Magnetism of metallacrown single-molecule magnets: From a simplest model to realistic systems

Electronic and magnetic properties of molecular nanomagnets are determined by competing energy scales due to the crystal field splitting, the exchange interactions between transition metal atoms, and relativistic effects. We present a comprehensive theory embracing all these phenomena based on first-principles calculations. In order to achieve this goal, we start from the ${\mathrm{FeNi}}_4$ cluster as a paradigm. The system can be accurately described on the ab initio level yielding all expected electronic states in a range of multiplicities from 1 to 9, with a ferromagnetic ground state. By adding the spin-orbit coupling between them we obtain the zero-field splitting. This allows to introduce a spin Hamiltonian of a giant spin model, which operates on a smaller energy scale. We compare the computed parameters of this Hamiltonian with the experimental and theoretical magnetic anisotropy energies of the monolayer Ni/Cu(001). In line with them, we find that the anisotropy almost entirely originates from the second-order spin-orbit coupling, the spin-spin coupling constitutes only a small fraction. Finally, we include the ligand atoms in our consideration. This component has a decisive role for the stabilization of molecules in experimental synthesis and characterization, and also substantially complicates the theory by bringing the superexchange mechanisms into play. Since they are higher-order effects involving two hopping matrix elements, not every theory can describe them. Our generalization of the corresponding perturbation theory substantiates the use of complete active space methods for the description of superexchange. At the same time, our numerical results for the $\left\{{\mathrm{CuFe}}_4\right\}$ system demonstrate that the Goodenough-Kanamori rules, which are often used to determine the sign of these exchange interactions, cannot deliver quantitative predictions due to the interplay of other mechanisms, e. g., involving multicenter Coulomb integrals. We conclude by comparing ab initio values of the exchange interaction constants for the $\left\{{\mathrm{CuCu}}_4\right\}$ and $\left\{{\mathrm{CuFe}}_4\right\}$ metallacrown magnetic molecules with experimental values determined by fitting of the magnetic susceptibility curves ${\chi }_{M}T\left(T\right)$, and attribute the remaining discrepancy between them to the role of virtual electron excitations into and out of the active space (dynamical correlations).

Figure 1

(a)–(c) The three systems of the present study and (d) a generic spin model. Ab initio calculations are performed on the ${\mathrm{FeNi}}_4$ cluster (a), and on the charged minimal models of metallacrowns [(b) and (c)]. They are constructed by depriving MCs of weakly bound solvent molecules and coordinated anions (${\mathrm{HNEt}}_3$, DMF, ${\mathrm{Cl}}^{-}$), and have a form ${\mathrm{AB}}_4{\mathrm{L}}_4$, where A and B are TM atoms. Notice that Ni ($d^9$) in ${\mathrm{FeNi}}_4$ is isoelectronic with ${\mathrm{Cu}}^{\mathrm{II}}$ in $\left\{{\mathrm{CuCu}}_4\right\}$ (b) and $\left\{{\mathrm{CuFe}}_4\right\}$ (c), and therefore these systems can be used for comparison. Color scheme: light blue, yellow, and magenta spheres stand for Cu, Fe, and Ni transition metal atoms, respectively. C, N, and O atoms are depicted as grey, blue, and red spheres, respectively. Hydrogen atoms have been removed for clarity. All three systems (a)–(c) are shown on the same scale. DMF stands for dimethylformamide, and L=${\mathrm{C}}_7{\mathrm{H}}_4{\mathrm{NO}}_3$ is the ligand complex.

Figure 2

Outline of this work. We specifically focus on the Heisenberg-Dirac-Van Vleck Hamiltonian ${\widehat{\mathcal{H}}}_{\text{ex}}$ and the zero-field splitting Hamiltonian ${\widehat{\mathcal{H}}}_{\text{ZFS}}$ bridging our ab initio calculations and experimental measurements. Other abbreviations are NMR/EPR (nuclear magnetic resonance/electron paramagnetic resonance), SQUID (superconducting quantum interference device), INS (inelastic neutron scattering), and FMR (ferromagnetic resonance). ${\widehat{\mathcal{H}}}_{\text{Z}}$ denotes the Zeeman Hamiltonian, whereas ${\widehat{\mathcal{H}}}_{\text{HFC}}$ is the hyperfine coupling Hamiltonian.

Figure 3

Bond lengths (Å) and angles (degrees) of a central fragment of the ${\mathrm{Cu}}^{\mathrm{II}}[12-{\mathrm{MC}}_{{\mathrm{Fe}}^{\mathrm{III}}\mathrm{N}\left(\mathrm{Shi}\right)}-4]$ metallacrown. Simplistic application of the Goodenough-Kanamori rule predicts that the magnetic interactions between ${\mathrm{Cu}}^{\mathrm{II}}$ and ${\mathrm{Fe}}^{\mathrm{III}}$ ions are antiferromagnetic when electrons can hop between the $d$ orbitals of TM ions via the $p_z$ orbital of O. In the opposite case, when the Fe–O–Cu angle approaches ${90}^{\circ }$, such hopping is not possible. In this case, a ferromagnetic interaction arises due to the hopping of $d$-electrons onto two orthogonal $p$ orbitals.

Figure 4

Correspondence between the rhombically distorted ${\mathrm{AB}}_4$ spin system and a spin triangle. Analytic solution is possible for $S_A=1/2$ and is given by Eq. (12).

Figure 5

Ordering of atomic orbitals in the $\text{Fe}{\mathrm{Ni}}_4$ cluster. Spin-polarized natural atomic orbitals localized on Ni and Fe atoms are depicted in (a) and (b), respectively. They constitute the active space. The Hamiltonian matrix elements relevant for the antiferromagnetic coupling of Ni atoms are indicated in (a). In accordance with the Hamitonian (8), indices associated with the central atom are denoted as $a$, and for the peripheral atoms we use $b, b^{\prime }$, etc. The state averaged calculation performed in CAS(10,9) yields the crystal field splitting in (b).

Figure 6

Electronic states of the ${\mathrm{FeNi}}_4$ cluster. (a) and (b) differ by the choice of reference spaces [CAS(8,8) and CAS(10,9) as depicted on the insets below]. (c) is a enlarged portion of the (b) showing spin-orbit splitting of the lowest energy spin multiplet. The strength of spin-orbit and spin-spin couplings is encoded in the width of lines connecting non- and relativistic calculations. The selection rule that $〈a{S}_{a}M\left|{\widehat{\mathcal{H}}}_{\text{SOC}}\right|b{S}_b{M}^{\prime }〉$ is different from zero only for $S_a-S_b=0,\pm 1$ [cf. Eq. (2)] is manifested in the fact that each relativistic state is predominantly composed of states with the same spin, i.e., $S_a=S_b$ with a small admixture of $S_a-S_b=\pm 1$. There are no states composed of $S_a=4$ and $S_b=2$ at the same time. On the inset of (c), we compare the computed energy fine structure with the eigenenergies of the ZFS Hamiltonian (28) using the computed value of the axial parameter $D$ and neglecting the rhombohedral splitting.

Figure 7

The magnetic susceptibility of the ${\mathrm{FeNi}}_4$ cluster computed using states from Fig. 6 (CAS(8,8) reference, circles), and Fig. 6 (CAS(10,9) reference, squares) and fitted by the spin model (8) with $s_A=2$ and $s_B=1/2$, full and dashed lines, respectively. The low and the high-temperature limits are consistent with ${\langle S^2\rangle }_0=20$ and ${\langle S^2\rangle }_{\infty }=69/8$, Eqs. (22) and (23). Insets (a) and (b) compare the density of states from ab initio CAS(8,8) calculations and from the spin model, respectively. Analogically, insets (c) and (d) compare the density of states from ab initio CAS(10,9) calculations and from the spin model, respectively. The solid line denotes the total contribution from all spin multiplicities, whereas color-shaded curves separately resolve different spins. Notice a larger density in (c) because more states are included.

Figure 8

Energy level scheme of the ${\mathrm{FeNi}}_4$ cluster in an undistorted square geometry neglecting the dynamical correlations. Nondegenerate and doubly degenerate states are shown as black and blue lines, respectively. The expected level spacings from the exact solution (13) of the spin Hamiltonian (8) are indicated. Since the root-mean-square error of the fitting ${\delta }_{\text{RMSE}}$ is very small, eigenenergies of the ab initio and the spin model (4) are indistinguishable on the shown energy scale.

Figure 9

The natural atomic orbitals on Fe and Cu atoms (a), and two representative localized MOs of the $p$ character on the ligand atoms computed using the Boys procedure [78] (b).

Figure 10

The inclusion of different Cu $d$-MOs into the active space gives rise to different electronic states. Energy differences of the two excited states (b) and (c) with respect to the ground-state configuration (a) are indicated. All calculations are performed at the CAS(21,21) level of theory and the total spin $S=21/2$.

Figure 11

Fitting of the DMRG calculations (circles) for two ${\mathrm{AB}}_2$ subsystems (green shaded areas) of the $\left\{{\mathrm{CuFe}}_4\right\}$ cluster with a spin model (8). The eigenvalues of the spin Hamiltonian corresponding to these isosceles triangles are shown as pluses. A large discrepancy in the fitted values of $J_1$ in the left vs right panels poses a question about the origin of the exchange coupling in this system.

Figure 12

Variation of the exchange constants in meV units for different included interactions within AB and ${\mathrm{AB}}_2$ spin clusters of the ${\mathrm{Cu}}^{\mathrm{II}}{\left(\mathrm{DMF}\right)}_2{\mathrm{Cl}}_2[12-{\mathrm{MC}}_{{\mathrm{Fe}}^{\mathrm{III}}\mathrm{N}\left(\mathrm{Shi}\right)}-4]{\left(\mathrm{DMF}\right)}_4$ molecule. Thick lines denote the included hopping and pairwise Coulomb matrix elements—black for nearest and blue for next-nearest neighbors—whereas dashed lines stand for the excluded pairwise interactions. DMRG solutions of the cluster models are mapped onto the analytic solutions for spin dimers (a) and trimers [(b)–(d)]. The energy levels of an ${\mathrm{AB}}_2$ cluster are given by Eq. (12) with $s_A=1/2$ and $s_B=5/2$.

Figure 13

Energy levels of the spin model (8) involving the ${\mathrm{Fe}}^{\mathrm{III}}$ high-spin $s_B=5/2$ configuration from the analytic solution (13). Black horizontal bars denote levels of the all-AFM setting ($J_1<0, J_2<0$), whereas red ones correspond to FM $J_1>0$ and AFM $J_2<0$ interactions. In order to emphasize symmetries between AFM and FM solutions, we use values of the exchange constants $J_1=\pm 5.5$ meV and $J_2=-0.5$ meV that only approximately correspond to the experimental exchange constants $J_1=-6.10$ meV and $J_2=-0.47$ meV ($s_B=5/2$) [9]. The lowest energy state in each scenario (chosen as reference) is a frustrated magnetic configuration with intermediate total spin.