NaRuO$_2$: Kitaev-Heisenberg exchange in triangular-lattice setting

Kitaev exchange, a new paradigm in quantum magnetism research, occurs for 90$^{\circ}$ metal-ligand-metal links, $t_{2g}^5$ transition ions, and sizable spin-orbit coupling. It is being studied in honeycomb compounds but also on triangular lattices. While for the former it is known by now that the Kitaev intersite couplings are ferromagnetic, for the latter the situation is unclear. Here we pin down the exchange mechanisms and determine the effective coupling constants in the $t_{2g}^5$ triangular-lattice material NaRuO$_2$, recently found to host a quantum spin liquid ground state. We show that, compared to honeycomb compounds, the characteristic triangular-lattice cation surroundings dramatically affect exchange paths and effective coupling parameters, changing the Kitaev interactions to antiferromagnetic. The quantum chemical analysis and subsequent effective spin model computations provide perspective onto the nature of the experimentally observed quantum spin liquid -- it seemingly implies finite longer-range exchange, and the atypical proximity to ferromagnetic order is related to sizable ferromagnetic Heisenberg nearest-neighbor couplings.

Exchange through 90 • metal-ligand-metal bonds represents one of the limiting cases in (super)exchange theory [1,2].In the simplest situation of half-filled dmetal orbitals, this geometry is associated with Heisenberg ferromagnetism.Away from half-filling and for degenerate orbitals, however, very intricate physics may arise, as pointed out by Jackeli and Khaliullin for t 5 2g magnetic centers with sizable spin-orbit coupling: highly anisotropic effective interactions involving only (pseudo)spin components normal to the M 2 L 2 square plaquette formed by two transition-metal (TM) ions and the two bridging ligands [3,4].In crystals of NaCl type and derivative rhombohedral structures which imply three different possible orientations of the M 2 L 2 plaquettes (see 1), this translates into directional dependence of the nearest-neighbor exchange: on differently oriented adjacent M 2 L 2 units -i.e., normal to either x, y, or z -different components of the magnetic moments interact, either Sx i and Sx j (on x-type, normal to the x-axis plaquettes), Sy i and Sy k (y type), or Sz i and Sz l (z type).The (super)exchange model of Jackeli and Khaliullin [3] thus formalizes Kitaev's effective Hamiltonian of bonddependent anisotropic magnetic couplings [5] proposed initially more like a heuristic device.It launched a whole new subfield in the research area of quantum magnetic materials [6,7], that of 5d 5 and 4d 5 honeycomb magnets [4], with subsequent extension to 3d 7 t 5 2g e 2 g hexagonal networks of edge-sharing ML 6 octahedra.
Here we explore the nature of nearest-neighbor couplings in a 4d 5 triangular-lattice magnet, NaRuO 2 , and evidence the presence of sizable bond-dependent Kitaev interactions.Interestingly, those are antiferromagnetic, different from the case of honeycomb 4d 5 and 5d 5 magnetic lattices.Preliminary numerical tests highlight the important role of cation species around the ligands mediating (super)exchange: the sign change is related to electrostatics having to do with adjacent TM ions that in the honeycomb setting are missing.Sizable antiferromagnetic off-diagonal intersite couplings are also predicted, along with somewhat larger ferromagnetic Heisenberg exchange.The latter seems to be consistent qualitatively with features seen in experiment: incipient ferromagnetism within a quantum disordered ground state [8,9].By quantum chemical computations at different levels of approximation, we additionally show that anisotropic Coulomb exchange defines a major interaction scale: it represents ∼45% of the kinetic exchange contribution in the case of the diagonal Kitaev coupling K and is 4 times larger than d-d kinetic exchange for the off-diagonal Γ .Coulomb exchange being ignored so far in Kitaev (super)exchange models, these results provide perspective onto what reliable quantitative predictions would imply: not only controlled approximations to deal with intersite virtual hopping [1][2][3] but also exact Coulomb exchange.

Results
On-site multiplet structure and intersite couplings in NaRuO 2 .NaRuO 2 is rather unique, a realization of j eff ≈ 1/2 t 5 2g moments on a triangular network of edge-sharing RuO 6 octahedra (see 2).The basic Ru 3+ 4d 5 multiplet structure in the specific delafossite crystalline setting of NaRuO 2 is illustrated in I, as obtained by quantum chemical [10] complete-active-space selfconsistent-field (CASSCF) [10,11] and post-CASSCF multireference configuration-interaction (MRCI) [10,12] embedded-cluster calculations (see Methods section for computational details and Supplementary Information for basis sets).
To separately visualize the effects of crystal-field (CF) splittings and spin-orbit coupling (SOC), both scalar relativistic (CASSCF, MRCI) and MRCI+SOC [13] results are listed.The t 2g -e g splitting is larger than in e. g.RuCl 3 [14], due to larger ligand effective charges and shorter TM-ligand bonds in NaRuO 2 .It is seen that the MRCI correlation treatment brings significant corrections to some of the CASSCF relative energies, in particular for the 6 A 1g term.Also important is the SOC (see the last two columns in I).However, the effects of trigonal fields are visible too: those remove the degeneracy of the 2 T 2g CF states (p-orbital-like l eff = 1/2  [15] and s = 1/2 quantum numbers in ideal cubic environment) and speak for significant deviations from textbook [3,15] j eff = 1/2 spin-orbit moments.
Even with sizable trigonal fields, existing estimates of several meV for the Kitaev interactions in honeycomblattice 4d t 5 2g compounds such as RuCl 3 (see e. g. [16] and discussion in [17]) and Li 2 RhO 3 [18] provide strong motivation for detailed ab initio investigation of the effective couplings in NaRuO 2 .To this end, further quantum chemical computations were performed for a block of two adjacent edge-sharing RuO 6 octahedra, following the procedure described in Refs.[14,19] (see also Methods section).Mapping the quantum chemical data onto the relevant effective spin model for C 2h symmetry of the [Ru 2 O 10 ] magnetic unit, by spin-orbit MRCI, we arrive to Kitaev, Heisenberg, and off-diagonal Γ coupling constants K = 2.0, J = −5.2,Γ ≡ Γ xy = 3.6, Γ ≡ Γ yz = Γ zx = 1.1 (meV).Interestingly, both K and Γ are antiferromagnetic, with Γ > K, but the largest interaction parameter is the Heisenberg J.The latter being ferromagnetic, it brings us away from the antiferromagnetic ground states (zigzag antiferromagnetic order, in particular) experimentally found in j ≈ 1/2 honeycomb systems, as depicted for instance in the phase diagrams computed for J-K-Γ triangular-lattice models in Ref. [20].
According to Ref. [20], quantum spin liquid (QSL) ground states can only be realized for relatively large, antiferromagnetic Γ.Our result for the strength of Γ represents the largest ab initio quantum chemical estimate so far in real material setting.Yet, a nearest-neighbor J that is even larger in magnitude and ferromagnetic (see the values listed above and at the top of II) implies that only longer-range Heisenberg couplings and/or higherorder interactions such as ring exchange [21] can pull the system to the QSL regime evidenced experimentally [8].
Signatures of a nearby ferromagnetic state in NaRuO 2 appear via the enhanced Van Vleck term in the magnetic susceptibility [8], which approaches values observed in nearly magnetic metals like Pd.Furthermore, despite possessing a charge gap and insulating ground state, clear quasiparticle excitations generate a Sommerfeld-like term in the low temperature heat capacity [8].This suggests the presence of strong spin fluctuations associated with a nearby magnetic state.These seemingly gapless spin fluctuations are robust, and are directly resolved in neutron scattering as diffuse continuum-like modes about the nuclear zone center and near Q=0 [8].Initial attempts at scaling S(Q,E) suggest the proximity of a nearby ferromagnetic quantum critical point, and small, nonmagnetic chemical perturbations to the network of Ru 3+ ions induce a glass-like freezing of the magnetic moments [9].
What makes K antiferromagnetic.Finding an antiferromagnetic Kitaev coupling in NaRuO 2 , opposite to the ferromagnetic K generally found in honeycomb Kitaev-Heisenberg magnets (see [14,18,22] for quantum chemical results and e. g. [16,23] plus discussion in [17] for altenative approaches), is intriguing.To shed light on this aspect, we performed the following numerical experiment: in a new set of quantum chemical computations, the two magnetic-plane cations in the immediate neighborhood of the bridging ligands, forming a line perpendicular to the link of nearest-neighbor Ru sites (see 2(c)), were removed and their charge redistributed within the point charge array modeling the extended solid state surroundings.Remarkably, without those nearby positive ions, three of the effective magnetic couplings change sign (see II).This suggests that strong orbital polarization effects at the ligand sites, induced by extra nearby positive charge on the triangular lattice (+3 effective nearby charges in NaRuO 2 versus +1 in honeycomb '213' compounds such as Na 2 IrO 3 and Li 2 RhO 3 or 0 in RuCl 3 ) have dramatic consequences on hopping matrix elements and superexchange processes.Similar additional tests   [22], produced in that case by cations residing between the magnetic layers.Ru 2 O 2 -unit correlations.Extended magnetic lattice.Insights into the interplay between spin-orbit couplings and electron correlation effects on a plaquette of two nearest-neighbor Ru ions and two bridging ligands can be obtained by analysing computational data obtained at lower levels of approximation, i. e., spin-orbit calculations including only the t 5 2g -t 5 2g electron configuration and CASSCF+SOC calculations that also account for excited-state t 4 2g -t 6 2g configurations since all possible occupations are considered within the six-orbital (three t 2g orbitals per Ru site) active space in the latter case.
As illustrated in III, anisotropic effective intersite interactions comparable in size with the isotropic Heisenberg J are already found at the single-configuration (SC) level, i. e., when accounting for just t 5 2g -t 5 2g Coulomb exchange (sometimes referred to as direct exchange [2]).Large anisotropic Coulomb exchange as found in the SC calculation represents physics not addressed so far in the literature -Kitaev magnetism is presently exclusively explained through TM-TM kinetic exchange and TM-L-TM superexchange.
CASSCF, through inclusion of intersite t 2g → t 2g excitations (i.e., TM-TM kinetic exchange), brings sizable corrections to K, J, and Γ in particular.By considering next, in the spin-orbit MRCI treatment, all possible single and double excitations out of the Ru t 2g and bridging-ligand orbitals, the most significant post-CASSCF corrections are found to occur for K and Γ.This suggests that more sophisticated calculations, e. g., MRCI+SOC based on larger active spaces in the prior CASSCF, would bring significant additional corrections to K and Γ mostly, i. e., would only enhance antiferromagnetic fluctuations on the extended lattice.The effects of sizable antiferromagnetic Γ (last column in III), longer-range Heisenberg couplings, and ring exchange [21] are so far unknown in triangular-lattice setting.
To obtain first impressions on the role of longer-range Heisenberg interactions, second-neighbor (J 2 ) and thirdneighbor (J 3 ), in particular, we performed density-matrix renormalization group (DMRG) calculations [24,25] for a J-K-Γ-Γ -J 2 -J 3 model on fragments of 19, 27, and 37 sites of the triangular lattice.In order to prevent artifacts, e. g., artificial stabilization (or destabilization) of particular magnetic states, we applied open boundary conditions.The validity of this material model is discussed in the Supplementary Information.Setting J, K, Γ, and Γ to -5.5, 4, 4, and 1.5 meV, respectively, we obtain a very rich phase diagram, see 3. Notably, a QSL phase is found for antiferromagnetic J 2 values and ferromagnetic J 3 .The corresponding structure factor has no Bragg peaks but exhibits a characteristic pattern.It indicates that the QSL emerges from the competition of adjacent ordered states.The most remarkable spot is the region where the QSL neighbors ferromagnetic order and a spiral phase, namely, around J 2 = 5, J 3 = −2 -as shown in 3, at J 2 = 5, J 3 = −2 the structure factor implies a broad spectral feature for momenta near |Q|=0, seemingly consistent with experimental observations [8].Interestingly, ferromagnetic and long-period 'spiral' do-mains in the presence of defects would also yield a broad spectral feature in the vicinity of |Q|=0.The rather sharp increase seen at low fields in the magnetization curve [8] would be reproduced in either case.

Conclusions
While the way nearby cations are structurally arranged can strongly affect on-site electronic-structure features such as subshell level splittings [19,26,27], single-ion anisotropies [26], and g factors [19,27], detailed ab initio investigations of the effect on intersite magnetic couplings are less numerous.Here we show that, compared to honeycomb compounds, the characteristic triangularlattice cation surroundings dramatically affect superexchange paths and effective coupling parameters.In particular, with respect to honeycomb j ≈ 1/2 systems, the Kitaev interaction constant changes its sign in triangularlattice NaRuO 2 .By providing insight into the signs and strengths of the nearest-neighbor magnetic interactions in this material, our work sets the frame within which the role of longer-range effective spin couplings should be addressed.Interestingly, while giving rise to antiferromagnetic order on honeycomb Kitaev-Heisenberg lattices, the latter appear to be decisive in realizing quantum disorder [8] in NaRuO 2 .Last but not least, we establish the role of anisotropic Coulomb exchange, a mechanism not addressed so far in Kitaev-Heisenberg magnetism.

Methods
Ru-site multiplet structure.All quantum chemical computations were carried out using the molpro suite of programs [28].Crystallographic data as reported by Ortiz et al. [8] were utilized.For each type of embedded cluster, the crystalline environment was modeled as a large array of point charges which reproduces the crystalline Madelung field within the cluster volume; we employed the ewald program [29,30] to generate the point-charge embeddings.
To clarify the Ru-site multiplet structure, a cluster consisting of one 'central' RuO 6 octahedron, the six inplane adjacent octahedra, and 12 nearby Na cations was designed.The quantum chemical study was initiated as complete active space self-consistent field (CASSCF) calculations [10,11], with an active orbital space containing the five 4d orbitals of the central Ru ion.Post-CASSCF correlation computations were performed at the level of multireference configuration-interaction (MRCI) with single and double excitations [10,12] out of the Ru 4d and O 2p orbitals of the central RuO 6 octahedron.Spin-orbit couplings (SOCs) were accounted for following the procedure described in Ref. [13].
Intersite exchange in NaRuO 2 .Clusters with two edge-sharing RuO 6 octahedra in the central region were considered in order to to derive the inter-site effective magnetic couplings.The eight in-plane octahedra directly linked to the two-octahedra central unit and 22 nearby Na ions were also explicitly included in the quantum chemical computations but using much more compact basis sets.
CASSCF computations were carried out with six (Ru t 2g ) valence orbitals and ten electrons as active (abbreviated as (10e,6o) active space); the t 2g orbitals of the adjacent TM ions were part of the inactive orbital space.In the subsequent MRCI correlation treatment, single and double excitations out of the central-unit Ru t 2g and bridging-O 2p levels were considered.We used the Pipek-Mezey methodology [31] to obtain localized central-unit orbitals.
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 SOCs were further accounted for [13], either at singleconfiguration (SC), CASSCF, or MRCI level, which finally yields a number of 36 spin-orbit states.The SC label in Table III in the main text stands for a CASCI in which intersite excitations are not considered.This is also referred to as occupation-restricted multiple active space (ORMAS) scheme [32].
Only one type of Ru-Ru links is present in NaRuO 2 .The unit of two nearest-neighbor octahedra displays C 2h point-group symmetry, which implies a generalized bilinear effective spin Hamiltonian of the following form for a pair of adjacent 1/2-pseudospins Si and Sj : The Γ αβ coefficients refer to the off-diagonal components of the 3×3 symmetric-anisotropy exchange matrix; α, β, γ ∈ {x, y, z}.An antisymmetric Dzyaloshinskii-Moriya coupling is not allowed, given the inversion center.
The lowest four spin-orbit eigenstates from the molpro output (eigenvalues lower by ∼200 meV or more than the eigenvalues of higher-lying excited states, as illustrated for example in Table I) were mapped onto the eigenvectors of the effective spin Hamiltonian (2), following the procedure described in Refs.[14,33] : 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 (2).Having two of the states in the same irreducible representation of the C 2h point group [18], such one-toone mapping translates into two possible sets of effective magnetic couplings.The relevant array is chosen as the one whose g factors fit the g factors obtained for a single RuO 6 t 5 2g octahedron.We used the standard coordinate frame usually employed in the literature, different from the rotated frame employed in earlier quantum chemical studies [14,18,22] that affects the sign of Γ.This is the reason the sign of Γ for RuCl 3 in Table II in the main text is different from the sign in Ref. [14] (see also footnote [48] in Ref. [17]).
Data Availability.The data that support the findings of the current study are available from the corresponding author upon reasonable request.
Quantum chemical computational details.All quantum chemical computations were carried out using the molpro suite of programs [1].Crystallographic data as reported by Ortiz et al. [2] were utilized.For each type of embedded cluster, the crystalline environment was modeled as a large array of point charges which reproduces the crystalline Madelung field within the cluster volume; we employed the ewald program [3,4] to generate the point-charge embeddings.
To clarify the Ru-site multiplet structure, a cluster consisting of one 'central' RuO 6 octahedron, the six inplane adjacent octahedra, and 12 nearby Na cations was designed.The quantum chemical study was initiated as complete active space self-consistent field (CASSCF) calculations [5,6], with an active orbital space containing the five 4d orbitals of the central Ru ion.Post-CASSCF correlation computations were performed at the level of multireference configuration-interaction (MRCI) with single and double excitations [5,7] out of the Ru 4d and O 2p orbitals of the central RuO 6 octahedron.Spin-orbit couplings (SOCs) were accounted for following the procedure described in Ref. [8].For the central Ru ion energy-consistent relativistic pseudopotentials (ECP28MDF) and Gaussian-type valence basis sets of effective quadruple-ζ quality (referred to as ECP28MDF-VTZ in the molpro library) [9] were employed, whereas we used all-electron triple-ζ basis sets for the O ligands of the central RuO 6 octahedron [10].The six adjacent in-plane cations were represented as closedshell Rh 3+ t 6 2g species, using quasirelativistic pseudopotentials (Ru ECP29MWB) [11] and (Ru ECP28MWB) (8s7p6d)/[5s4p3d] basis sets for electrons in the 4th shell [11,12].Large-core pseudopotentials were adopted for the 12 adjacent Na cations [13].
Clusters with two edge-sharing RuO 6 octahedra in the central region were considered in order to to derive the inter-site effective magnetic couplings.The eight in-plane octahedra directly linked to the twooctahedra central unit and 22 nearby Na ions were also explicitly included in the quantum chemical computations but using much more compact basis sets.We utilized energy-consistent relativistic pseudopotentials (ECP28MDF) and Gaussian-type valence basis sets of effective quadruple-ζ quality (ECP28MDF-VTZ) for the central Ru species [9].All-electron basis sets of quintuple-ζ quality were employed for the two bridging ligands [10] while all-electron basis sets of triple-ζ quality were applied for the remaining eight O anions [10] associated with the two octahedra of the reference unit.The eight adjacent transition-metal (TM) cations were represented as closed-shell Rh 3+ t 6 2g species, using nonrelativistic pseudopotentials (Ru ECP29MHF) and (Ru ECP28MHF) (8s7p6d)/[4s3p2d] basis sets [11,12]; the outer 22 O ligands associated with the eight adjacent octahedra were described through minimal all-electron atomic-natural-orbital (ANO) basis sets [14].Large-core pseudopotentials were considered for the 22 Na nearby cations [13].
CASSCF computations were carried out with six (Ru t 2g ) valence orbitals and ten electrons as active (abbreviated as (10e,6o) active space); the t 2g orbitals of the adjacent TM ions were part of the inactive orbital space.In the subsequent MRCI correlation treatment, single and double excitations out of the central-unit Ru t 2g and bridging-O 2p levels were considered.We used the Pipek-Mezey methodology [15] to obtain localized central-unit orbitals.
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 SOCs were further accounted for [8], either at singleconfiguration (SC), CASSCF, or MRCI level, which finally yields a number of 36 spin-orbit states.The SC label in Table III in the main text stands for a CASCI in which intersite excitations are not considered.This is also referred to as occupation-restricted multiple active space (ORMAS) scheme [16].
Only one type of Ru-Ru links is present in NaRuO 2 .The unit of two nearest-neighbor octahedra displays C 2h point-group symmetry, which implies a generalized bilinear effective spin Hamiltonian of the following form for a pair of adjacent 1/2-pseudospins Si and Sj : The Γ αβ coefficients refer to the off-diagonal components of the 3×3 symmetric-anisotropy exchange matrix; α, β, γ ∈ {x, y, z}.An antisymmetric Dzyaloshinskii-Moriya coupling is not allowed, given the inversion center.

arXiv:2212.09365v2 [cond-mat.str-el] 3 Jul 2023
The lowest four spin-orbit eigenstates from the molpro output (eigenvalues lower by ∼200 meV or more than the eigenvalues of higher-lying excited states, as illustrated for example in Table I) were mapped onto the eigenvectors of the effective spin Hamiltonian (S1), 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 lowestenergy spin-orbit eigenvectors are put in direct correspondence with the respective eigenvalues and matrix elements of (S1).Having two of the states in the same irreducible representation of the C 2h point group [19], such one-to-one mapping translates into two possible sets of effective magnetic couplings.The relevant array is chosen as the one whose g factors fit the g factors obtained for a single RuO 6 t 5 2g octahedron.We used the standard coordinate frame usually employed in the literature, different from the rotated frame employed in earlier quantum chemical studies [18][19][20] that affects the sign of Γ.This is the reason the sign of Γ for RuCl 3 in Table II in the main text is different from the sign in Ref. [18] (see also footnote [48] in Ref. [21]).
To demonstrate that various magnetic phases in the triangular-lattice effective-spin model can be properly identified for the clusters employed in our DMRG calculations (see Fig. S1(a) ), we plot in Fig. S1(c-e) the static spin structure factor for a 120 • structure (with J = 1), stripe order (computed for J = 1, J 2 = 0.3), and a four-sublattice 120 • structure (J = −1, J 3 = 10).In the expression for S(Q), N is the number of sites and r i is the position of site i.The Brillouin zone and the relevant ordering vectors are depicted in Fig. S1(b).As mentioned in the main text, in the spiral phase the propagation vector increases continuously when moving away from the ferromagnetic (FM) sector.To illustrate this, we show in Fig. S2 the evolution of S(Q) for variable J 3 and J 2 = 0 (J, K, Γ, and Γ being set to -5.5, 4, 4, and 1.5 meV, respectively).We find that the propagation vector increases continuously with increasing J 3 , from Q = 0 at the FM-spiral critical point J 3 ≈ 2.3 to Q = 2π/3, corresponding to the Bragg peak of a four-sublattice 120 • structure, at J 3 ≈ 7.
Cluster-size dependence of S(Q) for J 2 ≈ 5, J 3 ≈ −2.For longer-range Heisenberg couplings J 2 ≈ 5 and J 3 ≈ −2, inside the QSL phase, the structure factor exhibits a broad spectral feature for momenta near Q = 0, see discussion in the main text.Four ordered phases are con-

FIG. 1 .
FIG. 1.(a) Sketch of the NaCl-type lattice, with M-L bonds along x, y, or z.If two different cation varieties (alkaline, A, and TM, B, ions) form successive layers normal to the [111] axis, a rhombohedral ABL2 structure is realized, with a triangular network of edge-sharing octahedra in each layer.(b) On each B2L2 plaquette (ions not drawn), the Kitaev interaction couples only (pseudo)spin components perpendicular to the respective plaquette.All three components are shown only for the central magnetic site.

FIG. 2 .
FIG. 2. Delafossite layered structure (a) and the triangular network of edge-sharing octahedra of NaRuO2 (b); grey, red, and yellow spheres represent Ru, O, and Na ions, respectively.(c) To understand how the nearby surroundings affect magnetic couplings, Ru neighbors in the median plane (sites 3, 4) of a given Ru-Ru magnetic link (sites 1, 2) were removed in one set of quantum chemical computations.

FIG. 3 .
FIG.3.Ground-state phase diagram obtained by DMRG computations for the J-K-Γ-Γ -J2-J3 model (right) and typical static spin structure factors for each phase (left).Using the quantum chemical analysis as a guide (see data in III and related discussion), J, K, Γ, and Γ were set to to -5.5, 4, 4, and 1.5 meV, respectively.

TABLE I .
Ru 3+ 4d 5 multiplet structure, with all five 4d orbitals active in CASSCF.Each MRCI+SOC value refers to a Kramers doublet (KD); just the lowest and highest KDs are shown for each group of t 4 2g e 1 g spin-orbit states.Only the CF terms listed in the table entered the spin-orbit calculations.For simplicity, notations corresponding to O h symmetry are used.

TABLE III .
Nearest-neighbor magnetic couplings (meV) at different levels of theory: SC (only the t 5 2g -t 5 2g configuration considered), CASSCF (also t 4 2g -t 6 2g states included), and MRCI (single and double excitations out of the Ru 4d t2g and bridging-O 2p levels on top of CASSCF).forRuCl 3 , where +3 ions were placed in the centers of two edge-sharing Ru 6 hexagonal rings, confirm the trend: the extra positive charge in the neighborhood of the Ru 2 O 2 'magnetic' plaquette (i.e., [Ru 2 O 10 ] cluster of edge-sharing octahedra) invert the sign of the Kitaev coupling constant, from ferromagnetic in actual RuCl 3[14]to antiferromagnetic in the presence of positive charge in the centers of the two hexagons sharing the magnetic Ru-Ru link (see lowest lines in II).Strong polarization effects of this type were earlier pointed out in the case of the Kitaev-Heisenberg honeycomb iridate H 3 LiIr 2 O 6