Covalency and vibronic couplings make a nonmagnetic j=3/2 ion magnetic

For 4$d^1$ and 5$d^1$ spin-orbit-coupled 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 4$d^1$ and 5$d^1$ ions in molybdenum- and osmium-based double-perovskite systems and reveal different kinds of on-site many-body physics in the two families of compounds: while the sizable magnetic moments and $g$ factors measured experimentally are due to both metal $d$-ligand $p$ hybridization and dynamic Jahn-Teller interactions for 4$d$ electrons, it is essentially $d$-$p$ covalency for the 5$d^1$ 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 4$d$ and 5$d$ quantum materials. Cation charge imbalance in the double-perovskite 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 d-electron systems is the presence of sizable electron correlations, also referred to as Mott-Hubbard physics. The latter has been traditionally associated with firstseries (3d) transition-metal (TM) oxides. But recently one more ingredient entered the TM-oxide 'Mottness' paradigm -large spin-orbit couplings (SOC's) 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 to the more compact 3d states, the spin-orbit-induced level splittings can become large enough to break apart the 'nonrelativistic' t 2g bands into sets of well separated, significantly narrower subbands for which even a modest Hubbard U acting on the respective Wannier orbitals can then open up a finite Mott-Hubbard-like 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 SOC's in iridates [3][4][5][6][7][8], for example, 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-orbit-coupled t 1 2g and t 5 2g 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 1 2g configuration and a j = 1/2 ground state for t 5 2g [10][11][12]. While strongly spin-orbit-coupled t 5 2g oxides and halidesiridates, rhodates and ruthenates, in particular -have generated substantial experimental and theoretical investigations in recent years, much of the properties of 5d and 4d t 1 2g systems remain to large extent unexplored.
From textbook arguments [10][11][12], the t 1 2g j = 3/2 quadruplet should be characterized 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 , for example, develops no magnetic order despite a Curie-Weiss temperature of ≈-200 K [13,14] and features complex magnetic dynamics that persists down to the mK range, possibly due to either a valence-bond-glass [15] or spin-liquid [16] ground state. Also Ba 2 NaOsO 6 displays an antiferromagnetic Curie-Weiss temperature [18,19] but orders ferromagnetically below 7 K [20] whilst Ba 2 LiOsO 6 is a spin-flop 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 double-perovskite 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 d-level splittings, t 2g -e g and induced by SOC within the t 1 2g manifold, we analyze 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 hybridization 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 . While additional investigations are needed for clarifying the role of intersite cooperative couplings [22,23], our calculations thus emphasize 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 double-perovskite 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 electronlattice couplings (see Methods for computational details and Fig. 1 for a sketch of a three-dimensional double-perovskite crystal). Results for the splitting between the Mo 5+ t 1 2g j = 3/2 and j = 1/2 spin-orbit states, ∆ 3/2→1/2 , are provided in Table I at two levels of approximation, i. e., multiconfiguration complete-active-space self-consistent-field (CASSCF) and multireference configuration-interaction (MRCI) with single and double excitations on top of the CASSCF wave function [24]. Knowing the splitting ∆ 3/2→1/2 , the strength of the SOC parameter can be easily derived as λ = 2 3 ∆ CASSCF 3/2→1/2 [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 g factor. This obviously does not fit the nonmagnetic j = 3/2 quartet ground state assumed to arise in standard textbooks on crystal-field 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 g-factor values can in fact occur for j = 3/2 ions due to TM-O 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 g factor 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 hybridization. The latter is strong for high ionization states such as Mo 5+ (as the tails of the 4d-like valence orbitals indicate in the case the nearest-neighbor ligands are provided with atomic basis sets, see Fig. 2(a), 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 II, g factors 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 λ = 2 3 ∆ CASSCF 3/2→1/2 = 387 meV, lower than λ = 468 meV in tetravalent 5d 5 iridates [26].
Since the t 1 2g 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 total-energy profile for specified geometric configurations is provided in Fig. 2(c) for an embedded MoO 6 octahedron. It is seen that the minimum corresponds to about TABLE II. Os 7+ 5d-shell splittings (j = 3/2 to j = 1/2 and t2geg) and 'static' g factors in cubic Ba2LiOsO6 and Ba2NaOsO6. Only the 5d t2g orbitals were active in the CASSCF calculation for ∆ 3/2→1/2 ; all five 5d orbitals were active in the calculations for g and ∆t 2g →eg . For the latter, values including SOC are provided within parantheses. All energies in eV. 3% tetragonal compression, as compared to the cubic octahedron of the F m3m crystalline structure [14]. As expected, the magnetic moment rapidly increases in the presence of distortions, as illustrated in Table III and Fig. 2(d) .
Depending on further details related to the strength of the intersite couplings among 'JT centers', static deformations away from cubic symmetry may be realized in some systems, as observed for example in the Re 6+ 5d 1 double perovskite Sr 2 MgReO 6 [27] and rare-earth 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 1 2g configuration by, e.g., Kahn and Kettle [30]. The relevant vibrational modes that couple to the 2 T 2g (t 1 2g ) 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 potential-energy well is significantly shallower for these normal coordinates, as compared to z-axis-only 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 1970's for Mo 5+ t 1 2g 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 spinorbit MRCI calculations (see Table IV).
The vibronic model of Kahn and Kettle [30] provides specific expressions for the g factors. In particular, g can be parametrized as [30] where k cov is Stevens' covalency factor [21] and The parameters x and ρ are defined as [30] x = 3E JT /2hω and ρ = 3λ/2hω, wherehω is the E g -mode vibrational energy, and g ⊥ = 0 by symmetry [11,21,30]. Recent infrared transmission spectra indicate thathω ≈ 560 cm −1 ≈ 70 meV for the bond stretching phonons [17,31]. The effective parameter k cov we can easily evaluate from the static g values obtained in the MRCI spin-orbit treatment (see Tables I and  II) if vibronic interactions are neglected (k vib = 1 for 'frozen' cubic octahedra), with k cov ≡ 1 − g MRCI /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 IV, 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 . TABLE III. Mo 5+ t 1 2g electronic structure with 'static' tetragonal squeezing of the reference MoO6 octahedron. Only the t 1 2g configuration was considered in the reference CASSCF. ∆t 2g is t2g tetragonal splitting without SOC, δ1 and δ2 are excitation energies within the t 1 2g manifold with SOC accounted for (δ1 = 0 and δ2 = ∆ 3/2→1/2 for cubic octahedra, see Fig. 2(b)). MRCI results, all energies in eV.  [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 less than 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 to our calculations would be the results of electron spin resonance measurements of the g factors.

MoO6
It is also worth pointing out that using the Kahn-Kettle model even a E JT of 75 meV, 5 to 7 times larger than the values computed by MRCI for the osmates (see Table IV), 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 low-temperature magnetic moment derived from magnetization 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 to 200 meV enhances g of Eq. (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 .
(3) 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 about 0.13 eV in the absence of vibronic interactions (see Table I) to ≈0.20 eV with Jahn-Teller effects included (E JT = 40 meV). Experimentally the situation can be clarified by direct resonant inelastic x-ray scattering (RIXS) measurements on Ba 2 YMoO 6 . High-resolution 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 low-T infrared transmission spectra [17] and for Ba 2 NaOsO 6 from the integrated entropy through the magnetic phase transition at about 7 K [19]. According to the MRCI data in Table III Also of interest is an experimental confirmation of the un- usually large t 2g -e g gap we predict in the double-perovskite heptavalent osmates, 6 eV (see Table II). According to the results of additional computations we carried out, the source of this exceptional d-level splitting is the stabilization of the Os t 2g states due to the large effective charge (formally 7+) at the nearest-neighbor 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 alkaline-ion nearest-neighbor 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 d-electron 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 rare-earth 227 iridates R 2 Ir 2 O 7 [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 summarize, it is well known that nominal orbital degeneracy gives rise in 3d transition-metal 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 transition-metal ions with a formally degenerate t 1 2g electron configuration in the double-perovskite materials Ba 2 YMoO 6 , Ba 2 LiOsO 6 and Ba 2 NaOsO 6 . In particular, by using advanced quantum chemistry electronic-structure calculations, we reconcile the notion of a nonmagnetic spin-orbit-coupled t 1 2g 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 g factors found experimentally are due to strong TM d -ligand p hybridization and dynamic Jahn-Teller effects, providing new perspectives on the interplay between metal-ligand interactions and spin-orbit couplings in transition-metal 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 last two decades, vibronic couplings have unjustifiably received low attention in the case of these intriguing materials.

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 (ECP's), valence basis functions of triple-zeta quality and two f polarization functions for the reference Mo/Os ions [37,38] for which the dshell excitations are explicitly computed. All-electron triplezeta basis sets supplemented with two d polarization functions [39] were applied for each of the six adjacent O ligands. The eight Ba nearest neighbors were in each case modeled by Ba 2+ 'total-ion' pseudopotentials (TIP's) supplemented with a single s function [40]. For Ba 2 YMoO 6 , the six nearby Y sites were described by ECP's and valence basis functions of double-zeta quality [37]. In Ba 2 LiOsO 6 and Ba 2 NaOsO 6 , we employed TIP's 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 d-shell splittings, we used active spaces of either three (t 2g ) or five (t 2g plus e g ) orbitals. The CASSCF optimizations were carried out for an average of either the 2 T 2g (t 1 2g ) or 2 T 2g (t 1 2g ) + 2 E 2g (e 1 g ) 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 g factors were computed following a scheme proposed by Bolvin [47] and Vancoillie [48]. For a Kramers-doublet ground state {ψ,ψ}, the Abragam-Bleaney tensor [11] G = gg T can be expressed in matrix form as G kl = 2 u,v=ψ,ψ u|L k + g eŜk |v v|L l + g eŜl |u = m=x,y,z Λ km + g e km Λ lm + g e lm , where g e is the free-electron g factor and The matrix elements ofL were extracted from the MOLPRO outputs, while the matrix elements ofŜ were derived using the conventional expressions for the generalized Pauli matrices: The g factors were calculated as the positive square roots of the three eigenvalues of G.