Revisiting magnetic coupling in transition-metal-benzene complexes with maximally localized Wannier functions

Construction of maximally localized Wannier functions (MLWFs) has been implemented within the linear combination of pseudoatomic orbital method. Detailed analysis using MLWFs is applied to three closely related materials, single benzene (Bz) molecule, organometallic vanadium-Bz infinite chain, and ${\text{V}}_2{\text{Bz}}_3$ sandwich cluster. Two important results come out from the present analysis: (1) for the infinite chain, the validity of the basic assumption in the mechanism of Kanamori and Terakura for the ferromagnetic (FM) state stability is confirmed; (2) for ${\text{V}}_2{\text{Bz}}_3$, an important role played by the difference in the orbital energy between the edge Bzs and the middle Bz is revealed: the on-site energy of $p\delta$ states of edge Bzs is higher than that of middle Bz, which further reduces the FM stability of ${\text{V}}_2{\text{Bz}}_3$.

Figure 1

(Color online) Molecular orbitals of Bz around the HOMO and LUMO states. The energy eigenvalue and symmetry of each orbital are shown. Doubly degenerate orbitals are shown together on the same level.

Figure 2

(Color online) The MLWF around the carbon atom one obtained for a Bz molecule is plotted with $\text{isovalue}=0.1$. All six MLWFs are identical and their centers are shifted from nearby C atomic sites along each C-H bond by about $0.07\text{Å}$.

Figure 3

(Color online) Band structure of FM V-Bz infinite chain for (a) spin up and (b) spin down channels. Lines are from ordinary first-principles calculations while symbols are those from Wannier interpolation. The components of interpolated bands at $\Gamma$ point are labeled, too.

Figure 4

(Color online) Schematic pictures describing the mechanism in which the $p\text{−}d$ hybridization stabilizes the FM states. The DOS (a) without and (b) with $p\text{−}d$ hybridization is plotted. The energy gain due to the charge transfer associated with the magnetic relaxation of $p\delta$ states is indicated. Before $p\text{−}d$ hybridization, the on-site energies of $p\delta$ and $d\delta$ states are given in eV for both GGA and $\text{GGA}+U$ calculations. See the text for more details.

Figure 5

(Color online) FM stability energy (total energy of AFM state minus that of FM state, in eV) depends on the energy difference in $p\delta$ states $\left(\Delta {\epsilon }_p\right)$ on edge Bzs and middle Bz. The FM stability energy vanishes for the on-site energy difference of 0.67 eV where a vertical line is shown. The corresponding on-site energy difference deduced from the MLWF analysis is 0.83 eV.