Multiplet ligand-field theory using Wannier orbitals

We demonstrate how ab initio cluster calculations including the full Coulomb vertex can be done in the basis of the localized Wannier orbitals which describe the low-energy density functional (local-density approximation) band structure of an infinite crystal, e.g., the transition-metal $3d$ and oxygen $2p$ orbitals. The spatial extent of our $3d$ Wannier orbitals (orthonormalized $N$th-order muffin-tin orbitals) is close to that found for atomic Hartree-Fock orbitals. We define ligand orbitals as those linear combinations of the O $2p$ Wannier orbitals which couple to the $3d$ orbitals for the chosen cluster. The use of ligand orbitals allows for a minimal Hilbert space in multiplet ligand-field theory calculations, thus reducing the computational costs substantially. The result is a fast and simple ab initio theory, which can provide useful information about local properties of correlated insulators. We compare results for NiO, MnO, and SrTiO${}_3$ with x-ray absorption, inelastic x-ray scattering, and photoemission experiments. The multiplet ligand-field theory parameters found by our ab initio method agree within $\sim 10\%$ with known experimental values.

Figure 1

A NiO${}_6$ cluster used in multiplet ligand-field theory (MLFT) as a local representation of the rocksalt face-centered-cubic NiO solid. The Ni cation is surrounded by its six nearest-neighbor O ligands.

Figure 2

Left panel: NiO LDA band structure calculated with the large LAPW basis set (thin, black lines) and with the small Wannier-orbital basis set consisting of three O $p$ (green), three Ni $d\left(t_{2g}\right)$ (blue), and two Ni $d\left(e_g\right)$ (red) orthonormalized $N$MTOs per NiO (thick colored lines). Colors are mixed according to the hybridization between the Bloch sums of the three kinds of orbital. The Fermi level is taken as the zero of energy. Middle panel: Wannier-orbital projected densities of states. Right panel: The eight Wannier orbitals. Shown are constant-amplitude contours containing $90\%$ of the orbital charge with the color (red or blue) giving the sign. The Ni(O)-centered octahedra have O(Ni) at their corners.

Figure 3

Angular-averaged radial wave functions $R\left(r\right)$ for the Ni $e_g$ and Ni $t_{2g}$ Wannier orbitals compared with the Hartree-Fock radial wave function for a Ni${}^{2+}$ ion in a $3d^8$ configuration. For Ni${}^{+}$ $3d^8{s}^1$ and Ni $3d^8{s}^2,$ the radial functions are similar. The distance to the nearest oxygen is 2.09 Å, which is consistent with the sum of the ionic radii of 0.72 Å for Ni${}^{2+}$ and 1.40 Å for O${}^{2-}.$ The inset shows the Slater integrals Eq. (1) for the multipole Coulomb interactions.

Figure 4

Left panel: Orbital energy-level diagram for the NiO${}_6$ cluster on the same energy scale as the LDA band structure for the solid (Fig. 2) shown in the middle panel. The Fermi level is the zero of energy. Right panel: Constant-amplitude contours of the Ni $d$ Wannier orbitals and of the Ni-centered ligand orbitals. The latter are symmetrized linear combinations of the O $p$ Wannier orbitals.

Figure 5

As Fig. 2, but for fcc NiO $\left({\text{d}}^8\right)$, fcc MnO $\left({\text{d}}^5\right)$, and sc SrTiO${}_3$ $\left({\text{d}}^0\right)$. On the right-hand side, the oxygen orbitals are not shown.

Figure 6

Angular-averaged TM $t_{2g}$ Wannier orbitals in NiO, MnO, and SrTiO${}_3$. The distance to oxygen is 2.09 Å in NiO, 2.21 Å in MnO, but only 1.95 Å in SrTiO${}_3.$The ionic radii are 0.72 for Ni${}^{2+}$, 0.80 for Mn${}^{2+}$, 0.90 for Ti${}^{2+},$ but only 0.68 Å for Ti${}^{4+}$.

Figure 7

Comparison of the experimental (thick red) and MLFT (thin blue) TM $2p$ ($L_{2,3}$ edge) core-level x-ray absorption spectra for SrTiO${}_3$, MnO, and NiO. The experimental SrTiO${}_3$ spectra are reproduced from Uehara et al. (Ref. 84), the MnO spectra from Csiszar et al. (Refs. 85 and 86), and the NiO spectra from D. Alders et al. (Ref. 87).

Figure 8

Comparison of the experimental (thick red) and MLFT (thin blue) Ni $2p$ core-level photoemission spectra of NiO in the impurity limit. The experimental spectra are reproduced from Altieri et al. (Ref. 101).

Figure 9

Comparison of the experimental (thick red) and theoretical (thin blue) Mn $3p$ ($M_{2,3}$ edge) nonresonant inelastic x-ray scattering spectra of MnO at large momentum transfer ($\mathbf{q}$). The two panels give spectra for different directions of momentum transfer. They exhibit the generalized natural linear dichroism present for an octupole transition in cubic symmetry. The experimental spectra are reproduced from Gordon et al. (Ref. 109).

Figure 10

Comparison of the experimental (thick red) and MLFT (thin blue) nonresonant inelastic x-ray scattering intensity of low-energy $d$-$d$ excitations. The experimental spectra are reproduced from Verbeni et al. (Ref. 112).

Figure 11

NiO band structure (left) and Wannier orbitals (right) for three different basis sets. Top panels: Including both the Ni $\text{d}$ and O $\text{p}$ orbitals. Middle panels: Including only the Ni $\mathrm{d}$ orbitals. Bottom panels: Including only the O $\mathrm{p}$ orbitals.

Figure 12

Comparison of the LAPW band structures and densities of states for the full LDA potential (thin lines) and the warped MT potential (thick lines). When only a single line can be seen, the two band structures overlap within the linewidth of the plot. The colors indicate the partial-wave characters inside the MT spheres.