), and SGGA+U with $J=0$ eV (circular dots). Two $N_d$ values indicate two Ni atoms with distinct Ni-O bond lengths.

Figure 4

The magnetic moments per Ni atom in ${\mathrm{LaNiO}}_3$ obtained using different $J$ as a function of pressure using both (a) GGA and (b) LDA functionals and different methods indicated in the legends (DFT+U vs SDFT+U). The same orthonormalized projector method is used to construct the correlated subspace in all cases. $V_0$ is the zero-pressure volume computed using the given method and $J=1$ eV.

Figure 5

The total energy of ${\mathrm{LaNiO}}_3$ as a function of relative volume difference computed using GGA+U (top) and SGGA+U (bottom) with an orthonormalized projector definition of the correlated orbitals at $J$ values indicated. At each volume and for each method the structure is relaxed and the energy of the relaxed structure is presented. The zero of energy is chosen at a compression of $5\%$ in all curves.

Figure 6

Contributions to DFT+U energy functional [cf. Eq. (10)] computed for ${\mathrm{LaNiO}}_3$ as a function of relative volume. (a) The DFT contribution and (b) the correlation correction CC contribution ($E^{\mathrm{\Delta }KS}+E^{\mathrm{int}}$) are decomposed from Fig. 5 using the same relaxed structure at each volume and compared for different methods [DFT+U (the top panel) vs SDFT+U (the bottom panel)] and different $J$ values.

Figure 7

The occupancy of the correlated orbitals expressed as the number of $d$ electrons $N_d$ per Ni atom computed for ${\mathrm{LaNiO}}_3$ at $J$ values indicated for GGA+U (the top panel) and SGGA+U (the bottom panel).

Figure 8

The average Ni-O bond length difference $\delta a$ as a function of reduced volume computed for materials indicated using DFT+U as implemented with the energy functional of Eq. (10). Both maximally localized Wannier functions (filled symbols, solid lines) and orthonormalized projectors (open symbols, dashed lines) are compared. Interaction parameters of $U=5$ eV and $J=1$ eV are used for the projector-based calculations while the equivalent values $u=6.14$ eV and $j=0.71$ eV (Slater-Kanamori parametrization) are correspondingly used for the Wannier construction.

Figure 9

The spread of Ni $e_g$ maximally localized Wannier function (MLWF) as a function of the bond length difference $\delta a$ for ${\mathrm{NdNiO}}_3$ at $(V-V_0)/V_0=-12.5\%$. Ni A (square dots) means the Ni ion with the longer Ni-O bond length and Ni B (circular dots) means the one with the shorter bond length. The Wannier spread is defined by $\sqrt{\langle r^2\rangle -{\langle r\rangle }^2}$ and the value is averaged over two $e_g$ orbitals.

Figure 10

The partial density of states for the bond disproportionated ${\mathrm{NdNiO}}_3$ [$\delta a\simeq 0.06$ Å, $(V-V_0)/V_0=-12.5\%$] computed using DFT+U with the correlated orbital of the MLWF (top) and the projector (bottom). Ni A $d$ (with the longer Ni-O bond length), Ni B $d$ (with the shorter Ni-O bond length), and O $p$ data are shown for comparison. The density of states is spin averaged for up spin and down spin.

Figure 11

Average Ni-O bond length difference $\delta a$ graph as a function of volumes obtained using DFT+U. The normalization effect of the correlated orbital is investigated by comparing both the orthonormalized projector-correlated orbital (filled symbols, solid lines) and the un-normalized projector (open symbols, dashed lines). ${\mathrm{LuNiO}}_3$ (red square), ${\mathrm{NdNiO}}_3$ (green diamond), and ${\mathrm{LaNiO}}_3$ (blue circle) results are displayed for comparison.

Figure 12

Structural phase diagram of the rare-earth nickelates as a function of the volume compression and rare-earth ion, computed using DFT+U (square dots) and SDFT+U (circle dots). The unnormalized projector (open dots) and orthonormalized projector (filled dots) are compared for both methods. The MLWF-correlated orbital result (diamond dot) is also shown for DFT+U, yielding some similarity to the phase boundary to the orthonormalized projector. The experimental data (black dash-dotted lines) are also given for comparison [46] (see also Refs. [12, 13]).

Figure 13

Equilibrium volume $V_0$ calculated using SGGA (filled symbols) and LSDA (open symbols) for $R{\mathrm{NiO}}_3$ with $R$ = La, Sm, Nd, and Lu using DFT+U (square dots) and SDFT+U. For SDFT+U, orthonormalized projector (green diamond) and unnormalized projector (red circle) calculations are compared. The experimental volumes at the ambient pressure are depicted by a black dashed line with open pentagonal dots. LDA(+U) results are shown only for ${\mathrm{LaNiO}}_3$ since the pseudopotentials for other rare-earth ions except La are not available.