Origins Vs. fingerprints of the Jahn-Teller effect in d-electron ABX$_3$ perovskites

The Jahn-Teller (JT) distortion that can remove electronic degeneracies in partially occupied states and results in systematic atomic displacements is a common underlying feature to many of the intriguing phenomena observed in 3d perovskites, encompassing magnetism, superconductivity, orbital ordering and colossal magnetoresistance. Although the seminal Jahn and Teller theorem has been postulated almost a century ago, the origins of this effect in perovskite materials are still debated, including propositions such as super exchange, spin-phonon coupling, sterically induced lattice distortions, and strong dynamical correlation effects. Here we analyze the driving forces behind the Jahn-Teller motions and associated electronic fingerprints in a full range of ABX3 compounds. We identify (i) compounds that are prone to an electronically-driven instabilities (i.e. a pure JT effect) such as KCrF3, KCuF3 or LaVO3 and proceed to relax the structures, finding quantitatively the JTD in excellent agreement with experiment; (ii) compounds such as LaMnO3 or LaTiO3 that do not show electronically driven JTD despite orbital degeneracies, because their strongly hybridized B, d-X, p states supply but too weak JT forces to overcome the needed atomic distortions; (iii) although LaVO3 exhibits similar B, d-X, p hybridizations as LaTiO3, the former compound exhibits a robust electronic instability while LaTiO3 has zero stabilization energy, the reason being that LaVO3 has two electrons t2g2 relative to LaTiO3 with just one t2g1. (iv) We explain the trends in"orbital ordering"whereby electrons occupy orbitals that point to orthogonal directions between all nearest-neighbor 3d atoms. We thereby provide a unified vision to explain octahedra deformations in perovskites that, at odds with common wisdom, does not require the celebrated Mott-Hubbard mechanism.


I. INTRODUCTION
ABX 3 (X = O, F) perovskites [1,2] show a number of systematic atomic distortions relative to the ideal cubic perovskite structure made of corner-sharing, vertically positioned, all-parallel BX 6 octahedra with equal B-X bonds. The interpretation of much of the electronic and magnetic phenomenology surrounding such perovskites [1,2], including superconductivity, colossal magnetoresistance, orbital order- ing, and metal-insulator transitions is intimately related to the understanding of the causes versus consequences of the observed atomic distortions. The causes of such distortions in the specific case of Mott insulators are related to the broader issue of causes of gap formations in such compounds. In the electron correlation literature [3][4][5], gapping in Mott insulators has been directly related to interelectronic repulsion U in the symmetric, undistorted state, leading to the expectation that such gapping is a many-body correlation effect. Distortions can occur later as an independent effect. Recent literature predicting distortions in LaMnO 3 using dynamically correlated approaches [3] specifically attributed the causes of such distortions to dynamic correlations. Indeed, previous work claimed that the local density approximation (LDA) and generalized gradient approximation (GGA) "usually fail to predict the correct electronic and structural properties of materials where electronic correlations play a role" [9]; similar statements abound in the literature [3,5,6]. Nevertheless, such a belief was often grounded on a naive implementation of density functional theory (DFT) using non-spin-polarized simulations and/or a lack of symmetry-breaking modes and the use of simple exchange-correlation functionals that do not distinguish occupied from unoccupied orbitals. Recent works [7][8][9][10] have clarified the fact that strong dynamical correlations need not be the determining factor for gap opening or symmetry-lowering displacements in ABO 3 materials. Instead, gapping in oxide Mott insulators can be a consequence of symmetry breaking (positional, occupational, moment distribution) and can be obtained in single-determinant mean-field-like DFT calculations even at U = 0 eV [7,11]. Using U in DFT calculations is not the same as using U in a Hubbard Hamiltonian; in the former case U helps to reduce self-interaction error thereby making orbitals spatially more compact and thereby increasing the gain in energy from electronic degeneracy removal, thus better competing with energy cost from the atomic distortions. This realization suggests taking a new look at the causes and consequences of distortions in Mott insulators and related compounds.
Usually, distortion in ABX 3 perovskites produces inequivalent B-X bond lengths, such as the Q 2 motion (cf. Fig. 2 of Ref. [12]) that differentiates bond lengths along the x and y directions. This exists either as in-phase motion in consecutive planes along the z axis [here labeled Q + 2 motion; Fig. 1(c)] or as in antiphase along z [here labeled Q − 2 motion; Fig. 1(c)]. Although octahedral deformation tilts [e.g., the in-phase φ + z and antiphase φ − z ; displayed in Figs. 1(a), 1(b)] in different ABX 3 compounds often appear similar, they might have different origins. Possibilities include being (i) a consequence of the classic size mismatch between the A, B, and X atoms, as reflected by geometrical packing constructs such as the Goldschmidt tolerance factor [13], or (ii) a true Jahn-Teller [1,6,12,14] effect (JTE), reflecting an electronic instability associated with partial occupation of degenerate states, as described in the seminal work of Jahn and Teller [15]. Understanding predictively the causes of the observed displacements is central to interpreting numerous related phenomena, e.g., gap formation due to symmetry lowering, metal-insulator transitions [16], orbital orderings [5,17,18], and magnetic interactions [19] in ABX 3 perovskites. This is important since controlling octahedral deformations by using temperature, ferroelectricity, strain engineering, or heterostructures [20][21][22][23][24][25][26][27] may offer much needed knobs to control electronic devices.
Puzzles. Predictive understanding of the mechanisms of octahedral distortions is rather challenging, for a number of reasons: (a) Compounds with the same degenerate electronic configurations such as KCrF 3 [28], KCuF 3 [29], and LaMnO 3 [30] all being e 1 g or YVO 3 and LaVO 3 both being t 2 2g [31] show different types of octahedral deformations at low temperature-Q − 2 for KCrF 3 , KCuF 3 , or LaVO 3 and Q + 2 for YVO 3 and LaMnO 3 -raising the question of what creates Q + 2 and what creates Q − 2 . (b) Recent theoretical works predicted the surprising appearance of strong in-phase Q + 2 symmetry-lowering octahedral deformations in compounds such as SrTiO 3 , BaMnO 3 , or BiFeO 3 even when there are no degenerate states (t 0 2g , t 3 2g , or t 3 2g e 2 g configurations, respectively) [22,32].
(c) The textbook depiction of the Jahn-Teller force is often based on calculations based on the degenerate orbital alone, but one must consider the force resulting from the system's total energy, including not only the sum of one-electron orbital energy but also the electron-electron Coulomb and exchangecorrelation energies resulting from degeneracy removal.
Literature models for explaining the origin of octahedral deformations in ABX 3 materials span a large range of mechanisms, including electronic superexchange as codified by the Kugel-Khomskii model [18,19], electron-phonon coupling [5,33], lattice mode couplings involving rotations and forcing the Q + 2 mode [21,34,35], or specific dynamical correlation effects [3,6]. The Kugel-Khomskii approach represents a phenomenological approach to understanding the connection between orbital and spin order. However, it does not identify electronic and structural origins of what is a Jahn-Teller effect in perovskites, since both Q + 2 and Q − 2 modes are equally assumed to be Jahn-Teller distortions.
Since none of these existing models or visions to explain Jahn-Teller distortions have been applied to a full range of ABX 3 compounds with different e g -like and t 2g -like orbital occupations, at this time it is not clear whether different mechanisms are needed to explain different distortions in different compounds, whether Q + 2 and Q − 2 modes really reflect Jahn-Teller effects, and whether there is a basic theoretical framework that could explain them all.

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Distinguishing features of the current approach. We address this subject by studying several ABX 3 materials (X = O, F) with a transition-metal element exhibiting degenerate states (but without a propensity to undergo the disproportionation effect) using a mean-field-like Bloch-periodic DFT band theory with two stipulations. First, a polymorphous representation of the real-space structure that permits the existence of different local environments to atoms and spins is required. This entails using crystallographic cells that are not limited to the smallest primitive cell and therefore allow orbital, spatial, and spin symmetry breaking, should these modes lower the total energy. Second, breaking orbital symmetries of degenerate partners requires an exchange correlation functional that distinguishes occupied from unoccupied states, thereby affording significant cancellation of the self-interaction error and thus spatial compactness of 3d orbitals. Here we use DFT + U (but nonzero U is not essential, as other exchange correlation functionals make no use of an explicit U term [7]). This approach provides DFT with a fair opportunity to reveal whether the different patterns of symmetry breakingspin order, Jahn-Teller distortion and/or octahedral rotations, and orbital occupation symmetry breaking-lower the energy or not, thereby establishing a first-principles framework for predictive theory of displacements in ABX 3 perovskites.
A Landau-esque perturbation approach. To determine whether a potential deformation has an electronic origin or not we use a Landau-esque perturbation approach that examines whether symmetry lowering via nudged occupation numbers in a degenerate manifold of an initially high-symmetry cubic cell lowers the total energy or not. We examine the total energy for a number of prototypical cases encountered in 3d ABX 3 compounds: (a) the equal-occupation scenario of a degenerate level [such as (1/2, 1/2) for a single electron in a doubly occupied e g level] and (b) the orbital broken symmetry (OBS) scenario of degenerate partners, e.g., (1, 0). If (b) gives total energy lowering relative to (a) this signals the propensity for an electronic instability in the parent highsymmetry phase, which is then followed up by the complete relaxation of the unit cell displacements, providing our predicted Jahn-Teller distortions (possibly with coupling to octahedral tilting). If the unequal-occupation configuration (b) returns to the equal-occupation configuration (a) during selfconsistent calculation, this indicates that the high-symmetry system is electronically stable and thus not prone to develop an electronically enforced Jahn-Teller distortion. Distortions present may reflect other factors such as the classic steric effects induced by size mismatch.
The main conclusions are as follows: (a) The existence of electronic instabilities, i.e., a true JTE, is predicted in the high-symmetry Pm3m cubic phase of several ABX 3 compounds (KCrF 3 , KCuF 3 , LaVO 3 , KFeF 3 , and KCoF 3 ), whereas LaMnO 3 or LaTiO 3 have no such instability. Instabilities are manifested by the breaking of orbital degeneracies while lowering the total energy, and thereby concomitantly opening finite band gaps.
(b) Orbital ordering is always found to be a total-energylowering event. In such a state, electrons occupy an orbital that is pointing toward orthogonal directions between all nearestneighbor 3d atoms, such as alternation of d xy /d yz orbitals for a material with a t 2g degeneracy or alternation of d y 2 /d x 2 orbitals for a material with an e g -level degeneracy, thereby minimizing orbital interactions as codified by the phenomenological electronic superexchange model of Kugel and Khomskii. The above observations are independent of the imposed magnetic ordering, thus substantiating the view that the magnetic order is not the cause of the Jahn-Teller effect and related orbital ordering, but rather a consequence of these phenomena.
(c) The Q − 2 octahedral deformation mode is a Jahn-Teller distortion: Following the identification of electronic instabilities in cubic cells, we follow the quantum mechanical forces to establish the fully relaxed crystal structures, finding that the aforementioned Jahn-Teller distorted compounds develop a Q − 2 octahedra deformation mode, which is therefore identified as an electronically induced Jahn-Teller distortion. The significance of the development of this specific Q − 2 mode is that it is characterized by opposite octahedral deformations between nearest sites and is thus able to accommodate the electronically induced orbital ordering. Calculation of the magnitude of the displacements shows good agreement with the experimentally observed trends in a full range of ABX 3 compounds with t 2g and e g degenerate partners.
(d) At odds with the common perception, LaMnO 3 does not exhibit JTE whereas KCrF 3 does, despite isovalent (t 3 2g e 1 g ) configurations. Similarly, the t 1 2g configuration LaTiO 3 has no Jahn-Teller distortion, whereas KFeF 3 does. This is because the existence of strong hybridization in the oxide cases diminishes the orbital localization needed to create the Jahn-Teller distortion, whereas in the fluoride the hybridization is much weaker.
(e) The Q + 2 octahedral deformation, appearing in LaMnO 3 or LaTiO 3 , is not induced by an electronic instability and thus it is not a Jahn-Teller distortion. It is a consequence of octahedral rotations and tilts often appearing in the perovskite oxides due to pure semiclassical atomic size effects, i.e., geometric steric effects. The Q + 2 octahedral deformation mode thus cannot signal strong dynamical correlation effects in these ABX 3 compounds.
(f) Although LaVO 3 exhibits B d-X p hybridizations similarly to LaTiO 3 , the former compound exhibits a robust electronic instability in the cubic cell while LaTiO 3 has zero stabilization energy. The reason is that LaVO 3 has two electrons, t 2 2g , relative to LaTiO 3 with just one, t 1 2g . (g) Due to the coexistence of a JTE and of octahedral rotations in RVO 3 materials, the Q − 2 and Q + 2 modes compete and result in two distinct spin-orbital orders at low temperature depending on the octahedral rotation amplitude.
(h) Jahn-Teller distortions as well as semiclassical size effect distortions can contribute to the opening of band gaps in Mott insulators. We have previously identified [10] four gapping modalities; the JTE contributes just to modality (iii) as follows: (i) compounds with closed subshells that can open a gap due to octahedral crystal field splitting (CaMnO 3 , LaFeO 3 ), (ii) compounds opening gaps by lifting degeneracies through large symmetry-lowering displacements such as X 6 rotations (LaTiO 3 or LaMnO 3 ), (iii) compounds with two electrons in t 2g levels exhibiting Jahn-Teller-induced electronic instability able to cause gapping such as LaVO 3 (t 2 2g e 0 g ), and (iv) compounds with unstable single local electronic occupation patterns disproportionating into a double local environment, e.g., CaFeO 3 and YNiO 3 .

033131-3
We conclude that the electronically induced Jahn-Teller distortion mode Q − 2 and the geometrically induced steric Q + 2 octahedral deformation mode are fully captured by a static mean-field method. This is in line with recent theoretical works that have demonstrated that static mean-field methods capable of inducing broken symmetry such as density functional theory [7,10,21,34] in a polymorphous representation suffice to also explain (i) the trends in gapping and type of magnetic order across the ternary ABO 3 series [10] and the binary 3d oxide series [8,36]; (ii) the trends in disproportionation into two different local environments of the B site [9]; and (iii) the explanation of doping Mott insulators including cuprates [37,38], doping kagome structures [39], as well as "antidoping" oxides [40].

II. THE ELEMENTS OF THE METHOD
The main features of the theoretical framework used in the study are as follows: (a) Cell geometry and relaxation. We consider the following structure types: a high-symmetry Pm3m cubic cell (LaTiO 3 , LaVO 3 , LaMnO 3 , KFeF 3 , KCoF 3 , KCrF 3 , and KCuF 3 ) as well as the experimentally observed structures, namely I 4 /mcm (KCuF 3 , KCrF 3 ), Pbnm (LaTiO 3 , LaVO 3 , (KCoF 3 ), and I 2 /a (KFeF 3 ), for identifying the DFT ground state structure. We allow cell sizes larger than the minimal one (one formula unit per cell) so as to permit symmetry-breaking distortions, should they lower the total energy. Atoms have to be nudged initially off their high-symmetry positions in the cubic cell, followed by the calculation of the restoring Hellman-Feynman forces guiding full relaxation. In order to provide sufficient flexibility for developing general patterns of energy-lowering deformations we have used a polymorphous crystallographic cell corresponding to a ( √ 2a, √ 2a, 2a) cubic cell (i.e., 4 f.u. per cell). A polymorphous network [7,8,10] refers to cases where the macroscopically averaged symmetry (seen by XRD with intensities fitted to a minimal unit cell) is high while the local symmetries can be low. Polymorphous networks will generally show such distortions when positional and occupational degrees of freedom are nudged in a supercell calculation. In contrast, a monomorphous network refers to cases where a single structural motif exists, as seen in calculations that restrict the cell to a minimal size. The structural relaxation (lattice parameters and atomic positions) of ground state structures has also been performed until forces are lower than 0.5 meV/Å. We performed a symmetry-adapted mode analysis that allows us to extract the amplitudes of general distortion modes by projecting the distorted structure on the basis of the phonon eigendisplacements of the high-symmetry cubic cell [41,42].
(b) Orbital symmetry breaking. The occupation numbers of partially filled degenerate orbitals in the high-symmetry cubic cell are not restricted to equal occupations of the degenerate partners. For a compound with a single e g electron, we do not preselect a (1/2, 1/2) occupation pattern as generally done in standard band calculations, but allow also exploration of a (1, 0) occupation pattern. Analogously, for 2 electrons in the t 2g level we explore (2/3, 2/3, 2/3) versus (1, 1, 0) occupation patterns. Wave function symmetrization to a presumed symmetry is avoided so as to allow electrons to freely occupy energy-lowering configurations. We therefore initially nudge not only atomic positions but also the occupation patterns of electrons in specific degenerate partners, called orbital broken symmetry (OBS) states [43][44][45]. The self-consistent field with possible changes in d-orbital occupancies is then obtained starting from this initial guess.
(c) Spin order. We use the observed low-temperature spin order, e.g., AFMA (ferromagnetic planes coupled antiferromagnetically together), AFMC (AFM planes coupled ferromagnetically together), and AFMG (all nearest-neighbor cations are antiferromagnetically coupled). Since we have previously shown that a low-temperature spin-ordered phase inherits the physics of the high-temperature paramagnetic (PM) phase in ABO 3 materials [10], we did not attempt to model the PM distortions presently. For the study of the linear response of the ideal cubic phase, we prefer to use a simple FM order so as to avoid potentially strongly entangled spin-orbital situations as codified by the Kugel-Khomskii model [18,19], i.e., the chicken and the egg dilemma. This is reinforced by experimental observations of octahedral deformations appearing at higher temperatures than the AFM to PM (for example, in KCrF 3 , KCuF 3 , LaMnO 3 ).
(d) Exchange-correlation functional. To enable energylowering occupations one needs to use an exchangecorrelation (XC) functional that distinguishes occupied from unoccupied states. Local (LDA) and semilocal (GGA, meta-GGA) XC functionals do not make such a distinction. The simplest XC allowing this is DFT + U , where U is an on-site potential acting on a subset of orbitals-here the transition metal (TM) d states-and shifting to lower (upper) energies occupied (unoccupied) levels. We have thus employed this formalism in combination with the PBEsol [46] XC functional where U is an effective parameter U eff = U -J [47]. We did not optimize the effective U values for each material to achieve an optimal fit (this might be done if needed in the future), but opted instead to use fixed U = 3.5 eV for all 3d TM elements for simplicity.

A. Examining the propensity of the symmetric structure for electronically induced distortions
The Landau-esque symmetry-breaking test perturbs a highsymmetry cubic cell (here, Pm3m with lattice parameter a cub that is initially relaxed) and examines the total energy for (a) assumed equal occupations of the degenerate partners [no OBS, such as (1/2, 1/2) for a single electron in a doubly degenerate e g level] and (b) OBS, looking for energy-lowering configurations [e.g., (1, 0)]. The spin configuration used is FM. Energy differences between these configurations are provided in Table I.
We see that given the opportunity for orbital broken symmetry, the compounds LaVO 3 , KFeF 3 , KCoF 3 , KCrF 3 , and KCuF 3 show an energy gain E OBS−no OBS < 0, with a concomitant opening of the band gap, while LaTiO 3 and LaMnO 3 have E OBS−no OBS = 0; i.e., the initially imposed OBS relaxes back to the configuration with equally occupied degenerate orbitals, and the system stays metallic. These TABLE I. Detection of a spontaneous electronic instability in the high-symmetry cubic phase of ABX 3 perovskites. Energy differences between solutions with equal occupancy of degenerate levels (no OBS) and with the most stable OBS state in meV per formula unit obtained with GGA + U and HSE06 functionals (the latter uses a smaller cubic cell for computational cost reasons). The lattice parameter a is fixed to a relaxed cubic cell without OBS. A FM order is assumed. Results for the ground state AFM order are reported in parentheses, namely AFMG for LaTiO 3 , KFeF 3 , and KCoF 3 ; AFMC for LaVO 3 ; and AFMA for LaMnO 3 , KCrF 3 , and KCuF 3 .
observations are unchanged by using another nonlocal functional such as the HSE06 [48,49] hybrid functional method, having also a good adherence to self-interaction cancellation (see the method section at the end of Sec. IV for the protocol we used to perform OBS studies with the HSE06 functional). It is significant that the sign of the energy difference is unaltered when one uses instead of the FM spin configuration the ground state magnetic order, namely AFMG for LaTiO 3 , KFeF 3 , and KCoF 3 , AFMC for LaVO 3 , and AFMA for LaMnO 3 , KCrF 3 , and KCuF 3 ; see numbers in parentheses in Table I

The emergence of orbital order as a DFT total-energy-lowering Jahn-Teller effect
Once the degeneracies of the high-symmetry phase are removed, one notes the creation of deterministic localization patterns of the degenerate partner components (such as d x 2 −y 2 and d z 2 ) on different atomic sites. This "orbital ordering" has intrigued the 3d oxide community [12,19], raising various exotic interpretations [3,6,18,19,21,[33][34][35]. To understand the possible unexotic, mean-field total energy origins of such orbital ordering, we have investigated the energetics of different assumed orbital arrangements: (a) electrons in the degenerate levels are initially nudged in identical orbitals on all nearest transition metal sites, such as a d z 2 occupancy on all sites for a compound with a single e g electron; (b) the electron is initially nudged in a different degenerate partner between nearest TM sites in the (xy) plane but on the same partner for TM located in the consecutive plane along the z direction, such as alternation of d x 2 and d y 2 in the xy plane with a similar arrangement on the consecutive plane along z, thus forming a "columnar arrangement"; and (c) the electron is initially nudged in a different degenerate partner between all nearest-neighbor sites, such as alternation of d x 2 and d y 2 in all Cartesian directions, thus forming a 3D checkerboard. For a compound with 2 unpaired electrons for t 2g levels, cases (b) and (c) necessarily require that an identical orbital has to be occupied between nearest TM sites. We thus imposed one electron in a specific t 2g partner on all TM sites (d xy here) and then we alternate the occupancy of the remaining partners (d xz /d yz ). After initial nudging (intended to avoid accidental local minimum), the solution is iterated to self-consistency with full relaxation of the electronic structure but without any structural relaxation.
The results show that for all compounds displaying a spontaneous electronic instability, the lowest-energy state is always associated with configuration (c), e.g., alternation of d x 2 and d y 2 in all Cartesian directions, thus forming a 3D checkerboard (the total energies of the single, columnar, and checkerboard OBS states are provided in Appendix A). In such a minimum-energy state, electrons occupy an orbital that is pointing to orthogonal directions between all nearestneighbor transition elements, such as alternation of d xy /d yz orbitals for a material with a t 2g degeneracy, or alternation of d y 2 /d x 2 orbitals for a material with an e g -level degeneracy as shown by our partial charge density maps of states near the Fermi level [ Fig. 2, panels (a), (b), (d), and (e)]. This specific pattern is referred to as a G-type antiferro-orbital ordering [21,31,50] that minimizes orbital interactions between all nearest-neighbor TM sites on a perfectly cubic lattice. Significantly, these energy-lowering states all exhibit band gaps with respect to the cubic cell with equally occupied degenerate partners, even though no structural relaxation has been performed as yet (see Table I). We conclude here that in LaVO 3 , KFeF 3 , KCoF 3 , KCrF 3 , and KCuF 3 , the electronic structure in the cubic cell distorts in order to remove the orbital degeneracy, thereby producing an antiferro-orbital arrangement and opening a band gap. These compounds therefore exhibit the signatures of a Jahn-Teller effect that is directly related to gapping.

When electronic delocalization of degenerate orbitals prevents the JTE
One might wonder at this stage why compounds with identical electronic degeneracy differ in their ability to have a JTE. For example, LaMnO 3 does not exhibit a JTE whereas KCrF 3 does; similarly, LaTiO 3 has no JTE, whereas KFeF 3 does. To understand this, let us recall that the energy surface of a Jahn-Teller system, as expressed by Baraff et al. in Ref. [51], as a function of a displacement x can be represented as E (x) = −F JT x + 1 2 Kx 2 , consisting of a stabilizing electronic Jahn-Teller force F JT associated with electronic degeneracy removal, and an opposing harmonic restoring force of the surrounding bond characterized by the force constant K-let us note that there might be other linear terms but this is out of the scope of the present paper. That the existence of degeneracy of partially filled states does not automatically force a Jahn-Teller distortion is clear from the competition between these two terms: if the relevant (degenerate) orbitals are too hybridized or delocalized (weak F JT ), or the harmonic response of the bonds about to be deformed is too stiff, degeneracy will not lead to a Jahn-Teller distortion. Strongly (weakly) p-d hybridized systems will tend to have a weaker (stronger) Jahn-Teller force. To assess such tendencies, one must carry out a Jahn-Teller distortion inspecting the total (electron-electron, electron-ion, and ion-ion) energy terms, not just orbital energies. This is readily done in DFT supercell theory.
To assess the difference in hybridization between fluorides and oxides, Fig. 3 shows the projected density of states of isoelectronic pairs (LaTiO 3 versus KFeF 3 , as well as LaMnO 3 versus KCrF 3 ) in the high-symmetry cubic cell with equal occupancies of degenerate partners. As one can see, the fluorine-based compounds show minimal hybridizations between the B-d and X-p states, and the bandwidth associated with degenerate partners is rather narrow, thus resulting in a localized electronic structure and d-d-like band edges. On the other hand, for oxygen-based perovskites, the hybridization between B-d and X-p states and the bandwidth associated with degenerate partners increases continuously upon adding electrons to the d levels, until reaching a "charge transfer insulator" regime for LaMnO 3 -i.e., an anion-p-like valence band maximum and cation-d-like conduction band minimum. This thus suggests that oxides have more delocalized d states than fluorides. Thus, in the oxides the Jahn-Teller force can be too weak and might be overcome by the restoring force but in the fluorides the Jahn-Teller force is stronger and might win. This is what we see in Table I.  To next check how the degree of localization of degenerate orbitals induces Jahn-Teller displacements, we design enhanced localization in LaTiO 3 by increasing the lattice constant, i.e., using a (negative) hydrostatic pressure. Starting from this new cubic cell in which the bandwidth of the t 2g degenerate partners is reduced by 0.5 eV with respect to the unperturbed cubic cell, we again nudge the Ti 3+ electron to specific t 2g partners (see Appendix B). LaTiO 3 now develops a small electronic instability ( E OBS−no OBS = −6 meV/f.u.) producing an antiferro-orbital occupancy between all nearestneighbor Ti 3+ sites and opening a band gap of 0.1 eV. Therefore, upon decreasing B d-X p hybridizations and the t 2g -level bandwidth, LaTiO 3 undergoes a Jahn-Teller effect.
Finally, one may question the choice of the U value used in the present work. Using an enormous U value in DFT + U will always tilt the balance in favor of degeneracy removal. But this has to be judged on the basis of reasonableness of the choice of an arbitrary U . To this end we have used in the current work a rather low U value of 3.5 eV. Increasing for LaTiO 3 U to 5 eV can also result in a weak electronic instability but in fact, it is just associated with an increase of the cubic cell volume. However, such a large repulsion on Ti-3d levels was previously shown [34] to be totally irrelevant for studying LaTiO 3 . In LaMnO 3 no electronic instability is found for any U (5 eV, 7.5 eV). This is likely due to the intrinsic charge transfer insulator nature of this compound (the valence band being oxygen like); increasing U has here the effect of pushing down in energy Mn-d levels and thus increasing anion p-Mn d hybridizations.
We conclude that delocalized degenerate states can prevent an electronic instability and the ensuing Jahn-Teller effect despite the presence of orbital degenerate states. This also explains why SrVO 3 and CaVO 3 , showing strongly hybridized electronic structures, do not develop a JTE and thereby stay metallic in bulk.

Trends in Jahn-Teller energies with orbital occupancy
Although LaVO 3 exhibits B d-X p hybridizations similarly to LaTiO 3 , the former compound possesses a robust electronic instability in the cubic cell (an energy lowering of 297 meV; Table I), while LaTiO 3 has zero stabilization energy. This reflects the fact that the JT force is larger in systems with higher orbital occupancy-LaVO 3 with two electrons, t 2 2g↑ , relative to LaTiO 3 with just one, t 1 2g↑ . Similarly, the energy gain associated with breaking orbital degeneracies increases when going from t 1 2g in KFeF 3 to a t 2 2g in KCoF 3 . In general, one expects that triply degenerate t 2g states occupied by 2 electrons will have greater Jahn-Teller instability than t 2g states occupied by 1 electron because of electron repulsion. Specifically, (i) for a single electron in a triply degenerate t 2g level there is no repulsion within that shell, so electron delocalization on all three levels is possible, weakening JT. On the other hand, (ii) if there are two electrons in t 2g , electron repulsion exists even if one places electrons in different degenerate partners, leading to a driving force for degeneracy removal. We next turn to study the trends in the deformation amplitudes.

Electronically unstable cubic structures develop the Q − 2 octahedral deformation upon relaxation
While the literature often ascribes Q + 2 and Q − 2 modes to a JTE [2,12], we inspect here the type of distortion forced by the spontaneous electronic instability in the hypothetical cubic structure of LaVO 3 , KFeF 3 , KCoF 3 , KCrF 3 , and KCuF 3 . To that end, we allow these systems to change their structures by fully developing energy-lowering displacements. Application of structural relaxation techniques (following quantum mechanical forces to zero force configurations) reveals that all these systems develop specifically the antiphase Q − 2 octahedral deformation that we therefore consider as the fingerprint of the JTE in these systems. We tested the Q + 2 octahedral deformation mode but this octahedral deformation mode produces lower energy gains than the Q − 2 mode (see Appendix C).
Symmetry-adapted mode analysis of the relaxed DFT structures (starting from experimentally observed structures and magnetic orders) are presented in Table II. One can appreciate the development of the specific Q − 2 mode by the presence of a shifted single well potential energy surface associated with the Q − 2 mode starting from a cubic cell, signaling the presence of a Jahn-Teller force F JT (see Appendix D). Note that the shifted single wells are independent of the spin order-FM and AFM orders yield similar resultsand thus the magnetic order at low temperature may be a consequence of the specific orbital orderings forced by the electronic instability, but not its cause. The significance of the development of this specific Q − 2 mode [ Fig. 1(d)] is that it is characterized by opposite octahedral deformations between nearest sites, and is thus able to accommodate and amplify the incipient orbital orderings [Figs. 2(c) and 2(f)]. Specifically, the B-X bonds expand and contract in the plane defined by the alternating directions of the occupied orbitals.
We emphasize that the appearance of an antiphase rotation propagating orthogonally to the Jahn-Teller distortion leads to very small additional components to the symmetry-allowed Jahn-Teller distortion in which O apices move inward or outward (Appendix E explains that the antiphase rotation and the Q − 2 motion displacement patterns describe orthogonal planes in KCrF 3 for minimizing electronic superexchange). Although it is sometimes proposed to come from additional electron interactions [52], it is just a consequence of the X 6 rotation (see Appendix F).

Theory versus experimental observation for the Q − 2 octahedral deformation mode
The predicted specific symmetry of the Q − 2 mode is precisely what is needed to explain the cubic to tetragonal I 4 /mcm structural transition in KCrF 3 and KCuF 3 (observed at T = 973 K and 800 K [53,54], respectively), and of the Pbnm to P2 1 /b structural transition observed in RVO 3 compounds (R = Lu-La, Y) [31,55]. Table II II. Amplitudes associated with structurally distorted octahedra in the ground state of ABX 3 perovskites. Amplitudes of distortions (in Å) associated with the Q + 2 and Q − 2 modes (irreducible representations M + 3 and R − 3 , respectively) distorting octahedra of the optimized structures starting from a cubic cell with the A cation located at the corner of the cell. Experimental values extracted from structures available in literature are provided in parentheses. The Goldschmidt t factor is also reported, as well as the magnetic state observed experimentally at low temperature and assumed in the simulation for the relaxation of the ground states.

C. The Q + 2 motion in LaTiO 3 and LaMnO 3 is a consequence of classic octahedral rotations and tilting, not a fingerprint of an electronic Jahn-Teller effect
Consistently with the absence of electronic instabilities in the high-symmetry Pm3m cubic cell of LaTiO 3  Ref. [59]. This means that although a sizable Q + 2 octahedral deformation mode appears in the ground state structure of LaTiO 3 and LaMnO 3 , this mode is not produced by an electronic instability and does not lift orbital degeneracies or produce the metal-to-insulator transition, all being key aspects of the Jahn-Teller effect. Let us note that the amplitude of the Q 2 octahedral deformation modes reported in Table II are not really different between isoelectronic compounds such as  [60].
Nevertheless, these compounds develop two antiphase and one in-phase X 6 rotation (due to low-t factors; see Table II) that allow a specific lattice mode coupling between the Q + 2 mode plus the antiphase rotation φ − xy and the antipolar motion A X (sketched in Fig. 1 in the free-energy expansion as identified in Ref. [21]. The appearance of finite amplitudes of φ − xy and the A X modes automatically forces finite amplitudes of the Q + 2 mode in order to lower the total energy as we see in Fig. 4(c) for LaMnO 3 . A similar result is found for LaTiO 3 (not shown). Although the dependence of the Q + 2 mode on rotation amplitude in LaMnO 3 was already shown in Ref. [59], we show that the Q + 2 is nothing but a consequence of ABX 3 distortions originating from pure steric atomic size effects captured already by the 1926 pre-quantum Goldschmidt tolerance factor [13]. In analogy with improper ferroelectrics [61,62], the Q + 2 mode can be thought of as an improper mode, being a result of a specific octahedral rotation pattern: a combination of two antiphase and one in-phase rotation. In contrast to often articulated statements [3,5,30,56,63] this Q + 2 motion should not be confused with a Jahn-Teller distortion originating from an electronic instability. The improper mode origin of Q + 2 is consistent with previous theoretical reports of strong Q + 2 mode amplitude in compounds that lack degenerate levels [22], clearly signaling that this mode cannot originate from an electronic instability.

The appearance of a Q + 2 mode is not a statement of strong dynamical correlation effects
Leonov et al. [3] highlighted the importance of dynamical correlations in stabilization of the Q + 2 mode in LaMnO 3 . They plotted the potential energy surface as a function of the Q + 2 mode amplitude with dynamical mean field theory (DMFT) simulations starting from a cell with rotations and the A X mode, explaining that "in the calculation we change only the parameter ∂ JT (i.e., the amplitude of the Q + 2 mode) … and keep the value of the MnO 6 octahedron tilting and rotation fixed." As can be seen by comparing our mean-field DFT result of Fig. 4(c) with their DMFT results (Fig. 8 of Ref. [3]), the two agree closely, suggesting that in this case, adding dynamic electronic correlations does not bring any new features to the understanding of the octahedral deformation in ABX 3 materials.

The origin of the improper Q + 2 motion in LaMnO 3
Our first-principles results demonstrate that the Q + 2 mode is a simple consequence of octahedral rotations, but one wonders whether one can extract experimental evidence of this phenomena. At high temperature, LaMnO 3 adopts a rhombohedral R3c phase that is characterized by an antiphase rotation φ − xyz around all Cartesian axes and in which no octahedral deformation or related phenomena such as an orbital ordering are reported [30,64]. This is compatible with our models since (i) no electronic instability yielding a Q − 2 Jahn-Teller distortion mode is identified in LaMnO 3 and (ii) such X 6 octahedral tilt pattern does not allow A X and Q + 2 modes to develop by symmetry. Once LaMnO 3 transforms to the Pbnm cell around 750 K, one observes that experimental structures at various temperature taken from Ref. [30] develop an A X distortion whose amplitude increases with decreasing temperature, while φ − xy is not temperature dependent. Upon increasing of the A X mode amplitude on cooling, the Q + 2 mode amplitude increases (see Appendix G for symmetry-adapted modes of these structures). This thus confirms the "improper appearance" of the Q + 2 octahedral distortion mode.

The origin of the orbital ordering in RTiO 3 (R = Lu-La, Y)
Along with LaMnO 3 , LaTiO 3 is often taken as a textbook example of compounds exhibiting a Jahn-Teller effect. Just as in Ref. [34], we again show here that LaTiO 3 exhibits a very weak improper Q + 2 octahedral deformation mode. The single Ti-d electron is localized in an αd xy + βd xz + γ d yz orbital whose α, β, γ coefficients depend on the sterically induced octahedral rotation, thus resulting in the various orbital orderings as a function of the rare-earth ionic radius (i.e., of the amplitude of the induced octahedral rotations) as demonstrated by some of the present authors in Ref. [34].

D. Competing Jahn-Teller effect and sterically induced Q + 2 mode is at the core of the entangled spin-orbital properties in RVO 3 compounds
Although Q + 2 and Q − 2 distortions have totally different origins, they can coexist as long as the Q + 2 mode is allowed by symmetry. This is the case in LaVO 3 for which one observes finite amplitudes of the two modes in the low-temperature phase both at the theoretical and experimental levels (see Table II), although the Jahn-Teller distortion largely dominates. However, among the RVO 3 (R = Lu-La, Y) compounds, LaVO 3 shows the smallest octahedral rotations. One may question what happens for a compound showing larger X 6 rotations such as YVO 3 .
Consistently with previous literature [21,55], full structural relaxation (atomic position + cell parameters) of YVO 3 yields a Pbnm ground state showing only the Q + 2 mode at 0 K. Interestingly, through appearance of the Q + 2 mode, a columnar orbital pattern is stabilized instead of the checkerboard pattern associated with the Jahn-Teller effect. Consequently, superexchange paths are enabled along the z axis thereby creating AFM interactions along the "columns." Such a behavior is indeed verified in our simulations: when starting from a ( √ 2a, √ 2a, 2a) cubic cell of YVO 3 with specific initial electron nudging (similar to those presented in Sec. III A), we find after variational self-consistency that the columnar orbital pattern is more stable than the checkerboard arrangement when considering the G-type AFM order ( E che−col = +31 meV/f.u.), and conversely for the C-type AFM order ( E che−col = −74 meV/f.u.). In other words, each Q 2 mode is associated with a specific orbital pattern, and consequently to a precise spin order.
Nevertheless, the checkerboard orbital pattern remains the global energy minimum in the cubic cell (between the G and C type AFM orders), a similar conclusion being drawn with a FM order ( E che−col = −34 meV). Therefore, YVO 3 should exhibit the signature of a Jahn-Teller effect although the low-T phase only shows the Q + 2 mode. This is verified experimentally by the presence of an intermediate P2 1 /b symmetry characterized by the Q − 2 mode as shown by the symmetryadapted mode analysis presented in Table I of Ref. [21]. Thus, the question is now, what is the driving force of the transition to the purely orthorhombic Pbnm cell at 0 K for YVO 3 ?
To get insights on this peculiar transition, we start from the high-symmetry ( √ 2a, √ 2a, 2a) cubic cell of YVO 3 and we freeze fixed amplitudes of all distortions appearing in the ground state structure except the Q + 2 mode; i.e., we add only octahedral rotations and antipolar motions of ions. Then, we compute the potential energy surface associated with Q + 2 or Q − 2 modes (see Fig. 5). Upon increasing "orthorhombic distortions," two antagonistic effects are observed irrespective FIG. 5. Energy difference (in meV per 20-atom unit cell) as a function of the Q + 2 and Q − 2 mode amplitudes (in fractional units) at fixed rotation and antipolar displacement amplitude (in %) appearing in the ground state structure of YVO 3 using the AFMG (a) and AFMC (b) order. The unit cell is fixed to a ( √ 2a, √ 2a, 2a) cell whose lattice parameter yields the ground state volume. Here 1 is actually the amplitude appearing in the ground state structure.

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of the AFMG or AFMC magnetic order: (i) the Q + 2 mode describes a single well potential whose energy minimum is progressively shifted to larger amplitudes while (ii) the Q − 2 mode vanishes. The former observation is in line with the improper origin of the Q + 2 distortion discussed in Sec. III C but the latter finding indicates that orthorhombic distortions produce a crystal field sufficient to split the t 2g -level degeneracies [10] but also diminish electronic superexchange interaction at the core of the Jahn-Teller effect (see Appendix H).
Our results are closely compatible with the experimental phase diagram of rare-earth vanadates [31,55] and settle the issue that Q − 2 and Q + 2 have totally different origins, the former being the signature of the Jahn-Teller effect while the latter is just a consequence of the ubiquitous octahedral rotations appearing in perovskites. RVO 3 are then unique materials in the sense that they are the only oxide perovskites with a B-3d element showing a Jahn-Teller effect plus the sterically induced Q + 2 mode; thereby they are the only members possessing entangled spin-orbital properties associated with a JTE.

IV. CONCLUSIONS
We have explained the modalities enabling a Jahn-Teller effect in ABX 3 perovskites and identified its signature: strongly localized electronic states are the key factor for a spontaneous electronic instability to produce a Jahn-Teller distortion with a specific octahedral deformation pattern that is experimentally detectable and detected. In materials with larger anion-p and B-cation-d state hybridizations, there is no electronic instability that can break orbital degeneracies and the observed octahedral deformation has a distinct symmetry that is pushed by lattice mode couplings with rotations. The Q + 2 octahedral deformation, appearing in LaMnO 3 or LaTiO 3 , is not induced by an electronic instability and thus it is not a Jahn-Teller distortion. It is a consequence of octahedral rotations and tilts often appearing in the perovskite oxides due to pure semiclassical atomic size effects, i.e., geometric steric effects.
Our work provides a single theory explaining the Jahn-Teller effect and its specific signatures and reconciles the numerous experimental results of ABX 3 materials studied to date. Last but not least, our results settle the fact that dynamical correlations, and the Mott-Hubbard model, are absolutely not an essential aspect of the physics of ABX 3 materials showing insulating states despite the presence of degenerate states in the parent cubic cell, thereby defining DFT as a sufficient platform to study the physics of ABX 3 materials.
Some details of the method. DFT calculations are performed with the VASP package [43,44] and electrons are nudged using the modified VASP routine [45]. For exploration of orbital broken states using the HSE06 functional in cubic cells, a preconverged wave function is initially obtained using DFT + U with the orbital occupancy control routine and symmetry off for the wave function. Starting from this initial guess, we then perform HSE06 calculation allowing the solver to update orbital occupancies, still without symmetry of the wave function. We used PAW potentials with the outer 4s, 4p, and 3d B-cation electrons explicitly treated in the simulations. Since electronic instability is sensitive to the pressure, we determine the equilibrium volume of cubic cells. Full structural relaxation (lattice parameters plus cell internal coordinates)-reduced to lattice parameters for cubic cell relaxations-is performed until forces are lower than 0.5 meV/Å (increased to 0.01 meV/Å for cubic cell relaxations). We use a 6×6×4 (8×8×8) k-point mesh for the relaxation of the ( √ 2a, √ 2a, 2a) cubic cells (primitive cubic cells), and increase to 8×8×6 for plotting energy potential surfaces and seeking OBS, accompanied by an energy cutoff of 500 eV. Energy potential surfaces were plotted with symmetry of the wave function off, in order to allow electrons to occupy the lowest energy state.

APPENDIX A: TOTAL ENERGIES OF ORBITAL BROKEN SYMMETRY STATES IN CUBIC CELLS
We have performed total energy calculations starting from ( √ 2a, √ 2a, 2a) cubic cells and preconverged solutions in which electrons are nudged in specific degenerate partners, forming either (i) a single pattern with the same occupied orbital on all transition metal sites, (ii) a "columnar arrangement" with alternating orbitals in the xy plane (such as d x 2 /d y 2 for e g degeneracies or d xz /d yz for t 2g degeneracies) with similar stackings along the z axis, or (iii) a "3D checkerboard pattern" with alternating orbital occupancies between nearestneighbor transition metal sites. Results for KFeF 3 , KCoF 3 , KCrF 3 , KCuF 3 , and LaVO 3 are presented in Table III for GGA + U simulations.

APPENDIX B: DENSITY OF STATES OF LaTiO 3 UNDER NEGATIVE PRESSURE
In order to check the role of electron localization on the Jahn-Teller effect, we have compared the projected density of  states of LaTiO 3 with and without a negative pressure (i.e., a volume expansion). Under a volume expansion, LaTiO 3 still shows minimalistic hybridizations between O p-Ti d levels but the bandwidth W associated with t 2g levels is strongly reduced with respect to the unperturbed cubic cell (see Fig. 6). Following the reduction of the bandwidth, a narrow electronic instability able to break orbital degeneracies and to produce a G-type antiferro-orbital arrangement is observed ( E = −6 meV/f.u.).  from Pm3m to P 4 /mbm) mode and we performed a full structural relaxation of lattice parameters and atomic positions using a FM order. Energy differences with respect to the perfectly cubic cell (in meV per 20-atom unit cell) for the two lattice modes are reported in Table IV for LaVO 3 , KFeF 3 , KCoF 3 , KCrF 3 , and KCuF 3 . The Q − 2 lattice mode always produces the largest energy gain in these compounds, although Q + 2 and Q − 2 are nearly metastable in KCuF 3 (note that the AFMA order slightly increases the energy stabilization of the Q − 2 mode by 2 meV/f.u. in KCuF 3 ).

2
We report in Fig. 7 the energy gain associated with the condensation of the Q − 2 mode amplitude in a cubic cell (the 0 is set to the perfectly cubic cell) using a FM order. We note that in Fig. 7 no cell relaxation was performed and lattice parameters were fixed to the cubic cell, unlike the results reported in Table IV where full geometry relaxation was performed yielding a tetragonal cell. As one can see, KFeF 3 , KCoF 3 , LaVO 3 , KCrF 3 , and KCuF 3 are characterized by shifted single well energy potentials signaling the existence of a Jahn-Teller force F JT , a signature of a JT effect.

MOTION DISPLACEMENT PATTERNS DESCRIBE ORTHOGONAL PLANES IN KCrF 3 FOR MINIMIZING ELECTRONIC SUPEREXCHANGE
At low temperature, KCrF 3 shows a Jahn-Teller distortion (JTD) producing a F displacement pattern mostly along the x and z directions while the antiphase rotation produces F displacements along the x and y directions. Therefore, both motions produce displacements of ions in orthogonal planes, the xy and xz planes for the rotation and JTD, respectively. This situation is not fortuitous and it originates from electronic superexchange minimization. In the perfectly cubic cell, the Jahn-Teller effect produces a purely antiferro-orbital arrangement with orbital pointing along orthogonal directions between nearest-neighbor cation sites. This situation is sketched in Fig. 8(a) and it corresponds to the energy minimum of the electronic superexchange model, similarly to the Heisenberg model describing spin interactions (see chapter 6 of Ref. [2] for a detailed description of the Kugel-Khomskii model). Introducing a rotation producing atomic displacements coplanar with JTD atomic displacements, i.e., coplanar to the alternating orbital directions such as the situation sketched in Fig. 8(b), "orbitals directions" are not orthogonal anymore thereby weakening electronic superexchange. A rotation producing F displacements orthogonal to those described by the JTD [and so the alternating directions of orbitals such as the situation presented in Fig. 8(c)] preserves orthogonality between "orbital directions" and therefore the electronic superexchange is still minimized. We have checked this hypothesis in our simulations and we indeed observe that starting from a cubic cell in which we add the ground state rotation, the electronic instability associated with the situation presented in Fig. 8(c) is slightly larger than the situation presented in Fig. 8(b) by 6 meV/f.u. In conclusion, the antiphase rotation and the JTD propagate along orthogonal axes in KCrF 3 for the purpose of electronic superexchange minimization.

APPENDIX F: LOW-TEMPERATURE ORBITAL AND STRUCTURAL TRANSITION IN KCrF 3
In KCrF 3 , the energy gain associated with the φ − z antiphase octahedral rotation is one order of magnitude smaller ( E rot = −32 meV/f.u.) than the energy gain produced by the electronic instability ( E = −124 meV/f.u.), further amplified by the JT distortion ( E JTD = −140 meV/f.u.). Thus, the rotation appears at a temperature far below the JT effect. This is indeed observed experimentally: the material transits from a tetragonal to an I 2 /m monoclinic cell at T = 250 K, a temperature much lower than that of the JT effect (T = 973 K). In the I 2 /m symmetry, the octahedral rotation and the JT distortion have orthogonal atomic displacement patterns: X 6 groups rotate around the z axis-F motions along the x and y axes-while the JT distortion propagates along the y axis-F motions along the x and z axes. Such a peculiar orientation of propagation axes minimizes the electronic superexchange at the core of the JT effect (Appendix E) and enables by symmetry an extra component to the JT distortion in which X -apex anions move inward or outward the B cation, therefore very slightly modifying orbital shapes. However, unlike previous statements involving strong correlation effects, Slater superexchange, or covalent effects [52] for explaining the tetragonal to monoclinic transition, it is driven by nothing  other than a steric effect (a non magnetic (NM) simulation also provides a stabilization of the rotation mode). Let us note that the different orientations of occupied orbitals between the tetragonal (alternation of d x 2 and d y 2 orbitals) and monoclinic (alternation of d z 2 and d y 2 orbitals) cells simply originate from an arbitrary choice of different crystallographic parameters between the two symmetries.

APPENDIX G: SYMMETRY MODE ANALYSIS OF LaMnO 3 EXPERIMENTAL STRUCTURES
We have performed symmetry mode analysis of LaMnO 3 structures [30] extracted experimentally at different temperatures in order to get the amplitude associated with each individual distortion, most notably the antiphase φ − xy and inphase φ + z rotations, the antipolar A X motion, and the Q + 2 octahedral deformation. A high-symmetry cubic cell is set as the reference structure with A cations located at the corner. Results are presented in Table V.

APPENDIX H: COOPERATING AND COMPETING OCTAHEDRAL ROTATIONS AND JAHN-TELLER EFFECT IN LaVO 3
In the spirit of the electronic superexchange minimization analysis we have performed in KCrF 3 for explaining the interplay between the rotation and the JTD (Appendix E), we can perform such an analysis for LaVO 3 . For the purpose of electronic superexchange minimization, the JTD prefers to share coplanar atomic displacements with the in-phase rota-tions since such situation preserves orthogonality of orbital directions between consecutive planes along the propagation axis of the distortion. Our first-principles simulations confirm this hypothesis and starting from a cell with either a single in-phase or antiphase rotation, we observe a slightly larger energy gain by breaking orbital degeneracies for the in-phase rotations than for the antiphase rotations ( E = −9 meV/f.u. between the two solutions).
We nevertheless observe a surprising behavior with the amplitude of the in-phase rotation: the electronic instability breaking orbital degeneracies slightly increases upon increasing the rotation amplitude and then decreases for large rotation amplitude [ Fig. 9(a)]. This observation is strictly compatible with the experimental phase diagram of RVO 3 compounds [31]: the JT effect temperature as a function of the tolerance factor (i.e., the rotation amplitude) follows the strength of our computed electronic instability as a function of the rotation amplitude. This a priori unexpected behavior has three origins: (i) The rotation amplitude first splits the degeneracy of the threefold-degenerate t 2g partners and the d xy orbital is pushed to lower energies, leaving degenerate d xz /d yz orbitals at higher energies [see Fig. 9