New mechanism of kinetic exchange interaction induced by strong magnetic anisotropy

It is well known that the kinetic exchange interaction between single-occupied magnetic orbitals (s-s) is always antiferromagnetic, while between single- and double-occupied orbitals (s-d) is always ferromagnetic and much weaker. Here we show that the exchange interaction between strongly anisotropic doublets of lanthanides, actinides and transition metal ions with unquenched orbital momentum contains a new s-d kinetic contribution equal in strength with the s-s one. In non-collinear magnetic systems, this s-d kinetic mechanism can cause an overall ferromagnetic exchange interaction which can become very strong for transition metal ions. These findings are fully confirmed by DFT based analysis of exchange interaction in several Ln3+ complexes.

Here,Ĥ 0 ,Ĥ t ,Ĥ bi indicate the orbital energy level of the bridging atoms with respect to the f orbital level, the electron transfer Hamiltonian between metal site and the bridging atoms (ligand), andĤ bi is the Coulomb repulsion on f metal sites, respectively. In the Hamiltonian, i (= 1, 2) is the metal (lanthanide) center, L is the bridging ligand, m (= 3, 2, ..., −3) is the projection of the orbital angular momentum of f atomic orbital, r is the orbital energy level of the ligand, σ, σ is the spin projection, c imσ (ĉ † imσ ) andĉ Lrσ (ĉ † Lrσ ) are the annihilation (creation) operators on the mσ orbital of metal site and rσ orbital of the ligand L, respectively,n imσ =ĉ † imσĉ imσ andn Lrσ =ĉ † Lrσĉ Lrσ are the number operators, τ iL mr (= τ Li * rm ) is the transfer parameter between magnetic center and ligand, and u f is the Coulomb repulsion energy on f metal site. Since the transfer parameter between metal sites are small, we neglect the direct electron transfer between the metal centers. The Coulomb repulsion on the ligand and the intersite one between the metal center and the ligand as well as the Hund's rule coupling are neglected because they are smaller than u f . Based on the Hamiltonian, we describe the low-energy magnetic states. The transfer parameters are extracted from density functional theory (DFT) calculations and the Coulomb repulsions are determined to reproduce the experimental exchange parameters.
The model Hamiltonian Eq. (S1) is not the same as the model Hamiltonian used for the derivation of the exchange interactions in the main text. In the latter model Hamiltonian, the electron transfer between the metal sites and ligand is reduced to the transfer parameter between the metals. However, in the DFT based calculations of the exchange parameters, such simplification is not always possible because the f orbital levels and ligand levels are sometimes close to each other. Thus, we take Eq. (S3) for our calculations of the lanthanide complexes.

II. DFT CALCULATIONS
In order to find a density functional which is suitable for the electronic structure calculations of lanthanide complexes, we calculated (i) the exchange interactions and (ii) the binding energies with several methods (DFT and Hartree-Fock (HF) method), and compared the results with experimental data. We found that the hybrid functional including about 40 % of HF exchange is suitable to express one-orbital parameters ( r and τ iL mr in Eqs. (S2) and (S3)) of lanthanide complexes. For the quantum chemistry calculations, we used ORCA3.0 package [S5].

A. Isotropic exchange parameters for Gd 3+ complexes
In general, the spin-orbit coupled ground states of the lanthanide ion (Dy 3+ , Er 3+ , etc.) cannot be adequately treated within single Slater determinant approach. On the other hand, the ground state of half-filled Gd 3+ ion is in a good approximation described by pure spin state, and the state with maximal spin projection is a single Slater determinant. Thus, for the quantum chemistry calculations, we replaced the lanthanide ions (Dy 3+ and Er 3+ ions) by Gd 3+ ions. As the functional, we chose B3LYP with various HF exchange contributions (20 -60 %). The variation of the contribution of the HF exchange is to reduce the exaggerated electron transfer and to increase the underestimated Coulomb repulsion on site. The molecular structures were taken from the x-ray diffraction data. As the basis set, SVP was used.
The exchange parameter between metal sites was obtained within broken-symmetry approach [S6]. The exchange parameter J defined by Heisenberg Hamiltonian, is estimated using the high-spin and broken-symmetry (low-spin) states: Here, E HS and E BS are the ground electronic energies obtained from the high-spin (HS) and broken-symmetry (BS) calculations, and Ŝ 2 HS and Ŝ 2 BS are the expectation values of the magnitude of the total spin.
The obtained isotropic exchange parameters and those from experimental data are tabulated in Table S1. We find B3LYP functionals with 40-50 % of Hartree-Fock exchange well reproduce the the experimental ones.

B. Binding energy vs. photoemission measurements
The binding energies of tris-cyclopentadienyl lutetium (LuCp 3 ) were estimated and compared with experimental data. The experimental binding energies are taken from the photoelectron spectra (PES) of LuCp 3 in gas phase [S8]. We chose Lu 3+ ion complex because the 4f orbitals are completely filled in the ground state, which can be described by singe Slater determinant. For the calculations, we used the x-ray diffraction structure of CeCp 3 [S9] as the structure of LuCp 3 since their structures are similar to each other and the energy scale of the PES is much larger than the change in energy due to the structure. The binding energies were estimated applying the Koopmans' theorem.
The calculated binding energies are shown in Table S2. In comparison with the experimental data, the B3LYP (20 %) and the HF calculations underestimates and overestimates the 4f binding energies, respectively. Better agreement is obtained when the contribution of the HF exchange is about 40 %. Since both the exchange interaction and the binding energies obtained with the hybrid B3LYP functional with about 40 % of the HF exchange are close to the experimental data, we used the functional for the calculations of the transfer Hamiltonian.

III. DERIVATION OF ISING EXCHANGE PARAMETERS FOR LN 3+ COMPLEXES
In order to project the electronic states from the DFT calculations into the model Hamiltonian (S1), we local-ized the valence Kohn-Sham orbitals ). The Kohn-Sham orbitals are divided them into the magnetic 4f orbitals, the bridging ligand orbitals, and the other ligand orbitals, where the last orbitals are not important for the exchange interaction from each other. The localization on the magnetic core was measured by the Mulliken population on the f type orbitals of the metal centers and the p type orbitals of the bridging ligand. With the use of the localized orbitals as the basis set, we expressed the Kohn-Sham Hamiltonian matrix h KS for the magnetic core. Here, the localized f orbitals were transformed into the eigenstates of the atomic orbital angular momentuml z whose quantization axes agree with the ab initio main magnetic axes [S1-S3], |imσ . On the other hand, the ligand orbitals |Lrσ are chosen so that the bridging ligand part of h KS becomes diagonal. Then, the diagonal elements of the Hamiltonian matrix h KS are used as the orbital energy levels (the f orbital level is averaged) in Eq. (S2), and the elements in the off-diagonal block matrices between the metals and the bridging ligand as the transfer parameters in Eq. (S3). The matrix of model Hamiltonian (S1) was calculated using the ground ferromagnetic, the ground antiferromagnetic electron configurations (Fig. 1b in the main  text), and the configurations with one-electron transfer between metal sites and between metal and ligand as the basis set. Diagonalizing the Hamiltonian matrix, we obtain the ferromagnetic and antiferromagnetic states. When there is no energy level between these two states, we can project these states into Ising Hamiltonian (Eq. (3) in the main text). The Coulomb repulsion u f is determined to reproduce the experimental exchange parameter.
The obtained exchange parameters are shown in Table I in the main text. The Coulomb repulsion energies are 1.50, 1.40, 1.20 eV for the Tb, Dy, Ho complexes in the series (a), 1.48 and 1.44 eV for the complexes (b) and (c), respectively. In all cases, there is no level between the ground ferro-and antiferromagnetic levels. At a glance, u f looks too small (cf. Ref. S11), whereas it is not too small because the effect of the Coulomb repulsion is partly included in the Kohn-Sham orbital energies within the mean-field approximation. Thus, u f is the difference between the Hubbard U and the meanfield value. The ferromagnetic and the antiferromagnetic ground states are mainly contributed by the electron configurations without electron transfer (the type of Fig. 1b in the main text). Their contributions (probabilities) to the ground state are 98.3 %, 99.4 %, 98.9 % for the Tb, Dy, Ho complexes of the series (a), 99.3 % for the complex (b), 99.7 % for the complex (c), and 97 % for the Er complex. These high probabilities guarantee the validity of the description of the low-lying states by the pseudospin Hamiltonian. S1. The exchange parameters extracted from the broken-symmetry DFT calculations and experimental magnetic susceptibility (cm −1 ). The numbers in the first row indicate the contribution of the HF part in the exchange-correlation functional. The last column are the experimental data (Exp.) extracted using the Lines model or Heisenberg model. The experimental data of the Gd complexes (a), (b), (d) are taken from Refs. S1, S2, and S7, respectively. The effect of the magnetic dipolar interaction is removed only for the complex (a).

IV. S-S AND S-D EXCHANGE MECHANISMS IN d METAL COMPLEXES
Within the simple two-sites model for the non-collinear doublets in the main text, we calculated the exchange interaction parameter for d metal complexes assuming that the ground state is |J, ±J . The exchange parameter J for the d 6 , d 7 and d 8 ions are shown in Fig. S1(a). As in the case of the f metal ions, the exchange becomes ferromagnetic as the increase of angle φ.
Although the crystal field in transition metal complex is stronger than in lanthanide complex, the crystal field level with unquenched orbital can be obtained with suitable symmetry of the ligands. For example, axial (d 6 , d 7 ) or trigonal (d 8 ) crystal field splits the d levels into two doublets and one nondegenerate state [S12]. With the splitting, the ground doublets of the d 6 metal originates from the J-multiplet due to the strong Hund's rule coupling ( Fig. S1(b)). In the case of the d 7 and d 8 systems, the orbital momentum is unquenched in the presence of the spin-orbit coupling which exceeds the crystal field splitting of the orbital levels (∆ in Fig. S1(b)), which could be observed even in 3d metals systems. The non-aufbau population in the last two electron configurations is achieved for relatively small separation ∆ between orbital levels with non-zero m (compared to spinorbit coupling). This order of orbitals is realized, for example, in crystal fields with axial symmetry (d 6 , d 7 ) and/or trigonal symmetry (d 8 ) [S12].