Effect of extra electrons on the exchange and magnetic anisotropy in the anionic single-molecule magnet ${\mathrm{Mn}}_{12}$

To understand the effect of molecular environment on the electronic and magnetic properties of the single-molecule magnet ${\mathrm{Mn}}_{12}$, we explore two possible means for adding extra electrons to molecule. We explore both substitution of Mn ions by Fe ions and the inclusion of neighboring electronic donors. For both possibilities we calculate, within density-functional theory, the electronic structure, the total ground-state spin and ordering, the magnetic anisotropy barrier, the transverse magnetic anisotropy parameter $E$ which is responsible for some measured tunneling, and the tilting angle of the easy axis from the $z$ axis. Our calculations show that the total spin increases with increasing number of extra electrons except for the case of ${\mathrm{Mn}}_8{\mathrm{Fe}}_4$ where the resulting ground state has a low spin. The calculated energy gaps between the unoccupied and the occupied orbitals exhibit no clear trend as a function of number of extra electrons. The calculated magnetic anisotropy barrier decreases with an increasing number of extra electrons and can be directly traced to a quenching of Jahn-Teller distortions at the sites where the additional electrons are localized. The values of $E$ and the easy-axis tilting angles for the geometries with one- and two-extra electrons are significantly larger than those induced by solvent disorder.

Figure 1

(Color online) Schematic diagram of the fourfold symmetric tetra-Fe substituted ${\mathrm{Mn}}_{12}$ geometry, ${\mathrm{Mn}}_8{\mathrm{Fe}}_4{\mathrm{O}}_{12}{\left(\mathrm{HCOO}\right)}_{16}{\left({\mathrm{H}}_2\mathrm{O}\right)}_4$. The shading scheme of the atoms is as follows: darkness increases from H (white), to Mn, to Fe (largest spheres), to C, and to O (darkest). The three inequivalent metal ion sites are labeled as $\alpha ,\beta$, and $\delta$. The bottom figures illustrate occupancy of the $e_g$ and $t_{2g}3d$ orbitals for ${\mathrm{Fe}}^{3+}$ (or ${\mathrm{Mn}}^{2+}$) and ${\mathrm{Mn}}^{4+}$.

Figure 2

(Color online) (a) Initial spin configuration considered for the fourfold symmetric ${\mathrm{Mn}}_8{\mathrm{Fe}}_4{\mathrm{O}}_{12}{\left(\mathrm{HCOO}\right)}_{16}{\left({\mathrm{H}}_2\mathrm{O}\right)}_4$ geometry to determine the most favorable sites for extra electrons. The inequivalent sites are indicated by the three different symbols. (b) Schematic diagram of the exchange constants, $J_1,J_2,J_3,J_3^{\prime },J_4$, and $J_4^{\prime }$, where $J_3$ and $J_3^{\prime }$ ($J_4$ and $J_4^{\prime }$) are slightly different from each other because of a difference in the bond length by approximately 0.1 Å. Each spin is labeled from 1 to 12. Sites $\alpha$ correspond to spins 1-4, Sites $\beta$ to spins 5-8, and Sites $\delta$ to spin 9-12.

Figure 3

Calculated density of states projected onto the majjority and minority $\mathrm{Mn}\left(3d\right)$ orbitals at each of the three inequivalent sites ($\alpha ,\beta$, and $\delta$) for the fourfold symmetric $\left[{\mathrm{Mn}}_{12}{\mathrm{O}}_{12}{\left(\mathrm{HCOO}\right)}_{16}{\left({\mathrm{H}}_2\mathrm{O}\right)}_4\right]$ whose structure is exactly the same as Fig. 1 with Fe replaced by Mn. The solid vertical line denotes the Fermi energy level.

Figure 4

(Color online) Schematic diagram of the fourfold symmetric ${\mathrm{K}}_4\left[{\mathrm{Mn}}_{12}{\mathrm{O}}_{12}{\left(\mathrm{HCOO}\right)}_{16}{\left({\mathrm{H}}_2\mathrm{O}\right)}_4\right]$ geometry, where two of the K cations are located slightly above the $x\text{−}y$ plane and the rest two are slightly below the $x\text{−}y$ plane. The shading scheme is the same as that for Fig. 1 except that the largest spheres are now K cations. The distance between the closest Mn $\beta$ site ($\delta$ site) and K is 4.403 Å (5.323 Å).

Figure 5

(Color online) Schematic diagram of the twofold symmetric ${\mathrm{K}}_2\left[{\mathrm{Mn}}_{12}{\mathrm{O}}_{12}{\left(\mathrm{HCOO}\right)}_{16}{\left({\mathrm{H}}_2\mathrm{O}\right)}_4\right]$ geometry, where two K cations are close to the site $\delta$ in the $x\text{−}y$ plane. The shading scheme is the same as that for Fig. 4. The distance between the closest Mn and K is about 4.642 Å.

Figure 6

Calculated MAB versus number of extra electrons for the Fe substituted and K added ${\mathrm{Mn}}_{12}$ geometries and the ${\mathrm{Mn}}_{12}$ geometry.