Magnetism of a four-center transition-metal cluster revisited

We propose a surprisingly efficient procedure to map a sophisticated treatment of electronic correlations to a few-parameter model Hamiltonian. By pushing first-principles coupled-cluster methods to the limit we get a lucid picture of the electronic excitations responsible for the magnetic interactions in a paradigm ${\mathrm{Ni}}_4$ cluster. Using an extended Hubbard model, we demonstrate that the magnetism in this system cannot be explained solely on the basis of spin models but additionally requires consideration of the indirect mechanism involving hopping and on-site repulsion. We present a protocol to extract the corresponding parameters from experiment or calculations.

Figure 1

Many-body-state energies of the first-principles EOM-CCSD calculations in the ${\mathrm{Ni}}_4$ cluster. Pairs of singlet (in blue) and triplet (in red) states built from the same dominant virtual one-electron excitations (shown in the insets): (a) $(d_{xy},d_{yz})\to d_{x^2-y^2}$, (b) $d_{z^2}\to d_{x^2-y^2}$, and (c) $d_{xy}\to d_{x^2-y^2}$. Note that optical selection rules are not relevant to virtual molecular excitations. (d) Summary of the singlet-triplet splittings for the different singlet-triplet pairs as the symmetry of the cluster is gradually lowered $\left(D_{4h}\to C_{2v}\to C_s\right)$. The colors refer to the irreducible representations (see Fig. 4). Note that the doubly degenerate states $E_g$ and $E_u$ get split when leaving the perfect square geometry $\left(D_{4h}\right)$ [67].

Figure 2

Main virtual excitations building up the correlated many-body states. The numbers below the Hartree-Fock molecular orbitals indicate their ascending energetic ordering. The numbers above the arrows are the excitation coefficients of the triplets, and those below the arrow are the excitation coefficients of the pertinent singlet states. (a) The energetically lowest $B_{\text{2}g}$ states [see also Fig. 1]. (b) An excited state in which the virtual excitations have a $d\to p$ character and which is therefore excluded from our considerations for the spin Hamiltonian.

Figure 3

Top view of the $20d$-character molecular orbitals, combined to form irreducible representations of the $D_{4h}$ point group. The top row (stemming from $d_{x^2-y^2})$ shows virtual orbitals. The dashed lines represent nodal planes, the number of which is connected to the total energy of the molecular orbital within a group. The depicted orbitals are predicted through group theory and, as such, take into account neither $p$ or $s$ hybridization nor correlations.

Figure 4

(a) Energies of singlet (dashed lines) and triplet (solid lines) states of the extended Hubbard Hamiltonian as functions of the Coulomb repulsion parameter for the five irreducible representations created by the virtual excitations $d_{xy}\to d_{x^2-y^2}$ [see Fig. 1]. The indicated $\pm t_a$ values refer to the actual ab initio calculations. (b) Schematic of the definitions of the hopping and interaction parameters for the same excitations. The values are not extracted from the EOM-CCSD calculations but are directly the bare, noncorrelated integrals between the $d$ orbitals. (c) Singlet-triplet splittings for two sets of exchange parameters: $J_1=0.3$ eV, $J_2=0.8$ eV, $J_3=0.4$ eV (solid lines) and $J_1=0.56$ eV, $J_2=J_3=0$ eV (dotted lines). $t_b=-2.3$ eV and $t_a=0.93$ eV for both sets. The colors refer to the different irreducible representations (purple for $B_{2g}$, green for $E_u$, and orange for $A_{2g})$ and are the same as in (a). The values indicated with the colored triangles refer to the actual ab initio calculations [see Figs. 1–1]. The best matching is obtained for $U=10$ eV, while $U=0$ yields the value $\frac{1}{4}{J}_1+\frac{1}{2}{J}_2+\frac{1}{4}{J}_3$.