Ab initio based ligand field approach to determine electronic multiplet properties

A method is developed to calculate the ligand field (LF) parameters and the multiplet spectra of local magnetic centers with open $d$ and $f$ shells in solids in a parameter-free way. This method proceeds from density functional theory and employs Wannier projections of nonmagnetic band structures onto local $d$ or $f$ orbitals. Energies of multiplets and optical, as well as x-ray spectra are determined by exact numerical diagonalization of a local Hamiltonian describing Coulomb, LF, and spin-orbit interactions. The method is tested for several $3d$ and $5f$ compounds for which the LF parameters and multiplet spectra are experimentally well known. In this way, we obtain good agreement with the experiments for ${\mathrm{La}}_2{\mathrm{NiO}}_4, {\mathrm{CaCuO}}_2, {\mathrm{Li}}_2{\mathrm{CuO}}_2$, ZnO:Co, and ${\mathrm{UO}}_2$.

Figure 1

GGA band structures (thin black lines) and Fourier-transformed Wannier projections onto Ni $3d$ states (thick green lines) for the experimental (top) and simplified (bottom) structure of ${\mathrm{La}}_2{\mathrm{NiO}}_4$. Note that the experimental structure features two Ni sites per cell giving rise to ten $3d$ bands, while the simplified structure has only one Ni site per primitive cell, and hence only five bands. The right panels show the densities of states (DOS): total (gray-shaded), Ni $3d$ (green-shaded), and O $2p$ (blue). For the notation of $k$ vectors, see Ref. [28]. The Fermi level is at zero energy.

Figure 2

Comparison of experimental [9] and theoretical RIXS spectra for $\pi$ and $\sigma$ polarization and an income angle of $\mathrm{\Theta }={20}^{\circ }$ (The Gaussian broadening of the theoretical curve was set to 0.24 eV; see text for more details).

Figure 3

Band structure (thin black lines) and Wannier projection (thick green lines), as well as densities of states for the high-$T_c$ parent compound ${\mathrm{CaCuO}}_2$. The Fermi level ${\varepsilon }_F$ is set to zero.

Figure 4

Comparison of experimental (cf. Fig. 2. of Ref. [8]) and two theoretical (without and with exchange field) RIXS spectra for $\pi$ and $\sigma$ polarization and an income angle of $\mathrm{\Theta }={20}^{\circ }$ (The Lorentzian broadening of the theoretical curves was set to 0.23 eV; see text for more details).

Figure 5

Band structure (thin black lines) and Wannier projection (thick green lines), as well as densities of states for the quasi-one-dimensional cuprate compound ${\mathrm{Li}}_2{\mathrm{CuO}}_2$.

Figure 6

Band structure (thin black lines) and Wannier projection (thick purple lines), as well as densities of states for a ${\mathrm{CoZn}}_{15}{\mathrm{O}}_{16}$ ($2\times 2\times 2$) supercell with relaxed positions of Co and its oxygen neighbors.

Figure 7

Optical absorption spectra of the ${}^{4}T_2$ band in ZnO:Co with $\pi$ and $\sigma$ polarization as calculated from the elisa code with parameters fitted to the experimental result [59] (a) or Wannier parameters (b). The width of the Gaussian broadening increases linearly from 0.1 to 1 meV going from lower to higher energies.

Figure 8

The dependence of on-site energies of Wannier functions having $d$-function symmetry on the energy window for the compounds ${\mathrm{CaCuO}}_2$ (a), ${\mathrm{Li}}_2{\mathrm{CuO}}_2$ (b), and $\mathrm{ZnO}:\mathrm{Co}$ (c). The energy window is defined by a function being 1 between $E_{\min }$ and $E_{\max }$ and falling of as a Gaussian with a width $dE=1$ eV outside this window for (a) and (b) and with a width $dE=0.5$ eV for (c). The lower limit of the energy window which was really taken into account in our determination of LF parameters was $-3$ eV for (a) and (b) and $-0.75$ eV for (c).