Giant exchange interaction in mixed lanthanides

Combining strong magnetic anisotropy with strong exchange interaction is a long standing goal in the design of quantum magnets. The lanthanide complexes, while exhibiting a very strong ionic anisotropy, usually display a weak exchange coupling, amounting to only a few wavenumbers. Recently, an isostructural series of mixed (Ln = Gd, Tb, Dy, Ho, Er) have been reported, in which the exchange splitting is estimated to reach hundreds wavenumbers. The microscopic mechanism governing the unusual exchange interaction in these compounds is revealed here by combining detailed modeling with density-functional theory and ab initio calculations. We find it to be basically kinetic and highly complex, involving non-negligible contributions up to seventh power of total angular momentum of each lanthanide site. The performed analysis also elucidates the origin of magnetization blocking in these compounds. Contrary to general expectations the latter is not always favored by strong exchange interaction.


I. DFT CALCULATIONS
A. Extraction of the transfer parameter t for 1 -5 In order to derive the transfer parameters between the 4f orbital and the π * orbital of the bridging N 2 , the Kohn-Sham levels are projected into tight-binding Hamiltonian: where i(= 1, 2) is the index for the Ln 3+ site in the complex, N 3− 2 site is described by the type of the magnetic orbital π * ,γ is the orbital component xyz, σ =↑, ↓ is the projection of spin operator, f and π * (= f + ∆) are one electron orbital levels of the 4f orbital and the π * orbital, respectively, t is the transfer parameter between the 4f and the π * orbitals,ĉ † (ĉ) is an electron creation (annihilation) operator, andn is a number operator. The subscripts of the creation, annihilation, and number operators indicate the site, the orbital index for only lanthanide site, and spin projection. Because of the D 2h symmetry of the magnetic core part, only one 4f orbital (4f xyz ) overlaps with the π * orbital (Fig. 2b in the main text). Therefore, we only include the 4f xyz orbital for each lanthanide site in the model Hamiltonian.
Diagonalizing the tight-binding Hamiltonian (S1), the one-electron levels are obtained as where the subscript "a" and "s" indicate antisymmetric and symmetric orbitals, respectively.
Comparing these orbital levels with the DFT calculations, we obtain parameters f , t, and ∆.
The highest occupied Kohn-Sham orbital for the down spin in the low-symmetry DFT solutions correspond to the π * orbital. On the other hand, 4f atomic orbitals contribute to many Kohn-Sham orbitals. Thus, the 4f orbitals are localized as follows. Because of the inversion symmetry of the complexes, the 4f orbital part of each Kohn-Sham orbital ψ i is decomposed into the antisymmetric and symmetric parts: where, |1 and |2 indicate the 4f xyz orbitals on the first and the second lanthanide sites, respectively. The absolute values of C a,i and C s,i for the occupied Kohn-Sham orbitals for the up spin part are shown in Fig. S1. As the antisymmetric and the symmetric levels, we averaged the Kohn-Sham levels: In Eq. (S6), the sum is taken over occupied Kohn-Sham orbitals. With the use of the levels,

S3
the parameters t and ∆ are derived ( Table I in  The high-and low-spin states of the complex 1 were analyzed based on the Hubbard where γ is the component of the 4f orbital, u f and u π * are the intrasite Coulomb repulsions on Gd and N 2 sites, respectively, and v is the intersite Coulomb repulsion between the Gd and N 2 sites.
The high-spin state with the maximal projection is described by one electron configuration: where 1 and 2 are the lanthanide sites and ↑ and ↓ are spin projections. The 4f electrons which are not in the 4f xyz orbital are not explicitly written here. The total energy E HS is where E 0 is the total electronic energy except for the electrons in the 4f orbitals and π * orbitals, and n is the number of the 4f electrons in Gd 3+ ion. For the low-spin state (↑, ↓, ↑ type), the basis set is Here, the configurations with the electron transfer from the 4f to the π * are not included because these configurations do not contribute much to the low-energy states due to the large energy gap ∆ between the 4f and the π * levels. The lowest energy is . (S11)

S4
The energy difference between the low-and high-spin states are is the (averaged) electron promotion energy. Eq. (S15) shows that (i) the energy gap ∆ significantly reduces the promotion energy and (ii) the promotion energy increases with the number of the 4f electrons n. Using the transfer parameter t derived from the Kohn-Sham orbital, energy gaps between the high-spin state and low-spin state, and Eq. (S14), the averaged promotion energyŪ is derived.  decomposition threshold was set to 5×10 −8 Hartree. The obtained SO-RASSI wave functions were transformed into pseudo spin states (or pseudoJ states) [2][3][4][5] to analyze the magnetic data using SINGLE ANISO module [6].

A. Fragment calculations for Ln
The obtained crystal-field (CF) levels are shown in Table S2. In all cases, the lowest spin-orbit states are doubly degenerate (Kramers doublet (KD) for Ln = Gd, Dy, Er) or quasidegenerate (Ising doublet for Ln = Tb, Ho). The ground CF states |ψ are decomposed into the sum of the ground pseudoJ multiplets |JM [4,5]: The coefficients C M are shown in  are calculated (Table S4). The Er complex is not magnetically anisotropic as much as the other complexes (Tb, Dy, Ho). This is because the multiplets |JM with small M (|M | < J) are mixed more than the other systems.

B. Calculation of atomic J-multiplets of Ln 2+ ions
The excitation energies of the intermediate virtual electron transferred states were replaced by the excitation energies for isolated Ln 2+ ion (Ln = Gd, Tb, Dy, Ho, Er). To obtain the energies, the CASSCF and the SO-RASSI calculations were performed with ANO-RCC QZP basis set [1]. As in the case of the fragment calculations, all 4f orbitals are  Table S5.

III. ANALYSIS OF FIRST RANK EXCHANGE PARAMETERS
As shown in Table II where t mπ * is the electron transfer between the 4f with component m of orbital angular momentum and the π * orbital of N 2 , l 1 = 3 is the magnitude of the atomic orbital angular momentum ., x, C xξ l 1 m kq and C xξ k −q l 1 m are Clebsch-Gordan coefficients [9], α J and J are the LS-term and the total angular momentum of Ln 2+ , respectively, ∆E n+1 α J J is the excitation multiplet energies of Ln 2+ , and G Ln α J Jk xk andF N 2 k are functions of their subscripts. For the detailed description of x, G Ln α J Jk xk , andF N 2 k , see Ref. 8. Since the dependence of the exchange parameter (S18) on q and q appears only in {t × t} x kqk q (S19), the condition for the isotropy of J kq1q is revealed from the equation. First, we consider the cases where only the transfer between f ±m 0 orbitals (m 0 = 0, 1, 2, 3) and the  Table S9. We find that the condition (S17) is fulfilled when m 0 = 2, while it is not for other m 0 . In the case of m 0 = 1, the nonzero terms with q = q = ±1 are also the source of the anisotropic exchange. When more than one set of f orbitals m 0 contribute to the electron transfer, the exchange interaction becomes always anisotropic. Finally, since Eq. (S19) is independent of ions, the condition given above applies to the exchange interaction between any f electron ions and spin 1/2.