Superexchange interactions between spin-orbit-coupled $j\!\approx\!1/2$ ions in oxides with face-sharing ligand octahedra

TL;DR

This study uses ab initio calculations to reveal exceptionally strong antiferromagnetic interactions in face-sharing 5d and 4d oxide structures, highlighting their potential for novel quantum magnetic materials.

Contribution

It provides the first detailed ab initio analysis of superexchange interactions in face-sharing ligand octahedra, showing larger values than in other geometries.

Findings

Antiferromagnetic interactions up to 400 meV in idealized structures.

Significant electron-lattice coupling effects.

Potential for rich phase diagrams under strain and pressure.

Abstract

Using ab initio wave-function-based calculations, we provide valuable insights with regard to the magnetic exchange in 5$d$ and 4$d$ oxides with face-sharing ligand octahedra, BaIrO$_3$ and BaRhO$_3$. Surprisingly strong antiferromagnetic Heisenberg interactions as large as 400 meV are computed for idealized iridate structures with 90$^{\circ}$ Ir-O-Ir bond angles and in the range of 125 meV for angles of 80$^{\circ}$ as measured experimentally in BaIrO$_3$. These estimates exceed the values derived so far for corner-sharing and edge-sharing systems and motivate more detailed experimental investigations of quantum magnets with extended 5$d$/4$d$ orbitals and networks of face-sharing ligand cages. The strong electron-lattice couplings evidenced by our calculations suggest rich phase diagrams as function of strain and pressure, a research direction with much potential for materials of this type.

Tables

Table 1. Table 1: Energy splittings for the lowest four spin-orbit states of two face-sharing NN IrO 6 octahedra and the corresponding effective coupling constants at different levels of approximation, for d 0 = 2.63 subscript 𝑑 0 2.63 d_{0}\!=\!2.63 Å and θ 0 = 80 ∘ subscript 𝜃 0 superscript 80 \theta_{0}\!=\!80^{\circ} (all values in meV). The J 𝐽 J values without SOC by rAS, CAS, and CI are − - 1.2, 27.4, and 35.4 meV, respectively.

	rAS+SOC	CAS+SOC	CI+SOC
${\Psi }_S=(|\uparrow \downarrow ⟩-|\downarrow \uparrow ⟩)/\sqrt{2}$	15.2	0.0	0.0
${\Psi }_2=(|\uparrow \uparrow ⟩+|\downarrow \downarrow ⟩)/\sqrt{2}$	0.4	72.0	123.3
${\Psi }_3=(|\uparrow \uparrow ⟩-|\downarrow \downarrow ⟩)/\sqrt{2}$	0.0	74.0	126.5
${\Psi }_1=(|\uparrow \downarrow ⟩+|\downarrow \uparrow ⟩)/\sqrt{2}$	0.0	74.0	126.5
$\overline{J},\overline{\Gamma }$	$-$14.9, $-$0.7	72.0, 4.1	123.3, 6.3

Table 2. Table 2: Energy splittings for the lowest four spin-orbit states and the corresponding effective coupling constants for variable Ir-O-Ir angle, MRCI+SOC results (meV). The NN Ir-Ir distance is d 0 subscript 𝑑 0 d_{0} =2.63 Å. The Ir-O-Ir angle θ 𝜃 \theta is listed for each geometry; distances between Ir and the bridging O’s, d 𝑑 d (Ir-O), are provided within brackets.

	${70.5}^{\circ }$(2.27Å)	${75}^{\circ }$(2.16Å)	${80}^{\circ }$(2.04Å)	${85}^{\circ }$(1.94Å)	${90}^{\circ }$(1.86Å)
${\Psi }_S$	0.0	0.0	0.0	0.0	0.0
${\Psi }_2$	58.2	73.9	123.3	225.8	417.5
${\Psi }_1$	73.4	81.8	126.5	226.9	418.5
${\Psi }_3$	73.4	81.8	126.5	226.9	418.5
$\overline{J},\overline{\Gamma }$	58.2, 30.3	73.9, 15.9	123.3, 6.3	225.8, 2.1	417.5, 2.0

Table 3. Table 3: Energy splittings for the lowest four spin-orbit states of two face-sharing NN RhO 6 octahedra and the corresponding effective coupling constants at different levels of approximation, for d 0 ′ = 2.58 subscript superscript 𝑑 ′ 0 2.58 d^{\prime}_{0}\!=\!2.58 Å  and θ 0 ′ = 80 ∘ superscript subscript 𝜃 0 ′ superscript 80 \theta_{0}^{\prime}\!=\!80^{\circ} (all in meV). The J 𝐽 J values without SOC by rAS, CAS, and CI are − - 0.9, 19.0, and 29.4 meV, respectively.

	rAS+SOC	CAS+SOC	CI+SOC
${\Psi }_S=(|\uparrow \downarrow ⟩-|\downarrow \uparrow ⟩)/\sqrt{2}$	16.0	0.0	0.0
${\Psi }_3=(|\uparrow \uparrow ⟩-|\downarrow \downarrow ⟩)/\sqrt{2}$	0.0	25.4	54.0
${\Psi }_1=(|\uparrow \downarrow ⟩+|\downarrow \uparrow ⟩)/\sqrt{2}$	0.0	25.4	54.0
${\Psi }_2=(|\uparrow \uparrow ⟩+|\downarrow \downarrow ⟩)/\sqrt{2}$	1.1	26.5	55.5
$\overline{J},\overline{\Gamma }$	$-$14.9,$-$2.3	26.5,$-$2.2	55.5,$-$2.8

Table 4. Table 4: Energy splittings and the corresponding effective coupling constants for variable Rh-O-Rh angle, MRCI+SOC results (meV). The NN Rh-Rh distance is 2.58 Å. The Rh-O-Rh angle θ ′ superscript 𝜃 ′ \theta^{\prime} is listed for each geometry; distances between Rh and the bridging O’s, d 𝑑 d (Rh-O), are provided within brackets.

	${70.5}^{\circ }$(2.23Å)	${75}^{\circ }$(2.12Å)	${80}^{\circ }$(2.04Å)	${85}^{\circ }$(1.91Å)	${90}^{\circ }$(1.82Å)
${\Psi }_S$	0.0	0.0	0.0	0.0	0.0
${\Psi }_3$	3.7	15.0	54.0	135.7	173.4
${\Psi }_1$	3.7	15.0	54.0	135.7	173.4
${\Psi }_2$	1.5	15.2	55.5	137.8	164.0
$\overline{J},\overline{\Gamma }$	1.5, 4.4	15.2,$-$0.4	55.5,$-$2.8	137.8,$-$4.3	164.0, 18.8

Table 5. Table 5: Angle dependence of the trigonal CF splitting Δ t subscript Δ 𝑡 \Delta_{t} (eV) obtained from MRCI calculations for Ir 2 O 9 and Rh 2 O 9 fragments of face-sharing octahedra. The a 1 ​ g subscript 𝑎 1 𝑔 a_{1g} level is for all angles the lowest in energy.

$\theta /{\theta }^{\prime }$	${\Delta }_t^{\text{Ir}}$	${\Delta }_t^{\text{Rh}}$
70.5	$-$0.86	$-$0.69
75	$-$0.89	$-$0.67
80	$-$0.71	$-$0.53
85	$-$0.43	$-$0.29
90	$-$0.20	$-$0.10

Table 6. Table A1: Energy splittings for the lowest four spin-orbit states of two face-sharing NN IrO 6 octahedra and the corresponding effective coupling constants obtained from rAS+SOC, CAS+SOC, and MRCI+SOC calculations. For each geometry the Ir-Ir distance was fixed to 2.63Å. For θ 𝜃 \theta =90 ∘ , for instance, the J 𝐽 J values without SOC by rAS, CAS, and CI are 0.04, 64.7, and 116.4 meV, respectively. Distances between the Ir sites and the bridging O’s, d 𝑑 d (Ir-O), are provided within brackets. The data are presented as complementary to those in Table  2 (all values in meV).

$\theta$, $d$(Ir-O)	rAS+SOC	CASSCF+SOC	MRCI+SOC
${70.5}^{\circ }$ (2.27Å):
${\Psi }_S$	9.9	0.0	0.0
${\Psi }_2$	0.0	24.8	58.2
${\Psi }_1$	0.5	34.9	73.4
${\Psi }_3$	0.5	34.9	73.4
$\overline{J},\overline{\Gamma }$	$-$9.9, 1.1	24.8, 20.3	58.2, 30.3
${75}^{\circ }$ (2.16Å):
${\Psi }_S$	12.8	0.0	0.0
${\Psi }_2$	0.0	36.3	73.9
${\Psi }_1$	0.0	41.6	81.8
${\Psi }_3$	0.0	41.6	81.8
$\overline{J},\overline{\Gamma }$	$-$12.8, 0.0	36.3, 10.5	73.9, 15.9
${80}^{\circ }$ (2.04Å):
${\Psi }_S$	15.2	0.0	0.0
${\Psi }_2$	0.4	72.0	123.3
${\Psi }_1$	0.0	74.0	126.5
${\Psi }_3$	0.0	74.0	126.5
$\overline{J},\overline{\Gamma }$	$-$14.9, $-$0.7	72.0, 4.1	123.3, 6.3
${85}^{\circ }$ (1.94Å):
${\Psi }_S$	16.2	0.0	0.0
${\Psi }_2$	0.3	149.4	225.8
${\Psi }_1$	0.0	150.1	226.9
${\Psi }_3$	0.0	150.1	226.9
$\overline{J},\overline{\Gamma }$	$-$15.9, $-$0.6	149.4, 1.3	225.8, 2.1
${90}^{\circ }$ (1.86Å):
${\Psi }_S$	17.0	0.0	0.0
${\Psi }_2$	0.0	293.6	417.5
${\Psi }_1$	0.3	294.2	418.5
${\Psi }_3$	0.3	294.2	418.5
$\overline{J},\overline{\Gamma }$	$-$17.0, 0.7	293.6, 1.2	417.5, 2.0

Table 7. Table A2: Energy splittings for the lowest four spin-orbit states of two face-sharing NN IrO 6 octahedra and the corresponding effective coupling constants obtained from rAS+SOC, CAS+SOC, and MRCI+SOC calculations. The relative distances from the O ligands to the z 𝑧 z axis were fixed to 1.57 Å. The data are presented as complementary to those in Fig.  2 (all values in meV).

$d$(Ir-Ir)	rAS+SOC	CASSCF+SOC	MRCI+SOC
2.50Å:
${\Psi }_S$	22.0	0.0	0.0
${\Psi }_2$	0.5	188.6	329.7
${\Psi }_1$	0.0	198.1	345.0
${\Psi }_3$	0.0	198.1	345.0
$\overline{J},\overline{\Gamma }$	$-$21.5 $-$0.93	188.6, 18.9	329.7, 30.5
2.56Å:
${\Psi }_S$	18.3	0.0	0.0
${\Psi }_2$	0.4	106.7	187.2
${\Psi }_1$	0.0	111.2	194.3
${\Psi }_3$	0.0	111.2	194.3
$\overline{J},\overline{\Gamma }$	$-$17.9, $-$0.8	106.7, 8.9	187.2, 14.2
2.63Å:
${\Psi }_S$	15.2	0.0	0.0
${\Psi }_2$	0.4	72.0	123.3
${\Psi }_1$	0.0	74.0	126.5
${\Psi }_3$	0.0	74.0	126.5
$\overline{J},\overline{\Gamma }$	$-$14.9, $-$0.7	72.0, 4.1	123.3, 6.3
2.69Å:
${\Psi }_S$	12.6	0.0	0.0
${\Psi }_2$	0.3	54.7	90.5
${\Psi }_1$	0.0	55.5	91.7
${\Psi }_3$	0.0	55.5	91.7
$\overline{J},\overline{\Gamma }$	$-$12.4, $-$0.54	54.7, 1.7	90.5, 2.2
2.76Å:
${\Psi }_S$	10.4	0.0	0.0
${\Psi }_2$	0.2	44.5	72.1
${\Psi }_1$	0.0	44.8	72.4
${\Psi }_3$	0.0	44.8	72.4
$\overline{J},\overline{\Gamma }$	$-$10.3, $-$0.4	44.5, 0.6	72.1, 0.5

Table 8. Table A3: Energy splittings for the lowest four spin-orbit states of two face-sharing NN RhO 6 octahedra and the corresponding effective coupling constants obtained from rAS+SOC, CAS+SOC, and MRCI+SOC calculations. For each geometry the Rh-Rh distance was fixed to 2.58 Å . Distances between the Rh sites and the bridging O’s, d 𝑑 d (Rh-O), are provided within brackets. The data are presented as complementary to those in Table  4 (all values in meV).

$\theta$, $d$(Rh-O)	rAS+SOC	CASSCF+SOC	MRCI+SOC
${70.5}^{\circ }$ (2.23Å):
${\Psi }_S$	10.4	6.5	0.0
${\Psi }_3$	0.0	1.0	3.7
${\Psi }_1$	0.0	1.0	3.7
${\Psi }_2$	0.6	0.0	1.5
$\overline{J},\overline{\Gamma }$	$-$9.9 $-$1.1	$-$6.5, 2.0	1.5, 4.4
${75}^{\circ }$ (2.12Å):
${\Psi }_S$	13.3	0.0	0.0
${\Psi }_3$	0.0	0.1	15.0
${\Psi }_1$	0.0	0.1	15.0
${\Psi }_2$	0.8	0.4	15.2
$\overline{J},\overline{\Gamma }$	$-$12.5, $-$1.7	0.4, $-$0.6	15.2, $-$0.4
${80}^{\circ }$ (2.06Å):
${\Psi }_S$	16.0	0.0	0.0
${\Psi }_3$	0.0	25.4	54.0
${\Psi }_1$	0.0	25.4	54.0
${\Psi }_2$	1.1	26.5	55.5
$\overline{J},\overline{\Gamma }$	$-$14.9, $-$2.3	26.5, $-$2.2	55.5, $-$2.8
${85}^{\circ }$ (1.91Å):
${\Psi }_S$	17.7	0.0	0.0
${\Psi }_3$	0.0	82.4	135.7
${\Psi }_1$	0.0	82.4	135.7
${\Psi }_2$	1.3	84.0	137.8
$\overline{J},\overline{\Gamma }$	$-$16.4, $-$2.6	84.0, $-$3.2	137.8, $-$4.3
${90}^{\circ }$ (1.82Å):
${\Psi }_S$	16.5	0.0	0.0
${\Psi }_3$	0.0	165.4	173.4
${\Psi }_1$	0.0	165.4	173.4
${\Psi }_2$	0.9	190.5	164.0
$\overline{J},\overline{\Gamma }$	$-$15.6, $-$1.8	190.5, $-$50.2	164.0, 18.8

Table 9. Table A4: Energy splittings for the lowest four spin-orbit states of two face-sharing NN RhO 6 octahedra and the corresponding effective coupling constants obtained from rAS+SOC, CAS+SOC, and MRCI+SOC calculations. The relative distances from the O ligands to the z 𝑧 z axis are fixed to 1.54 Å. The data are presented as complementary to those in Fig.  2 (all values in meV).

$d$(Rh-Rh)	rAS+SOC	CASSCF+SOC	MRCI+SOC
2.45Å:
${\Psi }_S$	22.0	0.0	0.0
${\Psi }_3$	0.0	49.5	157.6
${\Psi }_1$	0.0	49.5	157.6
${\Psi }_2$	1.6	50.0	158.1
$\overline{J},\overline{\Gamma }$	$-$20.4 $-$3.2	50.0, $-$1.1	158.1, $-$1.1
2.51Å:
${\Psi }_S$	18.8	0.0	0.0
${\Psi }_3$	0.0	30.1	73.2
${\Psi }_1$	0.0	30.1	73.2
${\Psi }_2$	1.4	31.1	74.5
$\overline{J},\overline{\Gamma }$	$-$17.5, $-$2.7	31.1, $-$2.1	74.5, $-$2.5
2.58Å:
${\Psi }_S$	16.0	0.0	0.0
${\Psi }_3$	0.0	25.4	54.0
${\Psi }_1$	0.0	25.4	54.0
${\Psi }_2$	1.1	26.5	55.5
$\overline{J},\overline{\Gamma }$	$-$14.9, $-$2.3	26.5, $-$2.2	55.5, $-$2.8
2.64Å:
${\Psi }_S$	13.6	0.0	0.0
${\Psi }_3$	0.0	23.8	47.0
${\Psi }_1$	0.0	23.8	47.0
${\Psi }_2$	1.0	24.7	48.3
$\overline{J},\overline{\Gamma }$	$-$12.6, $-$1.9	24.7, $-$2.0	48.3, $-$2.6
2.71Å:
${\Psi }_S$	11.5	0.0	0.0
${\Psi }_3$	0.0	22.6	42.6
${\Psi }_1$	0.0	22.6	42.6
${\Psi }_2$	0.8	23.5	43.7
$\overline{J},\overline{\Gamma }$	$-$10.7, $-$1.6	23.5, $-$1.7	43.7, $-$2.2

Equations

$${\mathcal{H}}_{ij}=J_{ij}{\tilde{\mathbf{S}}}_i\cdot {\tilde{\mathbf{S}}}_j+{\tilde{\mathbf{S}}}_i\cdot {\mathbf{\Gamma }}_{ij}\cdot {\tilde{\mathbf{S}}}_j,$$

$$\overline{\stackrel{ˉ}{\mathbf{\Gamma }}}=\left(\begin{array}{ccc}\Gamma & 0& 0\\ 0& \Gamma & 0\\ 0& 0& -2\Gamma \end{array}\right).$$

$$E_S=-\frac{3}{4}J,$$

$$E_2=\frac{1}{4}J+\Gamma ,$$

$${\mathcal{H}}_{ij}=\bar{J}{\tilde{\mathbf{S}}}_i\cdot {\tilde{\mathbf{S}}}_j+\bar{\Gamma }{\tilde{S}}_i^z{\tilde{S}}_j^z,$$

$${\mathcal{H}}_{\text{CF},t}$$

$${\mathcal{H}}_U=\frac{U}{2}\sum _l\sum _{\mu {\mu }^{\prime }}\sum _{\sigma {\sigma }^{\prime }}{n}_{\mu l,\sigma }{n}_{{\mu }^{\prime }l,{\sigma }^{\prime }},$$

$$|{\varphi }_{\mu l};\sigma ⟩=\sum _{k,m}|{\psi }_{kl}(m)⟩U_{km,\mu \sigma }.\text{vspace}\ast -2mm$$

$$|a_l;\sigma =\pm 1/2⟩\to \mp \cos \gamma |{\psi }_{1l}(s=\pm 1/2)⟩,$$

$$|e_{1l};\sigma =\pm 1/2⟩\to \frac{1}{\sqrt{2}}\sin \gamma |{\psi }_{1l}(s=\pm 1/2)⟩,$$

$$|e_{2l};\sigma =\pm 1/2⟩\to \pm \frac{i}{\sqrt{2}}\sin \gamma |{\psi }_{1l}(s=\mp 1/2)⟩,$$

$${\mathcal{H}}_{hop}^{dd}=\sum _{\mu ,\sigma }{t}_{\mu \mu }^{dd}({\varphi }_{\mu A,\sigma }^{†}{\varphi }_{\mu B,\sigma }+H.c.).$$

$${\mathcal{H}}_{hop}^{dd}\simeq t^{dd}\sum _s({\psi }_{1A,s}^{†}{\psi }_{1B,s}+H.c.),$$

$${\hat{\mathcal{H}}}^Z=\sum _{l=A,B}{\mu }_B({\text{L}}_l+{\text{g}}_e{\text{S}}_l)\cdot \text{H},$$

Full text

I Introduction

The interest in the preparation and characterization of 5$d$ oxides and halides goes back to the 1950’s but some of the major implications of having a strong spin-orbit coupling (SOC), at least for certain 5$d^{n}$ electron configurations, have been only recently realized. The work of Kim et al. on the square-lattice 5$d^{5}$ iridate Sr2IrO4 Kim et al. (2008, 2009), for example, led to the concept of a spin-orbit driven (Mott-like) insulator while Jackeli and Khaliullin Jackeli and Khaliullin (2009) brought to the forefront of oxide research the honeycomb 5$d^{5}$ iridates, as possible hosts for Kitaev physics Kitaev (2006) and novel magnetic ground states and excitations Nussinov and van den Brink (2015). Both types of these iridate structures – square and honeycomb lattices – have been the topic of extensive investigations in recent years. The honeycomb compounds display edge-sharing ligand octahedra and advanced electronic-structure calculations indicate that the Kitaev exchange is indeed the largest intersite magnetic coupling Yamaji et al. (2014); Katukuri et al. (2014a). Remarkably large anisotropic interactions were also found for corner-sharing ligand cages in Sr2IrO4, in that case of Dzyaloshinskii-Moriya type, with strengths in the range of 10–15 meV Jackeli and Khaliullin (2009); Bogdanov et al. (2015).

In contrast to the cases of corner- and edge-sharing coordination, little is known with respect to the magnitude of the effective coupling constants for adjacent octahedra connected through a O3 facet. Representative materials of the latter type are the canted antiferromagnet BaIrO3 Lindsay et al. (1993); Cao et al. (2000); Brooks et al. (2005); Nakano and Terasaki (2006); Cheng et al. (2009); Laguna-Marco et al. (2010), the putative spin-liquid Ba3InIr2O9 Dey et al. (2017), the spin-gapped system Ba3BiIr2O9 Miiller et al. (2012), BaRhO3 Stitzer et al. (2004), and BaCoO3 Sugiyama et al. (2006). Here we provide ab initio results with regard to the strength of facet-mediated superexchange for IrO6 (RhO6) octahedra as found in the 5$d$ (4$d$) $t_{2g}^{5}$ system BaIrO3 (BaRhO3). We predict remarkably large antiferromagnetic (AFM) Heisenberg interactions in the range of 100 meV for Ir-O-Ir angles of about 80*∘* as found experimentally in BaIrO3 Stitzer et al. (2004). Moreover, for bond angles $\gtrsim$85*∘* the Heisenberg $J$ even exceeds 200 meV in our simulations. So strong AFM superexchange has been found so far only in one-dimensional corner-sharing cuprates Suzuura et al. (1996); Motoyama et al. (1996). Our findings point to a picture of unusually large, AFM couplings within the face-sharing octahedral units of BaIrO3. The strong dependence on bond angles of the effective magnetic interactions further resonates with available experimental data on BaIrO3 Cao et al. (2004, 2000); Cheng et al. (2009); Korneta et al. (2010); Laguna-Marco et al. (2014) and Ba3BiIr2O3 Miiller et al. (2012), that indicate subtle interplay between the electronic and lattice degrees of freedom.

II Material model

BaIrO3 features a distorted hexagonal structure with both face-sharing and corner-sharing IrO6 octahedra Yuan et al. (2016). Those connected by one single ligand form honeycomb-like planes; the linkage of adjacent honeycomb layers is ensured by inter-layer Ir ions located such that blocks of three face-sharing octahedra are formed along the $c$ axis, see Fig. 1. Since for any pair of nearest-neighbor (NN) octahedra the actual point-group symmetry is very low, we focus in our study on an idealized material model displaying $D_{3h}$ symmetry: [Ir2O9]10- units as depicted in Fig. 1 around which we additionally considered, for keeping overall charge neutrality, three Ba sites within the plane of the median O3 facet and two extra Ba ions along the $z$ axis. Although this material model is somewhat oversimplified, it should rather well describe the essential short-range electron interactions, as confirmed by similar investigations of edge-sharing $5d^{5}$ compounds Katukuri et al. (2014a).

One feature of 5$d$ transition-metal (TM) ions is that their valence orbitals are much more diffuse as compared to first-series TM species. The ligand field is therefore more effectively felt and for instance the Ir4+ ions tend to adopt low-spin $t^{5}_{2g}$ configurations. The more extended nature of the 5$d$ functions further gives rise to large intersite hoppings and large superexchange, as in e.g. Sr2IrO4 Jackeli and Khaliullin (2009); Bogdanov et al. (2015) and CaIrO3 Bogdanov et al. (2012). Under strong octahedral crystal fields (CFs) and spin-orbit interactions, with one single unpaired electron ($S$=1/2) in the $t_{2g}$ manifold (orbital angular momentum $L$=1), the 5$d^{5}$ (4$d^{5}$) valence electron configuration of Ir4+ (Rh4+) in BaIrO3 (BaRhO3) yields an effective $j$=1/2 Kramers-doublet ground state Jackeli and Khaliullin (2009); Abragam and Bleaney (1970). Deviations from a perfect cubic environment may lead to some degree of admixture between the $j$=1/2 and lower-lying $j$=3/2 spin-orbit states Abragam and Bleaney (1970). In BaIrO3 and BaRhO3, in particular, the trigonal distortion of the oxygen octahehra plays a quite important role in this regard, as illustrated in Appendix A through simple analytical expressions based on an effective ionic model. To estimate the strengths of the exchange interactions in BaIrO3 and BaRhO3, both isotropic and anisotropic, we here employ many-body ab initio techniques from wave-function-based quantum chemistry (QC), then map the magnetic spectrum obtained in the QC calculations onto an appropriate effective spin Hamiltonian, the form of the latter being dictated by the symmetry of the material model.

III Magnetic interactions

For the idealized M2O9 cluster (M=Ir, Rh) of face-sharing octahedra (Fig. 1) the overall symmetry is $D_{3h}$. Each particular superexchange path Mi-On-Mj ($n$ ​=​ $1,2,3$) implies a finite Dzyaloshinskii-Moriya (DM) vector $\mathbf{D^{\it{n}}_{\it{ij}}}$, since there is no inversion center for the M2O9 unit. However, given the $D_{3h}$ symmetry, these DM vectors lie within the plane of the O3-facet and are related to each other through rotations around the $C_{3}$ axis. This yields a vanishing DM coupling $\mathbf{D_{\it{ij}}}$ ​=​ $\sum_{n}\mathbf{D^{\it{n}}_{\it{ij}}}$ ​=​ 0. For a pair of NN 1/2 pseudospins $\bf{\tilde{S}_{\it{i}}}$ and $\bf{\tilde{S}_{\it{j}}}$ with this type of linkage, the most general bilinear spin Hamiltonian can be then cast in the form

[TABLE]

where $\mathbf{{\it J}}_{ij}$ is the isotropic Heisenberg exchange and $\mathbf{\Gamma}_{ij}$ is a symmetric traceless second-rank tensor that describes the symmetric exchange anisotropy. Considering the three-fold rotational symmetry around the M-M link, it is convenient to have one of the coordinates along the line defined by the two M sites. We therefore use the local frame indicated in Fig. 1, with both Ir ions on the $z$ axis. In this coordinate system the $\bar{\bar{\mathbf{\Gamma}}}$ tensor is diagonal and, for symmetry reasons, can be written as

[TABLE]

The eigenstates of such a two-site ${\tilde{S}}$=1/2 system are the singlet $\lvert\Psi_{\!S}\rangle\!=\!(\lvert\uparrow\downarrow\rangle-\lvert\downarrow\uparrow\rangle)\!/\!\sqrt{2}$ and the three triplet components $\lvert\Psi_{\!1}\rangle\!=\!(\lvert\uparrow\downarrow\rangle+\lvert\downarrow\uparrow\rangle)\!/\!\sqrt{2}$, $\lvert\Psi_{\!2}\rangle\!=\!(\lvert\uparrow\uparrow\rangle+\lvert\downarrow\downarrow\rangle)\!/\!\sqrt{2}$, $\lvert\Psi_{\!3}\rangle\!=\!(\lvert\uparrow\uparrow\rangle-\lvert\downarrow\downarrow\rangle)\!/\!\sqrt{2}$. The corresponding eigenvalues are

[TABLE]

Expression (1) can be then simplified to

[TABLE]

where $\mathbf{{\it{\bar{J}}}}\equiv{\it J}+{\Gamma}$ and ${\bar{\Gamma}}\equiv-3{\Gamma}$.

The first step in the actual QC calculations is defining a relevant set of Slater determinants in the prior complete-active-space self-consistent-field (CASSCF) treatment Helgaker et al. (2000). For two IrO6 (RhO6) octahedra, an optimal choice is having five electrons and three ($t_{2g}$) orbitals at each of the two magnetically active Ir (Rh) sites. The self-consistent-field optimization was carried out for an average of the lowest nine singlet and lowest nine triplet states associated with this manifold. Subsequent multireference configuration-interaction (MRCI) computations were performed for each spin multiplicity, either singlet or triplet, as nine-root calculations. All these states entered the spin-orbit treatment Berning et al. (2000), in both CASSCF and MRCI. Within the group of 36 spin-orbit eigenvectors associated with the $t^{5}_{2g}\!-\!t^{5}_{2g}$ manifold, the lowest-lying four “magnetic” states are separated by a significant energy gap from the other 32 states. The latter correspond to on-site $j\!\approx\!$ 3/2 to $j\!\approx\!$ 1/2 transitions, and are therefore left aside in the actual mapping procedure. In other words, given the strong SOC and large $j\!\approx\!$ 3/2 to $j\!\approx\!$ 1/2 excitation energies, the initial 36$\times$36 problem can be smoothly mapped onto a 4$\times$4 construction as defined by the effective Hamiltonian (1).

All computations were carried out with the molpro quantum-chemistry software Werner et al. (2012). In the MRCI treatment, single and double excitations from the six Ir (Rh) $t_{2g}$ orbitals and from the $2p$ shells of the bridging O ligand sites were taken into account. The Pipek-Mezey localization module Pipek and Mezey (1989) available in molpro was employed for separating the metal 5$d$ (4$d$) and O 2$p$ valence orbitals into different groups. To derive the magnitude of direct exchange, we additionally performed calculations in which the active space is again defined by ten electrons and six orbitals but intersite t_{2g}$$-$$t_{2g} excitations are forbidden by restricting to maximum five the number of electrons per TM site. We refer to these results as rAS (restricted active space, maximum one hole per site).

III.1 $\bm{[}$Ir2O9$\bm{]}$ unit

Effective magnetic couplings for Ir2O9 fragments of two face-sharing IrO6 octahedra are listed in Table 1, for an Ir-Ir interatomic distance $d_{0}\!=\!2.63$ Å and ligand coordinates that provide Ir-O-Ir angles $\theta_{0}\!=\!80^{\circ}$. These structural parameters, obtained by averaging the bond lengths and bond angles in the experimetally determined lattice configuration of BaIrO3 Stitzer et al. (2004), correspond to slightly elongated octahedra. For cubic (undistorted) octahedra, $\theta_{c}\!=\!70.52^{\circ}$.

Results at three different levels of approximation are shown: spin-orbit rAS (rAS+SOC), CASSCF (CAS+SOC), and MRCI (CI+SOC).

The rAS data account for only direct d$$-$$d exchange. For $d_{0}\!=\!2.63$ Å and $\theta_{0}\!=\!80^{\circ}$, the rAS ${\bar{J}}$ is $-$14.9 meV while the anisotropic ${\bar{\Gamma}}$ is $-$0.7 meV when including SOC. The magnitude of the ferromagnetic (FM) rAS ${\bar{J}}$ is similar to that computed in square-lattice 3$d^{9}$ Cu oxides Guo et al. (1988); Martin and Saxe (1988); van Oosten et al. (1996); Calzado et al. (2002) and in the corner-sharing iridate Ba2IrO4 Katukuri et al. (2014b). The anisotropic ${\bar{\Gamma}}$ is also FM at the rAS level and its magnitude is slightly larger as compared with the AFM rAS ${\bar{\Gamma}}$ of the corner-sharing iridate Ba2IrO4 Katukuri et al. (2014b). By CASSCF and MRCI, the singlet $\Psi_{\!S}$ becomes the ground state, well below the “triplet” components $\Psi_{\!1}$, $\Psi_{\!2}$, and $\Psi_{\!3}$. This indicates that the isotropic Heisenberg exchange $\bar{J}$ ($\bar{J}\!>\!0$) defines now the largest energy scale. In the CASSCF approximation, only t_{2g}$$-$$t_{2g} intersite excitations are accounted for, i.e., t^{6}_{\!2g}$$-$$t^{4}_{\!2g} configurations. The $\bar{J}$ value extracted by CAS+SOC, 72 meV, is twice as large as compared, e.g., to the CASSCF $J$’s in layered 3$d^{9}$ cuprates van Oosten et al. (1996); Guo et al. (1988); Martin and Saxe (1988); Calzado et al. (2002) and in the corner-sharing iridate Ba2IrO4 Katukuri et al. (2014b). In the configuration-interaction treatment, which includes TM $t_{2g}$ to $e_{g}$ and charge-transfer O $2p$ to Ir 5$d$ excitations as well, $\bar{J}$ is 123.3 meV, about 70% larger as compared to the CAS+SOC result. By accounting for correlation effects, the symmetric anisotropic coupling ${\bar{\Gamma}}$ is also significantly enlarged, from $-$0.7 meV by rAS+SOC to 6.3 meV by spin-orbit MRCI.

In the case of face-sharing ligand octahedra, the TM ions often form dimers, trimers, or chains Stitzer et al. (2004). This type of low-dimensional packing usually results in sizable distortions of the ligand cages. It is known that the effective spin interactions are strongly dependent on structural details such as bond angles Nishimoto et al. (2016); Katukuri et al. (2015); Yadav et al. (2016, 2018a) and bond lengths Yadav et al. (2018b). For better insight into the dependence of the NN magnetic couplings on such structural parameters, we performed additional calculations for distorted geometries with all ligands pushed closer to (or farther from) the Ir-Ir axis, which therefore yields larger (or smaller) Ir-O-Ir bond angles while keeping the overall $D_{3h}$ point-group symmetry. The resulting MRCI+SOC data are provided in Table 2. The overall trends for the magnetic couplings $\bar{J}$ and ${\bar{\Gamma}}$ are illustrated graphically in Fig. 2. It is seen that the angle dependence for both $\bar{J}$ and ${\bar{\Gamma}}$ can be rather well reproduced with parabolic curves. The Heisenberg $\bar{J}$ displays a steep increase with larger angle, i.e., from 58 meV at 70.5*∘* to 417 meV at 90*∘. On the other hand, the anisotropic coupling ${\bar{\Gamma}}$ shows a rapid decrease, from a remarkably large value of 30 meV at 70.5∘* to 2 meV at 90*∘*.

We further analyzed the dependence on the Ir-Ir interatomic distance $d$(Ir-Ir) of the magnetic interactions. In this set of calculations, the distance between the O ligands and the $z$ axis (along the Ir-Ir bond) was fixed to 1.57 Å, while $d$(Ir-Ir) was either increased or reduced by up to 5$\%$ with respect to the reference Ir-Ir separation $d$=$d_{0}$=2.63 Å. As shown in Fig. 2 (see also Appendix B, Table A2), both $\bar{J}$ and ${\bar{\Gamma}}$ have again pronounced parabolic dependence on $d$(Ir-Ir). In contrast to the variations as function of angle displayed in Fig. 2, here $\bar{J}$ and ${\bar{\Gamma}}$ follow the same trend. More specifically, both $\bar{J}$ and ${\bar{\Gamma}}$ rapidly increase with decreasing $d$(Ir-Ir).

We also performed calculations in which the six O ligands not shared by the Ir ions were displaced as well along the $z$ axis, such that each Ir site remains in the center of the respective octahedron. We found that the differences between the ${\bar{J}}$ values obtained from these computations and the corresponding ${\bar{J}}$’s in Fig. 2 are rather small, not more than 15%.

The face-sharing linkage and additional distortions applied to the two-octahedra clusters split the $t_{2g}$ levels into $a_{1g}$ and $e^{\pi}_{g}$ components. For all Ir2O9 units considered here, we find that the $a_{1g}$ sublevels lie at lower energy and that the $t_{2g}$ hole has $e_{g}^{\pi}$ character without accounting for SOC. The $a_{1g}$ orbitals belonging to NN sites have substantial direct overlap [see Fig 3], much larger than in the case of $e_{g}^{\pi}$ orbitals [see Fig 3]. The rather small AFM Heisenberg $J$ derived from the calculations without SOC (see caption of Table 1) is therefore the result of (relatively) weak direct exchange involving the higher-lying $e_{g}^{\pi}$ states. By accounting for spin-orbit interactions, however, the Heisenberg $J$ is enhanced to impressive values that are up to three times larger than the results obtained without SOC (72 vs 27 meV at the CASSCF level, 123 vs 35 meV by MRCI, see Table 1). This strong increase of the Heisenberg $J$ is the consequence of mixing $a_{1g}$ character to the spin-orbit ground-state wave-function.

III.2 $\bm{[}$Rh2O9$\bm{]}$ unit

In order to make a informative comparison between 5$d$ and 4$d$ oxides, we also performed calculations for the effective magnetic couplings on [Rh2O9] fragments consisting of two face-sharing RhO6 octahedra, with a Rh-Rh interatomic distance $d^{\prime}_{0}\!=\!2.58$ Å and Rh-O-Rh bond angles $\theta_{0}^{\prime}\!=\!80^{\circ}$. As for the material model of face-sharing 5$d^{5}$ octahedra, the structural parameters of the Rh2O9 cluster were chosen according to the average bond lengths and bond angles of the BaRhO3 compound. We used in this regard the crystallographic data reported in Ref. Stitzer et al. (2004). QC results are presented in Table 3. Interestingly, the $\bar{J}$ value obtained by spin-orbit rAS is the same as for the Ir2O9 cluster (Table 1). However, the $\bar{J}$’s obtained by CAS+SOC and CI+SOC are significantly smaller as compared with those in Table 1. Still, $\bar{J}$ remains much larger than the magnetic couplings in the edge-sharing 4$d^{5}$ compounds Li2RhO3 and $\alpha$-RuCl3 Katukuri et al. (2015); Yadav et al. (2016).

Energy splittings within the group of the four low-lying $d^{5}\!-d^{5}$ states and the resulting effective coupling constants for different Rh-O-Rh angles are listed in Table 4. Furthermore, the dependence of $\bar{J}$ and ${\bar{\Gamma}}$ on the Rh-O-Rh bond angles and on the Rh-Rh interatomic distances are illustrated in Fig. 2 and Fig. 2, respectively (for more details see Appendix B, Table A3 and Table A4). As indicated in Fig. 2, $\bar{J}$ displays nearly linear behavior with variable angle, increasing from 1.5 meV at 70.5*∘* to 164 meV at 90*∘. ${\bar{\Gamma}}$ changes sign from AFM to FM coupling close to 75∘*, with a minimum of $-$6 meV at $\!85^{\circ}$, and then changes back to AFM values for larger angles. On the other hand, with variable $d$(Rh-Rh) [Fig. 2], ${\bar{\Gamma}}$ is always FM, with a minimum of $-$2.8 meV at 2.56 Å, and $\bar{J}$ features a similar trend as for Ir sites in Fig. 2.

IV Discussion

We analyze in more detail in this section the relative values of the different contributions to intersite exchange, i.e., direct t_{2g}$$-$$t_{2g} exchange, t_{2g}$$-$$t_{2g} electron/hole hopping, and indirect hopping via the bridging oxygens. In first place, it is clear that a systematically small portion of FM potential exchange to the overall $\bar{J}$ is here of secondary importance. The contribution coming from direct hopping can be straightforwardly estimated from the CASSCF $J$ since only intersite M($t_{2g}$)–M($t_{2g}$) excitation processes ($t^{6}_{2g}$–$t^{4}_{2g}$ polar configurations) are taken into account at the CASSCF level. In the CI treatment, superexchange paths including the bridging-ligand $2p$ and TM $e_{g}$ orbitals are also added on top of direct hopping, providing a more comprising description of intersite exchange mechanisms. In the case of BaIrO3, for instance, when the Ir-O-Ir bond angle is $80^{\circ}$, the exchange calculated at the CASSCF level (without SOC), $J$ = 27.4 meV, is already 77$\%$ of the CI result, 35.4 meV (see Table 1). While this fraction is significantly reduced if the Ir-O-Ir bond angle is modified towards $90^{\circ}$ (see Table A1), indicating that the $d-p-d$ superexchange contribution starts to rise as a result of shorter Ir-O bonds, the data computed for 80*∘* bond angles show that, given the large direct-hopping integrals, the direct AFM d$$-$$d superexchange may surpass the d$$-$$p$$-$$d superexchange. The two mechanisms should be considered in any case on equal footing for high-quality estimates. In the context of recent discussions on the role of the various types of intersite exchange Kugel et al. (2015); Khomskii et al. (2016), our QC data provide a more quantitative picture on the different contributions.

Given the facet-sharing geometry, the direct d$$-$$d electron/hole hopping between $a_{1g}$ orbitals is considerable. This hopping interferes with the indirect hopping via the bridging-oxygen O3 group, providing a total transfer integral $t$. Since the exchange coupling $J$ is mainly controlled by the square of $t$ ($J\sim t^{2}/U$, where $U$ is the on-site Coulomb repulsion), a large $\bar{J}$ value of up to $\approx$400 meV (see Table A1 and A2) is not surprising. Both the direct (\sim$$t^{dd}) and indirect (\sim$$t^{dpd}) transfer processes can occur trough the $e^{\pi}_{g}$ and $a_{1g}$ channels independently. As discussed in Appendix A, the total transfer integral $t$ can be then decomposed as $t=w_{e}t_{e}+w_{a}t_{a}$, where $t_{e,a}=t_{e,a}^{dd}+t_{e,a}^{dpd}$. The corresponding channel weights, $w_{e}$ and $w_{a}$, are controlled by the ratio $\Delta_{t}/\lambda$, with $\lambda$ being the spin-orbit coupling, 0.47 eV for Ir and 0.15 eV for Rh Katukuri et al. (2014c); the dependence of the trigonal splitting $\Delta_{t}$ on bond angles is illustrated in Table 5. The different terms entering the total transfer integral $t$ are expected to behave differently when varying the geometry of the M-O3-M structure. The large direct overlap between two NN $a_{1g}$ orbitals suggests that the direct hopping $t_{a}^{dd}$ contributes significantly to $t_{a}$, as evidenced in Fig. 3. In contrast, the $e^{\pi}_{g}$ orbitals are tilted with respect to the $z$ axis, thus giving rise to weaker direct overlap and more significant d$$-$$p$$-$$d couplings (see Fig. 3), i.e., a more important role of $t^{dpd}_{e}$ in $t_{e}$. It is the interplay between these processes, $d-d$ and $d-p-d$ superexchange, that is mainly responsible for the strong variations as function of bond angles and bond lengths. From a wider perspective, it is clear that the equilibrium geometrical configuration and the associated $J$ value depend on interactions and degrees of freedom that also involve the extended crystalline surroundings. An interesting aspect to be considered is inter-site couplings within the entire M3O12 block of three face-sharing octahedra along the $c$ axis (see Fig.1). One question concerns the possibility of cooperative M$-$M dimerization as driving force for the charge density wave observed in BaIrO3 Cao et al. (2000). Two-site bond formation on three-center units with a spin 1/2 at each magnetic site and long-range ordering of these ‘dimers’ has been earlier proposed in the quasi-1D system NaV2O5 Hozoi et al. (2002, 2003).

To summarize, we employ quantum chemistry methods to provide valuable insights on the effective magnetic interactions in $5d$ and $4d$ oxides with face-sharing oxygen octahedra, BaIrO3 and BaRhO3. The same methodology has previously been used to derive magnetic coupling constants in good agreement with experimental estimates in the perovskite iridate CaIrO3 Bogdanov et al. (2012); Moretti Sala et al. (2014), in square-lattice Ba2IrO4 Katukuri et al. (2014b) and Sr2IrO4 Bogdanov et al. (2015), and in pyrochlore iridates Yadav et al. (2018a). The large AFM Heisenberg interactions computed here for face-sharing octahedra are remarkable since they exceed the values computed so far for corner-sharing Motoyama et al. (1996); van Oosten et al. (1996); Bogdanov et al. (2012); Katukuri et al. (2014b) and edge-sharing systems Nishimoto et al. (2016). One peculiar exception with regard to edge-sharing $4d^{5}$ NN ligand cages is RuCl3 under high pressure Bastien et al. (2018), where a strong stabilization of the singlet state is also found for certain Ru$-$Ru bonds. The present findings on face-sharing octahedra as encountered in BaIrO3 and BaRhO3 and recent results on RuCl3 Bastien et al. (2018) only provide additional motivation for even more detailed electronic-structure calculations on both edge- and face-sharing compounds, with main focus on the subtle interplay among strong spin-orbit interactions, direct $d-d$ orbital overlap and bonding, and couplings to the lattice degrees of freedom.

V Acknowledgements

Calculations were performed at the High Performance Computing Center (ZIH) of the Technical University Dresden (TUD). We thank U. Nitzsche for technical assistance. We acknowledge financial support from the German Science Foundation (Deutsche Forschungsgemeinschaft, DFG — HO-4427/2 and SFB-1143) and thank D. I. Khomskii, V. M. Katukuri, N. A. Bogdanov, and S.-L. Drechsler for instructive discussions.

Appendix A Appendix A: Effective spin model

To put in perspective the general trends obtained in the QC calculations for the isotropic exchange coupling $J$, an effective two-site model is analyzed here. Two different mechanisms are considered: ($a$) direct $t_{2g}-t_{2g}$ hopping \sim$$t^{dd} and ($b$) indirect processes via the bridging oxygens, \sim$$t^{dpd}. The three bridging oxygens within the median $xy$ mirror plane are denoted as On, with $n\in\{1,2,3\}$. For each metal ion at sites $l=A,B$, the trigonal CF term ${\cal H}_{\textrm{CF},t}$ splits the $t_{2g}$ orbital states into a two- and an one-dimensional subspace with basis states $|e_{1l}\rangle$, $|e_{2l}\rangle$ and $|a_{l}\rangle$, respectively Sugano et al. (1970). In what follows, these hole states are denoted as $|\phi_{\mu l}\rangle$, with $\mu=1,2$ for $|e_{1,2}\rangle$, $\mu=3$ for $|a\rangle$, and the creation (annihilation) operator $\phi^{\dagger}_{\mu l,\sigma}$ ($\phi_{\mu l,\sigma}$), where the spin variable $\sigma=\pm 1/2$ is added. Restricted to this low-energy orbital space, a single hole is described by the effective orbital angular momentum operator $L=1$ Sugano et al. (1970) in the spin-orbit term ${\cal H}_{\textrm{SO}}=\lambda\sum_{l}{\bf L}_{l}{\bf S}_{l}$; here, ${\bf S}$ is the spin-1/2 operator and $\lambda>$0. Altogether, relevant intra-atomic interactions are collected in the effective Hamiltonian ${\cal H}_{\textrm{at}}={\cal H}_{\textrm{CF},t}+{\cal H}_{\textrm{SO}}+{\cal H}_{U}$, where

[TABLE]

Here, $[{\bf L}_{l}]_{\mu\mu^{\prime}}=\langle\phi_{\mu l}|{\bf L}_{l}|\phi_{\mu^{\prime}l}\rangle$ while $\Delta_{t}$ is the trigonal CF splitting between $|e_{1,2}\rangle$ and $|a\rangle$ ‘​ local ​’ states. As stated in the main text, without SOC the calculated hole ground state of Ir4+/Rh4+ ions is of $e$-orbital character, which means $\Delta_{t}<0$. The term ${\cal H}_{U}$ includes the leading on-site Coulomb interaction

[TABLE]

where $n_{\mu l,\sigma}=\phi^{\dagger}_{\mu l,\sigma}\phi_{\mu l,\sigma}$. The approximation of assuming the Coulomb $U$ in the above expression to be independent of the orbital indices $\mu$ and $\mu^{\prime}$ simplifies the calculation of the isotropic exchange $J$ but excludes obtaining an estimate for the weaker anisotropic exchange $\Gamma$.

Within the one-hole sector and in the cubic limit $\Delta_{t}\to 0$, ${\cal H}_{\textrm{at}}$ is reduced to ${\cal H}_{\textrm{SO}}$. As well known Sugano et al. (1970), the ‘original’ six $L=1$ atomic states $|L_{z}=0,\pm 1;\sigma=\pm 1/2\rangle$ are split by ${\cal H}_{\textrm{SO}}$ into the Kramers doublet $|j=1/2;m=\pm 1/2\rangle$ and the quartet $|j=3/2;m=\pm 1/2,\pm 3/2\rangle$, whose eigenvalues are $-\lambda$ and $\lambda/2$, respectively; here, the site index $l$ is omitted for brevity. When lowering the CF symmetry to trigonal, i.e., $\Delta_{t}\not=0$, states with the same $m(=\pm 1/2)$, $|j=1/2;m\rangle$ and $|j=3/2;m\rangle$, are admixed. By solving the corresponding 2$\times$2 problem, the resulting doublet wave-functions are $|\psi_{1}(m=\pm 1/2)\rangle=c_{1}|1/2;m\rangle\pm c_{2}|3/2;m\rangle$ and $|\psi_{2}(m=\pm 1/2)\rangle=\mp c_{2}|1/2;m\rangle+c_{1}|3/2;m\rangle$, where $c_{1,2}=\left[1/2(1\pm A/\sqrt{A^{2}+B^{2}})\right]^{1/2}$ and $A=3-\delta$, $B=2\sqrt{2}\delta$, $\delta=2\Delta_{t}/3\lambda$. The corresponding eigenvalues are $E_{1,2}=(-\lambda/4)\left[1+3\delta\pm 3\sqrt{1-2\delta/3+\delta^{2}}\right]$ and, since $\lambda>0$, $E_{1}<E_{2}$. The energy of the remaining doublet, $|\psi_{3}(m)\rangle=|j=3/2;m=\pm 3/2\rangle$, is $E_{3}=\lambda/2$.

In general, the initial and new basis states are related by an unitary transformation with the rotation matrix $U_{km,\mu\sigma}$ (here, the site index $l$ is restored):

[TABLE]

Close inspection of the above expressions for the $E_{k}$ energy levels ($k=1,2,3$) shows that the ground-state doublet ($k=1$) is well separated from the excited ones ($k=2,3$) for any $\Delta_{t}$; the low-energy magnetic properties of the system are therefore described by pseudospin-1/2 states $|\psi_{1l}(m)\rangle$. Projection on the low-energy subspace consists in retaining in Eq.(A3) the term $k=1$ only, which reads with the replacement $m\to s$ as

[TABLE]

where $\cos\gamma=(c_{1}-\sqrt{2}c_{2})/\sqrt{3}$ and $\sin\gamma=(\sqrt{2}c_{1}+c_{2})/\sqrt{3}$. In the following, the creation (annihilation) of state $|\psi_{1l}(s)\rangle$ is associated with the operator $\psi^{\dagger}_{1l,s}$ ($\psi_{1l,s}$). Projected onto the pseudospin-1/2 subspace, the Coulomb interaction ${\cal H}_{U}$ takes the Hubbard-like form ${\cal H}_{U}=U\sum_{l}n_{1l,\uparrow}n_{1l,\downarrow}$. Actually, the unitary transformation (A3) yields $\sum_{\mu,\sigma}n_{\mu l,\sigma}=\sum_{k,m}n_{kl,m}$, where only the term $k=1$ is kept.

In case of face-sharing octahedra, the relatively short $\textrm{M}_{A}-\textrm{M}_{B}$ distance dictates inclusion of the direct $t_{2g}-t_{2g}$ hopping term

[TABLE]

The precise structure of ${\cal H}_{hop}^{dd}$ is determined by symmetry arguments that require that ($a$) the off-diagonal hopping is zero, i.e., $t^{dd}_{\mu\mu^{\prime}}=0$ if $\mu\not=\mu^{\prime}$ and ($b$) there are two independent hopping integrals, namely, $t^{dd}_{11}=t^{dd}_{22}\equiv t^{dd}_{e}$ and $t^{dd}_{33}\equiv t^{dd}_{a}$. Projection onto the low-energy subspace then leads to

[TABLE]

where $t^{dd}=t^{dd}_{e}\sin^{2}\gamma+t^{dd}_{a}\cos^{2}\gamma$. Obviously, variation of the MA$-$MB distance $d_{\text{MM}}$ gives rise to strong variation of the hopping integral $t^{dd}$. The $a$-channel contribution \sim$$t^{dd}_{a} is expected to be most sensitive to varying $d_{\text{MM}}$. For instance, according to Harrison (1980) $t^{dd}_{a}\sim d_{\textrm{MM}}^{-5}$ .

The treatment of indirect hopping processes via the bridging oxygens is a challenging problem. The M-O3-M unit should be viewed as a complex molecular-like structure, where superexchange couplings must be analyzed in terms of symmetry-adapted molecular orbitals of the O3 bridging group. A detailed analysis shows that in the low-energy subspace the indirect hopping term ${\cal H}_{hop}^{dpd}$ has the same structure as ${\cal H}_{hop}^{dd}$, Eq.(A6), with the replacement $t^{dd}\to t^{dpd}=t^{dpd}_{e}\sin^{2}\gamma+t^{dpd}_{a}\cos^{2}\gamma$. The hopping integrals $t^{dpd}_{e,a}$ due to second-order processes that occur through intermediate ligand-hole states in the $e$- and $a$-channel, respectively, can be expressed in factorized form as $t^{dpd}_{e,a}\approx[(t^{dp})^{2}/\Delta_{\textrm{CT}}]F_{e,a}(\theta)$. Here, $t^{dp}$ and $\Delta_{\textrm{CT}}$ define the characteristic $p-d$ hopping and charge-transfer energy scales. While the factor $F_{e,a}(\theta)$ is strongly dependent on the angle $\theta$, the parameter $t^{dp}$ is most sensitive to the metal-oxygen distance $d_{\textrm{MO}}$. According to Harrison (1980), $t^{dp}\sim d_{\textrm{MO}}^{-7/2}$. Transitions of first-order (\sim$$t^{dd}_{a,e}) and second-order type (\sim$$t^{dpd}_{a,e}) contribute in each sector independently to give the total transfer integral $t=t_{e}\sin^{2}\gamma+t_{a}\cos^{2}\gamma$, where $t_{e,a}=t_{e,a}^{dd}+t_{e,a}^{dpd}$. The resultant hopping Hamiltonian ${\cal H}_{hop}$ takes the same form as in Eq.(A6), but with the replacement $t^{dd}\to t$. The weight factors of the $e$- and $a$- channels are $w_{e}=\sin^{2}\gamma$ and $w_{a}=\cos^{2}\gamma$, respectively. As discussed above, these factors are controlled by the ratio $\Delta_{t}/\lambda$.

It is seen that the generic Hamiltonian ${\cal H}={\cal H}_{at}+{\cal H}_{hop}$ derived above takes the form of an effective ‘single-orbital’ Hubbard model operating in the pseudospin-1/2 subspace of NN metal ions. It can be treated perturbatively in the strong correlation regime t/U$$\ll$$1, meaning that excited polar states with two holes on the same metal ion are well separated from the low-energy magnetic excitations. In this regime, one immediately obtains as second-order estimate for the isotropic exchange $J=4t^{2}/U$.

Appendix B Appendix B: Intersite magnetic couplings

All computations were performed with the molpro quantum chemistry package Werner et al. (2012). Energy-consistent relativistic pseudopotentials were used for the Ir Figgen et al. (2009) and Rh Peterson et al. (2007) ions. For the Ir/Rh sites, the valence orbitals were described by basis sets of tripe-zeta quality supplemented with two $f$ polarization functions Figgen et al. (2009); Peterson et al. (2007). For the ligand O’s bridging the two magnetically active Ir (Rh) ions, quintuple-zeta valence basis sets and four $d$ polarization functions were applied Dunning (1989). The other O’s were modeled by triple-zeta valence basis sets Dunning (1989). The five Ba ions were modeled by Ba2+ ‘total-ion’ pseudopotentials (TIP’s) supplemented with a single $s$ function Lim et al. (2006). We used interatomic distances as derived by E. Stitzer et al. Stitzer et al. (2004).

The mapping of the ab initio quantum chemistry data onto the effective spin model defined by  (1) implies the lowest four spin-orbit states associated with the different possible couplings of two NN pseudospins-$1/2$. In order to safely identify the singlet and triplet components Bogdanov et al. (2015), we also consider the Zeeman coupling

[TABLE]

where $\textbf{L}_{l}$ and $\textbf{S}_{l}$ are angular-momentum and spin operators at a given Ir/Rh site, while $\textit{g}_{e}$ and ${\mu}_{\!B}$ stand for the free-electron Land$\acute{\text{e}}$ factor and Bohr magneton, respectively. Each of the resulting matrix element computed at the quantum chemistry level is assimilated to the corresponding matrix element of the effective spin Hamiltonian. This one-to-one correspondence between ab initio and effective-model matrix elements enables a clear assignment of each magnetically active spin-orbit CASSCF/MRCI state and determination of all couplings constants Bogdanov et al. (2015). Effective coupling constants at the rAS+SOC, CAS+SOC, and CI+SOC levels are listed in Table A1, Table A2, Table A3, and Table A4, complementary to tables and figures in the main text.