Abstract
For 4d ^{1} and 5d ^{1} spin–orbitcoupled electron configurations, the notion of nonmagnetic j=3/2 quartet ground state discussed in classical textbooks is at odds with the observed variety of magnetic properties. Here we throw fresh light on the electronic structure of 4d ^{1} and 5d ^{1} ions in molybdenum and osmiumbased doubleperovskite systems and reveal different kinds of onsite manybody physics in the two families of compounds: although the sizable magnetic moments and gfactors measured experimentally are due to both metal d–ligand p hybridisation and dynamic Jahn–Teller interactions for 4d electrons, it is essentially d−p covalency for the 5d ^{1} configuration. These results highlight the subtle interplay of spin–orbit interactions, covalency and electron–lattice couplings as the major factor in deciding the nature of the magnetic ground states of 4d and 5d quantum materials. Cation charge imbalance in the doubleperovskite structure is further shown to allow a fine tuning of the gap between the t _{2g } and e _{ g } levels, an effect of much potential in the context of orbital engineering in oxide electronics.
Introduction
A defining feature of delectron systems is the presence of sizable electron correlations, also referred to as Mott–Hubbard physics. The latter has been traditionally associated with firstseries (3d) transitionmetal (TM) oxides. But recently one more ingredient entered the TM oxide ‘Mottness’ paradigm—large spin–orbit couplings (SOCs) in 4d and 5d quantum materials.^{ 1,2 } It turns out that for specific t _{2g }shell electron configurations, a strong SOC can effectively augment the effect of Hubbard correlations:^{ 1 } although the 4d and 5d orbitals are relatively extended objects and the Coulomb repulsive interactions are weakened as compared with the more compact 3d states, the spin–orbitinduced level splittings can become large enough to break apart the ‘nonrelativistic’ t _{2g } bands into sets of wellseparated, significantly narrower subbands for which even a modest Hubbard U acting on the respective Wannier orbitals can then open up a finite Mott–Hubbardlike gap.^{ 1 } On top of that, SOC additionally reshuffles the intersite superexchange.^{ 3 } The surprisingly large anisotropic magnetic interactions that come into play via the strong SOCs in iridates,^{ 3,4,5,6,7,8 } e.g., are responsible for an exotic assortment of novel magnetic ground states and excitations.^{ 2,3,9 }
For large t _{2g }−e _{ g } splittings, the spin–orbitcoupled ${t}_{2g}^{1}$ and ${t}_{2g}^{5}$ electron configurations can be in first approximation viewed as ‘complementary’: in the simplest picture, the dshell manifold can be shrunk to the set of j=1/2 and j=3/2 relativistic levels, with a j=3/2 ground state for the TM ${t}_{2g}^{1}$ configuration and a j=1/2 ground state for ${t}_{2g}^{5}$ .^{ 10,11,12 } Although strongly spin–orbitcoupled ${t}_{2g}^{5}$ oxides and halides—iridates, rhodates and ruthenates, in particular—have generated substantial experimental and theoretical investigations in recent years, much of the properties of 5d and 4d ${t}_{2g}^{1}$ systems remain to large extent unexplored.
From textbook arguments,^{ 10,11,12 } the ${t}_{2g}^{1}$ j=3/2 quadruplet should be characterised by a vanishing magnetic moment in cubic symmetry, due to perfect cancellation of the spin and angular momentum contributions. But this assertion leaves unexplained the wide variety of magnetic properties recently found in 4d ^{1} and 5d ^{1} cubic oxide compounds.^{ 13,14,15,16,17,18,19,20 } Ba_{2}YMoO_{6}, e.g., develops no magnetic order despite a Curie–Weiss temperature of ≈−200 K (refs 13,14) and features complex magnetic dynamics that persists down to the mK range, possibly due to either a valencebondglass^{ 15 } or spinliquid^{ 16 } ground state. Also, Ba_{2}NaOsO_{6} displays an antiferromagnetic Curie–Weiss temperature^{ 18,19 } but orders ferromagnetically below 7 K (ref. 20), whereas Ba_{2}LiOsO_{6} is a spinflop antiferromagnet.^{ 20 }
Here we carry out a detailed ab initio investigation of the Mo^{5+} 4d ^{1} and Os^{7+} 5d ^{1} relativistic electronic structure in the doubleperovskite compounds Ba_{2}YMoO_{6}, Ba_{2}LiOsO_{6} and Ba_{2}NaOsO_{6}. In addition to providing reliable results for the energy scale of the dlevel splittings, t _{2g }−e _{ g } and induced by SOC within the ${t}_{2g}^{1}$ manifold, we analyse the role of TM d−O p orbital mixing plus the strength of electron–lattice couplings. It is found that strong metal d−O p hybridisation generates a finite magnetic moment even for perfectly cubic environment around the TM site, providing ab initio support to the phenomenological covalency factor introduced in this context by Stevens.^{ 21 } The TM d ^{1} magnetic moment is further enhanced by tetragonal distortions, against which the octahedral oxygen cage is unstable. According to our results, such Jahn–Teller (JT) effects are particularly strong for the Mo 4d ^{1} ions in Ba_{2}YMoO_{6}. Although additional investigations are needed for clarifying the role of intersite cooperative couplings,^{ 22,23 } our calculations emphasise the high sensitivity of the effective magnetic moments to both metal–ligand covalency effects and local JT physics. The material dependence for the ratio among the strengths of the spin–orbit interaction, the JT coupling parameter and the effective covalency factor that we compute here provide a solid basis for future studies addressing the role of intersite interactions on the doubleperovskite fcc lattice.
Results
Quantum chemistry calculations were first performed to resolve the essential features of the electronic structure of the cubic lattice configuration, without accounting for electron–lattice couplings (see Materials and methods for computational details and Figure 1 for a sketch of a threedimensional doubleperovskite crystal). Results for the splitting between the Mo^{5+} ${t}_{2g}^{1}$ j=3/2 and j=1/2 spin–orbit states, Δ_{3/2→1/2}, are provided in Table 1 at two levels of approximation, i.e., multiconfiguration completeactivespace selfconsistent field (CASSCF) and multireference configuration interaction (MRCI) with single and double excitations on top of the CASSCF wave function.^{ 24 } Knowing the splitting Δ_{3/2→1/2}, the strength of the SOC parameter can be easily derived as $\lambda =\frac{2}{3}{\Delta}_{\mathrm{3/2}\to \mathrm{1/2}}^{\mathrm{CASSCF}}$ .^{ 11 } The resulting λ of 89 meV is somewhat smaller than earlier estimates of 99 meV for Mo^{5+} impurities in SrTiO_{3}.^{ 25 } A most interesting finding, however, is that despite the cubic environment the quantum chemistry calculations yield a nonvanishing magnetic moment and a finite gfactor. This obviously does not fit the nonmagnetic j=3/2 quartet ground state assumed to arise in standard textbooks on crystalfield theory^{ 11,12 } from exact cancellation between the spin and the orbital moments.
At a qualitative level, it has been argued by Stevens^{ 21 } that finite gfactor values can in fact occur for j=3/2 ions due to TMO covalency on the TM O_{6} octahedron. For better insight into the nature of such effects, we therefore performed a simple numerical experiment in which the six ligands coordinating the reference Mo^{5+} 4d ^{1} ion are replaced by −2 point charges with no atomic basis functions. In that additional set of computations the magnetic moment and the gfactor do vanish, in agreement with the purely ionic picture of Kotani, Abragam and Bleaney.^{ 10,11 } This shows that one tuning knob for switching magnetism on is indeed the TM 4d−O 2p orbital hybridisation. The latter is strong for high ionisation states such as Mo^{5+} (as the tails of the 4dlike valence orbitals indicate in the case the nearestneighbor ligands are provided with atomic basis sets, see Figure 2a), gives rise to partial quenching of the orbital moment and makes that the exact cancellation between the spin and the orbital moments no longer holds.
We find that this effect is even stronger for the formally 7+ Os ion in Ba_{2}LiOsO_{6} and Ba_{2}NaOsO_{6}. As shown in Table 2, gfactors as large as 0.4 are computed in this case. The quantum chemistry results also allow us to estimate the strength of the effective Os^{7+} 5d ^{1} SOC constant, with $\lambda =\frac{2}{3}{\Delta}_{\mathrm{3/2}\to \mathrm{1/2}}^{\mathrm{CASSCF}}\mathrm{=387}\phantom{\rule{.25em}{0ex}}\mathrm{meV}$ , lower than λ=468 meV in tetravalent 5d ^{5} iridates.^{ 26 }
As the ${t}_{2g}^{1}$ electron configuration is susceptible to JT effects, we carried out further investigations on the stability of an ideal TM O_{6} octahedron against tetragonal (z axis) distortions. A totalenergy profile for specified geometric configurations is provided in Figure 2c for an embedded MoO_{6} octahedron. It is seen that the minimum corresponds to ~3% tetragonal compression, as compared with the cubic octahedron of the Fm3m crystalline structure.^{ 14 } As expected, the magnetic moment rapidly increases in the presence of distortions, as illustrated in Table 3 and Figure 2d.
Depending on further details related to the strength of the intersite couplings among ‘JT centres’, static deformations away from cubic symmetry may be realised in some systems, as observed for example in the Re^{6+} 5d ^{1} doubleperovskite Sr_{2}MgReO_{6} (ref. 27) and rareearth molibdates.^{ 28,29 } If the local JT couplings and intersite interactions are relatively weak, one may be left on the other hand in a dynamic JT regime, as earlier pointed out for the particular ${t}_{2g}^{1}$ configuration by, e.g., Kahn and Kettle.^{ 30 } The relevant vibrational modes that couple to the ^{2} T _{2g } $\left({t}_{2g}^{1}\right)$ electronic term are those of E _{ g } symmetry ((3z ^{2}−r ^{2}) and (x ^{2}−y ^{2})like). From the quantum chemistry calculations, we find that the potentialenergy well is significantly shallower for these normal coordinates, as compared with z axisonly compression. The value we computed for the Mo^{5+} ion in Ba_{2}YMoO_{6}, ≈40 meV, is comparable to the estimate made in the 1970s for Mo^{5+} t _{2g } ^{1} impurity ions within the SrTiO_{3} matrix, ≈60 meV.^{ 25 } For the osmates, the depth of this potential well is much reduced, with E _{JT} values in the range of 10–15 meV by spin–orbit MRCI calculations (Table 4).
The vibronic model of Kahn and Kettle^{ 30 } provides specific expressions for the gfactors. In particular, g _{} can be parametrised as^{ 30 } $$\begin{array}{}\text{(1)}& {g}_{\parallel}\mathrm{=2}\left(1{k}_{\mathrm{cov}}{k}_{\mathrm{vib}}\right),\end{array}$$ where k _{cov} is Stevens’ covalency factor^{ 21 } and $$\begin{array}{}\text{(2)}& {k}_{\mathrm{vib}}=\mathrm{exp}\left[x\mathrm{/(1}+\rho )\right].\end{array}$$
The parameters x and ρ are defined as^{ 30 } $x=3{E}_{\mathrm{JT}}\mathrm{/2}\overline{h}\omega $ and $\rho =3\lambda \mathrm{/2}\overline{h}\omega $ , where $\overline{h}\omega $ is the E _{ g }mode vibrational energy and g _{⊥}=0 by symmetry.^{ 11,21,30 } Recent infrared transmission spectra indicate that $\overline{h}\omega \phantom{\rule{.03em}{0ex}}\approx \phantom{\rule{.03em}{0ex}}560$ cm^{−1}≈70 meV for the bondstretching phonons.^{ 17,31 } The effective parameter k _{cov} we can easily evaluate from the static g _{} values obtained in the MRCI spin–orbit treatment (Tables 1 and 2) if vibronic interactions are neglected (k _{vib}=1 for ‘frozen’ cubic octahedra), with ${k}_{\mathrm{cov}}\phantom{\rule{.03em}{0ex}}\equiv \phantom{\rule{.03em}{0ex}}1\phantom{\rule{.03em}{0ex}}\phantom{\rule{.03em}{0ex}}{g}_{\parallel}^{\mathrm{MRCI}}\mathrm{/2}$ . This yields covalency reduction factors of 0.90 for Ba_{2}YMoO_{6} and 0.80 for the osmates.
Estimates for g _{} are provided in Table 4, using the Kahn–Kettle vibronic model and the quantum chemistry results for λ, k _{cov} and E _{JT}. It is seen that a large ρ/x ratio (i.e., large λ/E _{JT}) makes that g _{} is generated mostly through covalency effects in the osmates, with minor contributions from vibronic couplings. On the other hand, the small ρ/x ratio in Ba_{2}YMoO_{6} gives rise to a strong enhancement of g _{} through vibronic effects, with a factor of nearly 4 between (1−k _{cov} k _{vib}) and (1−k _{cov}). This way, the interesting situation arises that the TM magnetic moment is mainly due to vibronic effects in Ba_{2}YMoO_{6} and predominantly to strong covalency in Ba_{2}LiOsO_{6} and Ba_{2}NaOsO_{6}.
Discussion
Experimentally, the measured magnetic moments are indeed significantly smaller in Ba_{2}LiOsO_{6} and Ba_{2}NaOsO_{6} (refs 19,20) as compared with Ba_{2}YMoO_{6} (refs 13,14,17,25). With regard to the estimates we make here for g _{}, possible sources of errors concern the accuracy of the calculated E _{JT} when using the experimental crystal structure as reference and correlation and polarisation effects beyond a single TM O_{6} octahedron.^{ 32,33 } The latter effects would only increase E _{JT}. With respect to the former aspect, it is known that by advanced quantum chemistry calculations, the lattice constants of TM oxides can be computed with deviations of <0.5% from the measured values,^{ 32 } which implies rather small corrections to E _{JT}. Interestingly, recent findings of additional phonon modes at low temperatures^{ 17 } indicate static distortions of the MoO_{6} octahedra in Ba_{2}YMoO_{6} and indeed a rather large E _{JT}. More detailed investigations on this matter are left for future work. Valuable experimental data that can be directly compared with our calculations would be the results of electron spin resonance measurements of the gfactors.
It is also worth pointing out that using the Kahn–Kettle model even a E _{JT} of 75 meV, five to seven times larger than the values computed by MRCI for the osmates (Table 4), still yields a rather moderate g _{} factor of 0.65 for the Os 5d ^{1} ion. Such g _{} factors of 0.4–0.6 compare quite well with the lowtemperature magnetic moment derived from magnetisation and muon spin relaxation measurements on Ba_{2}NaOsO_{6}, ≈0.2 μ _{B}.^{ 19,20 } For the Mo 4d ^{1} ion in Ba_{2}YMoO_{6}, the computed g _{} factor is much more sensitive to variations of E _{JT}—increasing E _{JT} from, e.g., 40–200 meV enhances g _{} of Equation (1) from ≈0.6 to ≈1.6.
One other remarkable prediction of Kahn and Kettle^{ 30 } is that the splitting of the j=3/2 and j=1/2 states is increased through vibronic couplings, by a factor $$\begin{array}{}\text{(3)}& \gamma =1+x\frac{3+\rho}{\mathrm{3(}{\rho}^{2}\mathrm{1)}}.\end{array}$$
This effect turns out to be small in the osmates, given the small x and large ρ in those compounds. But we compute a strong modification of the j=3/2 to j=1/2 excitation energy for Ba_{2}YMoO_{6}, from ~0.13 eV in the absence of vibronic interactions (Table 1) to ≈0.20 eV with JT effects included (E _{JT}=40 meV). Experimentally, the situation can be clarified by direct resonant inelastic Xray scattering (RIXS) measurements on Ba_{2}YMoO_{6}. Highresolution RIXS measurements could also address the occurrence of static distortions at low temperatures, suggested for Ba_{2}YMoO_{6} on the basis of extra phonon modes in the lowT infrared transmission spectra^{ 17 } and for Ba_{2}NaOsO_{6} from the integrated entropy through the magnetic phase transition at ~7 K.^{ 19 } According to the MRCI data in Table 3, a reduction by 0.5–1.5% of the interatomic distances on one set of O–Mo–O links already gives a splitting of 20–50 meV of the lowlying spin–orbit states. Splittings of this size should be accessible with lastgeneration RIXS apparatus.
Also of interest is an experimental confirmation of the unusually large t _{2g }−e _{ g } gap we predict in the doubleperovskite heptavalent osmates, ≳6 eV (Table 2). According to the results of additional computations we carried out, the source of this exceptional dlevel splitting is the stabilisation of the Os t _{2g } states due to the large effective charge (formally 7+) at the nearestneighbor Os sites. The latter are situated on the axes along which the lobes of the t _{2g } orbitals are oriented; in contrast, the lobes of the e _{ g } functions point towards the monovalent species (Li^{1+} or Na^{1+}). For example, test CASSCF calculations in which the size of the point charges placed at the 12 Os and 6 alkalineion nearestneighbor sites are modified from the formal ionic values 7+ and 1+ (12×7+6×1=90) to 5+ and 5+ (12×5+6×5=90) show a reduction of about 2 eV of the t _{2g }−e _{ g } level splitting. Similar effects, with relative shifts and even inversion of the delectron energy levels due to charge imbalance at nearby cation sites, were recently evidenced in Sr_{2}RhO_{4} and Sr_{2}IrO_{4},^{ 8,26 } the rareearth 227 iridates R _{2}Ir_{2}O_{7} (ref. 34) and Cd_{2}Os_{2}O_{7}.^{ 35 } The mechanism has not been thoroughly explored so far experimentally but seems to hold much potential in the context of orbital engineering in TM compounds.
To summarise, it is well known that nominal orbital degeneracy gives rise in 3d TM oxides to subtle couplings between the electronic and lattice degrees of freedom and very rich physics. Here we resolve the effect of electron–lattice interactions on the magnetic properties of heavier, 4d and 5d TM ions with a formally degenerate t _{2g } ^{1} electron configuration in the doubleperovskite materials Ba_{2}YMoO_{6}, Ba_{2}LiOsO_{6} and Ba_{2}NaOsO_{6}. In particular, using advanced quantum chemistry electronicstructure calculations, we reconcile the notion of a nonmagnetic spin–orbitcoupled t _{2g } ^{1} j=3/2 ground state put forward by Kotani, Abragam, Bleaney and others^{ 10,11,12 } with the variety of magnetic properties recently observed in 4d ^{1} and 5d ^{1} double perovskites. Our analysis shows that the sizable magnetic moments and gfactors found experimentally are due to strong TM d–ligand p hybridisation and dynamic JT effects, providing new perspectives on the interplay between metal–ligand interactions and SOCs in TM oxides. It also highlights the proper theoretical frame for addressing the remarkably rich magnetic properties of d ^{1} double perovskites^{ 2,15,16,19,20 } in particular. Over the past two decades, vibronic couplings have unjustifiably received low attention in the case of these intriguing materials.
Materials and methods
All ab initio calculations were carried out with the quantum chemistry package Molpro.^{ 36 } Crystallographic data as derived in ref. 14 for Ba_{2}YMoO_{6} and in ref. 18 for Ba_{2}LiOsO_{6} and Ba_{2}NaOsO_{6} were employed.
We used effective core potentials, valence basis functions of triplezeta quality and two f polarisation functions for the reference Mo/Os ions^{ 37,38 } for which the dshell excitations are explicitly computed. Allelectron triplezeta basis sets supplemented with two d polarisation functions^{ 39 } were applied for each of the six adjacent O ligands. The eight Ba nearest neighbours were in each case modelled by Ba^{2+} ‘totalion’ pseudopotentials supplemented with a single s function.^{ 40 } For Ba_{2}YMoO_{6}, the six nearby Y sites were described by effective core potentials and valence basis functions of doublezeta quality.^{ 37 } In Ba_{2}LiOsO_{6} and Ba_{2}NaOsO_{6}, we employed totalion pseudopotentials for the six nearest Li and Na cations and sets of one s and one p functions.^{ 41 } The farther solidstate surroundings enter the quantum chemistry calculations at the level of a Madelung ionic potential. How the complexity and accuracy of quantum chemistry calculations for an infinite solid can be systematically increased is addressed in, e.g., refs 32,33,42,43.
For the CASSCF calculations of the dshell splittings, we used active spaces of either three (t _{2g }) or five (t _{2g } plus e _{ g }) orbitals. The CASSCF optimisations were carried out for an average of either the ^{2} T _{2g } ( ${t}_{2g}^{1}$ ) or ^{2} T _{2g }( ${t}_{2g}^{1}$ )+^{2} E _{2g }( ${e}_{g}^{1}$ ) eigenfunctions of the scalar relativistic Hamiltonian. All O 2p and Mo/Os 4d/5d electrons on the reference TM O_{6} octahedron were correlated in the MRCI treatment. The latter was performed with single and double substitutions with respect to the CASSCF reference, as described in refs 44,45. The spin–orbit treatment was carried out according to the procedure described in ref. 46.
The gfactors were computed following a scheme proposed by Bolvin^{ 47 } and Vancoillie.^{ 48 } For a Kramersdoublet ground state $\{\mathit{\psi},\overline{\mathit{\psi}}\}$ , the Abragam–Bleaney tensor^{ 11 } G=gg ^{ T } can be expressed in matrix form as $$\begin{array}{}\text{(4)}& \begin{array}{ccc}\hfill {G}_{kl}& \hfill =\hfill & 2\sum _{u,v=\psi ,\overline{\psi}}\u3008u{\stackrel{\u02c6}{L}}_{k}+{g}_{e}{\stackrel{\u02c6}{S}}_{k}v\u3009\u3008v{\stackrel{\u02c6}{L}}_{l}+{g}_{e}{\stackrel{\u02c6}{S}}_{l}u\u3009\hfill \\ \hfill & \hfill =\hfill & \sum _{m=x,y,z}\left({\Lambda}_{km}+{g}_{e}{\sum}_{km}\right)\left({\Lambda}_{lm}+{g}_{e}{\sum}_{lm}\right),\hfill \end{array}\end{array}$$ where g _{ e } is the freeelectron gfactor and $$\begin{array}{}\text{(5)}& \begin{array}{ccc}\hfill {\Lambda}_{kx}& \hfill =\hfill & 2Re\left[\phantom{\rule{0.4em}{0ex}}\u3008\overline{\psi}\left{\stackrel{\u02c6}{L}}_{k}\right\psi \u3009\phantom{\rule{0.4em}{0ex}}\right],\phantom{\rule{1em}{0ex}}{\sum}_{kx}\mathrm{=2}Re\left[\phantom{\rule{0.4em}{0ex}}\u3008\overline{\psi}\left{\stackrel{\u02c6}{S}}_{k}\right\psi \u3009\phantom{\rule{0.4em}{0ex}}\right],\hfill \\ \hfill {\Lambda}_{ky}& \hfill =\hfill & 2Im\left[\phantom{\rule{0.4em}{0ex}}\u3008\overline{\psi}\left{\stackrel{\u02c6}{L}}_{k}\right\psi \u3009\phantom{\rule{0.4em}{0ex}}\right],\phantom{\rule{1em}{0ex}}{\sum}_{ky}\mathrm{=2}Im\left[\phantom{\rule{0.4em}{0ex}}\u3008\overline{\psi}\left{\stackrel{\u02c6}{S}}_{k}\right\psi \u3009\phantom{\rule{0.4em}{0ex}}\right],\hfill \\ \hfill {\Lambda}_{kz}& \hfill =\hfill & 2\left[\phantom{\rule{0.4em}{0ex}}\u3008\psi \left{\stackrel{\u02c6}{L}}_{k}\right\psi \u3009\phantom{\rule{0.4em}{0ex}}\right],\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}{\sum}_{kz}\mathrm{=2}\left[\phantom{\rule{0.4em}{0ex}}\u3008\psi \left{\stackrel{\u02c6}{S}}_{k}\right\psi \u3009\phantom{\rule{0.4em}{0ex}}\right].\hfill \end{array}\end{array}$$
The matrix elements of $\stackrel{\u02c6}{L}$ were extracted from the Molpro outputs, whereas the matrix elements of $\stackrel{\u02c6}{S}$ were derived using the conventional expressions for the generalised Pauli matrices: $$\begin{array}{}\text{(6)}& \begin{array}{ccc}\hfill {\left({\stackrel{\u02c6}{S}}_{z}\right)}_{MM\prime}& \hfill =\hfill & M{\delta}_{MM\prime},\hfill \\ \hfill {\left({\stackrel{\u02c6}{S}}_{x}\right)}_{MM\prime}& \hfill =\hfill & \frac{1}{2}\sqrt{(S+M)(SM+\mathrm{1)}}{\delta}_{M\mathrm{1,}M\prime}\hfill \\ \hfill & \hfill \hfill & +\frac{1}{2}\sqrt{(SM)(S+M+\mathrm{1)}}{\delta}_{M+\mathrm{1,}M\prime},\hfill \\ \hfill {\left({\stackrel{\u02c6}{S}}_{y}\right)}_{MM\prime}& \hfill =\hfill & \frac{i}{2}\sqrt{(S+M)(SM+\mathrm{1)}}{\delta}_{M\mathrm{1,}M\prime}\hfill \\ \hfill & \hfill \hfill & +\frac{i}{2}\sqrt{(SM)(S+M+\mathrm{1)}}{\delta}_{M+\mathrm{1,}M\prime}.\hfill \end{array}\end{array}$$
The gfactors were calculated as the positive square roots of the three eigenvalues of G.
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Acknowledgements
We thank V. Kataev for discussions. Part of the computations were carried out at the High Performance Computing Center (ZIH) of the Technical University Dresden. We acknowledge financial support from the German Research Foundation (Deutsche Forschungsgemeinschaft, DFG)—SFB1143 and HO4427/2.
Author information
Affiliations
Institute for Theoretical Solid State Physics, IFW Dresden, Dresden, Germany
 Lei Xu
 , Nikolay A Bogdanov
 , Jeroen van den Brink
 & Liviu Hozoi
Department of Physics, University of Oxford, Clarendon Laboratory, Oxford, UK
 Andrew Princep
MaxPlanckInstitut für Physik komplexer Systeme, Dresden, Germany
 Peter Fulde
Institute for Theoretical Physics, TU Dresden, Dresden, Germany
 Jeroen van den Brink
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Contributions
L.X. carried out the ab initio quantum chemistry calculations, with assistance from N.A.B., A.P., P.F. and L.H. The mapping of the ab initio data onto the effective vibronic model was performed by L.X., L.H. and P.F. L.H., P.F. and J.vdB. designed the project. L.X. and L.H. wrote the paper, with contributions from all other coauthors.
Competing interests
The authors declare no conflict of interest.
Corresponding authors
Correspondence to Lei Xu or Liviu Hozoi.
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