Introduction

Magnetism has constantly been a source of new fundamental concepts in solid-state and statistical physics. It is also important to technological applications: many devices around us are (electro)magnetic. Magnetism is typically illustrated through Heisenberg’s textbook model of interacting atomic magnetic moments. Recently, however, it has become clear that for certain magnetic materials Heisenberg’s isotropic interaction picture is not applicable; their behavior can only be described through highly anisotropic spin models. The latter may imply completely different interaction strengths for different magnetic-moment projections and, seemingly counterintuitive, directional dependence of the leading anisotropic coupling1,2,3 for symmetry-equivalent pairs of moments. While that opens entire new perspectives in magnetism, provides the grounds for new, exotic states that are now being revealed for the first time, and hints to potential technological applications like quantum computation1, how such anisotropies arise is not yet fully clarified: we know those may be dominant in particular systems but do not understand in detail the underlying physics and how to tune such interactions in the lab.

Here we shed fresh light on the exchange mechanisms underlying symmetric anisotropic magnetic interactions, both diagonal (i.e., Kitaev1,2) and off-diagonal, by using ab initio quantum chemical computational methods. To do so, we exploit the ladder of controlled approximations that quantum chemistry offers: single-configuration schemes, multiconfiguration theory, and multireference configuration-interaction. Honeycomb α-RuCl3, in particular, the relatively high-symmetry crystalline structure recently discovered under a pressure of ≈1.3 GPa4, and triangular-lattice NaRuO25 were chosen as benchmark Kitaev–Heisenberg material platforms. We establish that a decisive contribution to the Kitaev effective coupling constant K comes from (anisotropic) Coulomb exchange, a mechanism ignored so far in the literature. In the case of the off-diagonal (x-z/y-z) interaction \({\Gamma }^{{\prime} }\), which can give rise to spin liquid ground states by itself 6, anisotropic Coulomb exchange is even dominant, as much as ~90% of the effective coupling parameter computed by multireference configuration-interaction. Our analysis provides unparalleled specifics as concerns \({t}_{2g}^{5}\)–\({t}_{2g}^{5}\) Kitaev–Heisenberg magnetic interactions and perspective onto what reliable quantitative predictions would imply: not only controlled ab initio approximations to explicitly tackle intersite virtual excitations but also exact Coulomb exchange; the latter is available in self-consistent-field Hartree–Fock theory, the former in post–Hartree–Fock wave-function-based quantum chemical methods.

Results

Kitaev magnetism refers to anisotropic magnetic interactions \(K{\tilde{S}}_{i}^{\gamma }{\tilde{S}}_{j}^{\gamma }\) that are ‘bond’ dependent, i.e., for a given pair of adjacent 1/2 pseudospins \({\tilde{{{{{{{{\bf{S}}}}}}}}}}_{i}\) and \({\tilde{{{{{{{{\bf{S}}}}}}}}}}_{j}\), the easy axis defined through the index γ can be parallel to either x, y, or z1. This can be easily visualized for layered configurations of edge-sharing ML6 octahedra derived from the rocksalt crystalline arrangement (see Fig. 1), either triangular-lattice AMO23,7 or honeycomb A2MO32 (and MCl3) structures, where M, A, and L are transition-metal, alkaline, and ligand ions, respectively: for each of the magnetic ‘bonds’ emerging out of a given magnetic site M, the easy axis (x, y, or z) is normal to the square plaquette defined by two adjacent transition ions and the two bridging ligands. Kitaev’s honeycomb spin model has quickly become a major reference point in quantum magnetism research: it is exactly solvable and yields a quantum spin liquid (QSL) ground state in which the spins fractionalize into emergent Majorana quasiparticles1. The latter are neutral self-adjoint fermions that are simultaneously particle and antiparticle. QSL ground states have been proposed for the Kitaev–Heisenberg honeycomb systems H3LiIr2O68 and α-RuCl39,10 and also in a triangular-lattice magnet with seemingly sizable anisotropic intersite couplings, NaRuO25. However, in the case of the honeycomb compounds, what is described as QSL behavior either occurs in the presence of coupling-parameter disorder induced by unavoidable randomness of the H+ cations (H3LiIr2O6) or requires an external magnetic field (α-RuCl3).

Fig. 1: Orthogonal M2L2 plaquettes in rocksalt structure and derivatives.
figure 1

a Rocksalt-type lattice. The M-L bonds are along either x, y, or z. b With two different cation species (A, B) forming successive layers normal to the [111] axis, a rhombohedral ABL2 structure is realized—each layer features a triangular network of edge-sharing octahedra. Honeycomb A2BL3 structures are obtained when one of the cation species (A) occupies additional sites within the layer of the other (B), corresponding to the centers of B6 hexagonal rings; in α-RuCl3, all A sites are empty. c On each B2L2 plaquette (ions not drawn), the Kitaev interaction couples only spin components normal to the respective plaquette. All three spin projections are shown only for the central magnetic site.

Full size image

Having QSL phases materialized on both hexagonal and triangular networks of Ru2L2 plaquettes makes Ru quite special. For insights into \({t}_{2g}^{5}\)–\({t}_{2g}^{5}\) anisotropic exchange in both α-RuCl3 and NaRuO2 crystallographic settings, detailed quantum chemical calculations were carried out for Ru2Cl10 and Ru2O10 magnetic units as found in the respective materials. The adjacent in-plane RuL6 octahedra coordinating those two-octahedra central units were also explicitly included in the quantum chemical computations but using more compact atomic basis sets. The finite quantum mechanical cluster was embedded within a large array of point changes which reproduces the crystalline Madelung field within the cluster volume. To generate this collection of point charges we employed the ewald package11,12. Complete-active-space self-consistent-field (CASSCF) optimizations13,14 were initially performed with six (Ru t2g) valence orbitals and ten electrons as active (abbreviated hereafter as (10e,6o) active space). Subsequently, two other types of wave-functions were generated, using in each case the orbitals obtained from the (10e,6o) CASSCF calculations: (i) single-configuration (SC) \({t}_{2g}^{5}\)–\({t}_{2g}^{5}\) (i.e., the \({t}_{2g}^{4}\)–\({t}_{2g}^{6}\) and \({t}_{2g}^{6}\)–\({t}_{2g}^{4}\) configurations which were accounted for in the initial CASSCF were excluded in this case by imposing appropriate orbital-occupation restrictions) and (ii) multireference configuration-interaction (MRCI)13,15 wave-functions having the (10e,6o) CASSCF as kernel and additionally accounting for single and double excitations out of the central-unit Ru t2g and bridging-ligand valence p (either O 2p or Cl 3p) orbitals. By comparing data at these different levels of approximation—SC, CASSCF, and MRCI—it is possible to draw conclusions on the role of various exchange mechanisms. The CASSCF optimization was performed for the lowest nine singlet and lowest nine triplet states associated with the (10e,6o) setting. Those were the states for which spin-orbit couplings (SOCs) were further accounted for16, at either SC, CASSCF, or MRCI level, which yields in each case a number of 36 spin-orbit states.

A unit of two nearest-neighbor octahedra exhibits C2h point-group symmetry, in both α-RuCl34 and NaRuO25, implying a generalized bilinear effective spin Hamiltonian of the following form for a pair of adjacent 1/2-pseudospins \({\tilde{{{{{{{{\bf{S}}}}}}}}}}_{i}\) and \({\tilde{{{{{{{{\bf{S}}}}}}}}}}_{j}\):

$${{{{{{{{\mathcal{H}}}}}}}}}_{ij}^{(\gamma )}=J{\tilde{{{{{{{{\bf{S}}}}}}}}}}_{i}\cdot {\tilde{{{{{{{{\bf{S}}}}}}}}}}_{j}+K{\tilde{S}}_{i}^{\gamma }{\tilde{S}}_{j}^{\gamma }+\mathop{\sum}\limits_{\alpha \ne \beta }{\Gamma }_{\alpha \beta }({\tilde{S}}_{i}^{\alpha }{\tilde{S}}_{j}^{\beta }+{\tilde{S}}_{i}^{\beta }{\tilde{S}}_{j}^{\alpha }).$$
(1)

The Γαβ coefficients denote the off-diagonal components of the 3 × 3 symmetric-anisotropy exchange tensor, with α, β, γ ∈ {x, y, z}. The lowest four spin-orbit eigenstates from the ab initio quantum chemical output (eigenvalues lower by ~ 0.2 eV with respect to the eigenvalues of higher-lying excited states) are mapped for each different set of calculations onto the eigenvectors of the effective spin Hamiltonian (1), following the procedure described in refs. 17,18: those four expectation values and the matrix elements of the Zeeman Hamiltonian in the basis of the four lowest-energy spin-orbit eigenvectors are put in direct correspondence with the respective eigenvalues and matrix elements of (1).

Nearest-neighbor effective magnetic couplings as obtained at three different levels of theory (SOC included) for α-RuCl3 under pressure are listed in Table 1. A very interesting finding is the vanishingly small J value in the spin-orbit MRCI computations. This yields a fully anisotropic K-Γ-\({\Gamma }^{{\prime} }\) effective model for the nearest-neighbor magnetic interactions and in principle increases the chances of realizing a QSL ground state, compared to the system at ambient pressure. In the latter case, magnetic field is needed to induce QSL behavior9,10.

Table 1 Nearest-neighbor magnetic couplings (meV) in high-symmetry α-RuCl34, results of spin-orbit calculations at various levels of theory
Full size table

Even more remarkable are the large anisotropic Coulomb exchange contributions obtained by SC calculations with SOC. The diagonal Kitaev coupling K, for example, is basically the same at the lowest two levels of approximation (first column in Table 1), SC (only the \({t}_{2g}^{5}\)–\({t}_{2g}^{5}\) electron configuration considered) and CASSCF (10e,6o) (\({t}_{2g}^{5}\)–\({t}_{2g}^{5},{t}_{2g}^{4}\)–\({t}_{2g}^{6}\), and \({t}_{2g}^{6}\)–\({t}_{2g}^{4}\) configurations treated on the same footing, where the latter type of states bring kinetic Ru (t2g)–Ru(t2g) exchange). This indicates that intersite Ru (t2g) →  Ru (t2g) excitations (i.e., kinetic exchange) do not really affect K. What matter as concerns the size of the Kitaev coupling are (i) Coulomb exchange, with a contribution of –1.75 meV, and (ii) Ru-Cl2-Ru superexchange2,3,19,20,21, excitations involving the Ru 4d eg levels2,3,19, and so called dynamical correlation effects13 accounted for by MRCI, with a combined contribution of –2 meV. Especially striking is the diagnosis carried out for the off-diagonal \({\Gamma }^{{\prime} }\) effective interaction parameter: out of a spin-orbit MRCI value of 0.45 meV, 0.42 corresponds to anisotropic Coulomb exchange. A pictorial representation of the various contributions to K, J, Γ, and \({\Gamma }^{{\prime} }\) in high-symmetry α-RuCl34 is provided in Fig. 2.

Fig. 2: Exchange contributions to the intersite effective magnetic couplings in high-symmetry α-RuCl34.
figure 2

Coulomb exchange (single-configuration (SC) results, in red), Ru(t2g)--Ru(t2g) kinetic exchange (as the difference between complete-active-space self-consistent-field (CASSCF) and SC data, in blue), plus contributions related to Ru(t2g) → Ru(eg) excitations2,3,19, Ru-Cl2-Ru superexchange2,3,19,20,21, and so called dynamical correlation effects13 (as the difference between multireference configuration-interaction (MRCI) and CASSCF, in green).

Full size image

MRCI+SOC computations for adjacent edge-sharing RuO6 octahedra in triangular-lattice NaRuO2 indicate that the largest nearest-neighbor coupling parameter is the isotropic Heisenberg J, –5.2 meV; the other effective interactions, K, Γ, and \({\Gamma }^{{\prime} }\), amount to 2, 3.6, and 1.1 meV, respectively, by spin-orbit MRCI. For better visualization, since the most important anisotropic Coulomb exchange contributions arise also in this system for K and \({\Gamma }^{{\prime} }\), we depict in Fig. 3 only the structure of these two magnetic couplings and omit the J and Γ effective interactions, which have significantly larger absolute values. Plots for the latter are provided in Supplementary Fig. 1. It is seen that anisotropic Coulomb exchange represents again the second largest contribution to the Kitaev K and the leading underlying mechanism in the case of \({\Gamma }^{{\prime} }\). Interestingly, for K and J, adding the Coulomb-exchange contributions obtained by SC quantum chemical calculations (−1.0 and −0.7 meV, respectively) to estimates obtained from effective-model (super)exchange theory (K = 2.9 and J = −4.2 meV)22, brings the latter in rather good agreement with the MRCI+SOC values (2.0 and −5.2 meV, respectively), although this is not the case for Γ and \({\Gamma }^{{\prime} }\).

Fig. 3: Exchange contributions to K and \({\Gamma }^{{\prime} }\) in NaRuO25.
figure 3

Coulomb exchange (single-configuration (SC) results, in red), Ru(t2g)--Ru(t2g) kinetic exchange (as the difference between complete-active-space self-consistent-field (CASSCF) and SC data, in blue), plus contributions related to Ru(t2g) → Ru(eg) intersite excitations, Ru-O2-Ru superexchange, and dynamical correlations (as the difference between multireference configuration-interaction (MRCI) and CASSCF, in green).

Full size image

Discussion

Anisotropic Coulomb exchange as found in the SC calculations (also referred to as direct exchange20 in isotropic context or potential exchange) has been mentioned as possibly relevant player in Dzyaloshinskii–Moriya cuprate context23 but not addressed in existing Kitaev–Heisenberg literature (see, e.g., the effective-model studies of refs. 19,22,24,25,26,27). Finding that up to ~45% of the Kitaev effective coupling constant K has to do with Coulomb exchange and that the off-diagonal anisotropic coupling \({\Gamma }^{{\prime} }\), which may give rise to spin-liquid ground states by itself 6, comes more than 90% from Coulomb exchange (last column in Table 1) obviously challenges present views and notions in Kitaev–Heisenberg quantum magnetism research. This is just another example illustrating the need for ab initio quantum chemical methods in order to achieve even a qualitatively correct picture of the essential underlying physics. Recent quantum chemical results that lead to the same conclusion refer to the role of fluctuations involving the third and fourth electronic shells in renormalizing antiferromagnetic interactions in copper oxide compounds28.

To provide additional reference points, we computed the isotropic Coulomb exchange integrals (i.e., without accounting for SOC) for different distributions of the Ru t2g holes in SC \({t}_{2g}^{5}\)–\({t}_{2g}^{5}\) arrangement (see Supplementary Table 1). For holes in plaquette-plane 4d orbitals having overlapping lobes along the Ru-Ru axis (i.e., for xy-like t2g functions), for example, the Coulomb exchange integral amounts to –25.4 meV. For comparison, in 3d9 copper oxide compounds, the Coulomb exchange matrix element lies in the region of –4 meV for edge-sharing geometry29 and –10 meV for corner-sharing ligand octahedra30,31 (from SC \({d}_{{x}^{2}-{y}^{2}}^{1}\)–\({d}_{{x}^{2}-{y}^{2}}^{1}\) calculations). For the latter type of linkage, being aware of experimental estimates of 100–250 meV for the Heisenberg J, an isotropic Coulomb exchange contribution of –10 meV can be neglected. But this is not the case for edge-sharing octahedra, in either Ru or Cu compounds.

How exactly SOC and Coulomb interactions commix to yield large anisotropic Coulomb exchange integrals remains to be analyzed in detail in future work. The important point however is that, at the SC (\({t}_{2g}^{5}\)–\({t}_{2g}^{5}\)) level, there is a Coulomb exchange matrix element for each possible pair of holes—dxy-dxy, dxy-dyz etc. SOC mixes up those different Slater determinants, and the resulting spin-orbit wave-functions are not spin eigenstates. Additionally, spin-orbit interactions remove the degeneracy of the ‘triplet’ states in the two-site magnetic problem. The spin-orbit fine structure in the two-site magnetic problem can be reduced to an effective pseudospin model only by introducing anisotropic Coulomb exchange matrix elements (i.e., the SC values provided in Table 1).

To summarize, we resolve the exchange mechanisms giving rise to anisotropic magnetic interactions on hexagonal and triangular networks of edge-sharing RuL\({}_{6}\,{t}_{2g}^{5}\) octahedra. Different from present assumptions and models relying exclusively on inter-atomic hopping (i.e., on indirect exchange), the quantum chemical analysis indicates major direct exchange contributions, to both K and \({\Gamma }^{{\prime} }\). This redefines the conceptual frame within which anisotropic intersite interactions should be addressed. In light of the ab initio quantum chemical data, various estimates, interpretations, and predictions based only on kinetic-exchange and superexchange mechanisms might require reevaluation—what is represented in red color in Figs. 2 and 3 is simply ignored in existing effective-model theories and studies. Our findings provide solid reference points for reliable electronic-structure investigations of other, closely related \({t}_{2g}^{5}\) Kitaev materials and also of \({t}_{2g}^{5}{e}_{g}^{2}\,j \, \approx \,1/2\) magnets.

Methods

All quantum chemical computations were carried out with the molpro suite of programs32. Atomic coordinates as determined by Stahl et al.4 and Ortiz et al.5 were used for α-RuCl3 and NaRuO2, respectively. We employed energy-consistent relativistic pseudopotentials (ECP28MDF) and Gaussian-type valence basis sets (BSs) of effective quadruple-ζ quality (ECP28MDF-VTZ)33 for the ‘central’ Ru species. All-electron BSs of quintuple-ζ quality were utilized for the two bridging ligands (Cl34 in α-RuCl3 and O35 in NaRuO2) and of triple-ζ quality for the remaining eight anions34,35 linked to the two octahedra of the reference, cental unit. The four adjacent cations in α-RuCl3 and eight adjacent transition ions in NaRuO2 were represented as closed-shell Rh\({}^{3+}\,{t}_{2g}^{6}\) species, using Ru ECP29 pseudopotentials and [3s3p3d] Ru BSs33 from the molpro library. The outer 16 Cl ligands associated with the four adjacent octahedra in α-RuCl3 and the outer 22 O ligands associated with the eight adjacent octahedra in NaRuO2 were described through minimal atomic-natural-orbital (ANO) BSs36. Large-core pseudopotentials were considered for the 22 Na nearby cations37 in NaRuO2.

We used the standard coordinate frame usually employed in the literature, different from the rotated frame employed in earlier quantum chemical studies18,38,39 that affects the sign of Γ (see also footnote [48] in ref. 40). The SC label in Table I in the main text indicates a CASCI in which intersite excitations are not considered. This is also referred to as occupation-restricted multiple active space (ORMAS) scheme41.