Correlated electron behavior of metal-organic molecules: Insights from density functional theory combined with many-body effects using exact diagonalization

A proper theoretical description of the electronic structure of the $3d$ orbitals in the metal centers of functional metalorganics is a challenging problem. We apply density functional theory and an exact diagonalization method in a many-body approach to study the ground-state electronic configuration of an iron porphyrin (FeP) molecule. Our study reveals that the consideration of multiple Slater determinants is important, and FeP is a potential candidate for realizing a spin crossover due to a subtle balance of crystal-field effects, on-site Coulomb repulsion, and hybridization between the Fe-$d$ orbitals and ligand N-$p$ states. The mechanism of switching between two close-lying electronic configurations of Fe-$d$ orbitals is shown. We discuss the generality of the suggested approach and the possibility to properly describe the electronic structure and related low-energy physics of the whole class of correlated metal-centered organometallic molecules.

Figure 1

Real and imaginary parts of the hybridization function for Fe in FeP calculated with PBE in the non-spin-polarized mode. A smearing parameter of 0.01 eV has been used for this plot for the sake of visualization. The geometry of FeP is shown in the inset with the atoms labeled by their types.

Figure 2

The phase diagram depicting the spin states of FeP with the tuning of the static crystal field $E_{\mathrm{cryst}}$ and the hybridization strength $V$. The phase boundary is accompanied by the allowed values of $E_{\mathrm{cryst}}$ for fixed values of $V$. The blue curve is a result of fitting with a tight-binding model described in the Supplemental Material [16]. AS and BS indicate antibonding and bonding regions. Calculated values of $E_{\mathrm{cryst}}$ and $V$ from DFT (non-spin-polarized PBE) are shown in orange circles along with the corresponding Fe-N bond lengths.

Figure 3

Schematic diagram depicting the ${\mathrm{C}}_{222}$ and ${\mathrm{C}}_{231}$ configurations. $E_{\mathrm{cryst}}$ describes the energy separation between $d_{x^2-y^2}$ and the rest of the $d$ orbitals treated as degenerate. $V$ denotes the strength of hybridization between the $d_{x^2-y^2}$ and the bath state with an energy $E_b$. The specific orbitals corresponding to the occupation of the energy levels are also indicated.

Figure 4

Upper panel: Energy differences between low-spin (LS) and high-spin (HS) states calculated by DFT++ [double counting (DC) treated by placing chemical potential in the middle of the gap and DC calculated using the fully localized limit (FLL)] and PBE+$U$ (double counting by Dudarev and Lichtenstein) methods as a function of Fe-N bond lengths in FeP. The error bars correspond to the variation of on-site energy [model Hamiltonian in Eq. (1)] in DFT++ calculations. Lower panel: Magnetic anisotropy energy (MAE) as a function of Fe-N bond lengths calculated with the double-counting correction in the fully localized limit.