Michael Pollet

Alla Artemenko

Spin State Diagrams of 3d n Cations in Tetragonally Distorted Octahedral Sites

The possible existence of an intermediate spin state is now a few decades fascinating solid state chemists and physicists, experimentalists, and theoreticians. In this article, we revise some recent results on the stability diagrams of spin states, low, high, or intermediate, in distorted environments and extend their approach to redraw more realistic diagrams for d 4 ,d 5 , and d 6 ions in a tetragonally distorted 6-fold oxygen environment (D 4h , D 2d , and C 4ν ). The model relies on a point charge model and further uses effective parameters to account for the cubic field drift on spin state change and for suitable values for solid state of the expectation values of the 3d-radial wave functions; additionally the model uses rational parameters to characterize the distortion; finally, we also consider the possible existence of states' combinations to propose reliable stability diagrams. Whatever the representation involved in the distortion, the existence domain of the intermediate spin state appears very small and more likely replaced with mixtures of cations in low and high spin states; the opportunity of induced distortive ordering is discussed.

INTRODUCTION

Physical properties of compounds containing transition metal cations closely depend on the electronic configuration of the ions. In a solid state chemistry approach and looking for new and exciting properties, one requires as a first step of the development to anticipate the electronic configuration of the transition metal cation in specific crystallographic environments. There are mostly two schools for the description of the physical properties of compounds containing transition metal cations: a band model approach school, which considers localization effects with effective parameters and an atomic (or ionic) model approach school, which considers additional parameters like, e.g., covalent bonding also using effective parameters. In a seminal work, Hubbard attempted to reconcile both schools as "the d electrons of transition metals exhibit behaviour characteristics of both the ordinary band model and the atomic model." [START_REF] Hubbard | Electron Correlations in Narrow Energy Bands[END_REF] For both schools, the use of effective parameters is clearly compulsory to be able to account for the properties of real materials. In a series of articles, Pouchard et al. [START_REF] Pouchard | Spin State Behavior in Some Cobaltites (III) and (IV) with Perovskite or Related Structure[END_REF] (hereafter simply referred as Pouchard) have developed models in the crystal field approximation to picture how simple distortions, and thus the ligand environment, impact on the spin state of transition metal cations. In particular, for d [START_REF] Ballhausen | Introduction to Ligand Field Theory[END_REF][START_REF] Watson | Iron Series Hartree-Fock Calculation II[END_REF][START_REF] Richardson | Approximate Radial Functions for First-Row Transition-Metal Atoms and Ions. I. Inner-Shell, 3d and 4s Atomic Orbitals[END_REF] ions, their results clearly evidence the existence of domains where an intermediate spin state can be stabilized. Such unusual state coming with the promise of new/interesting properties and/or physics, it has already been long debated and although several authors claimed they could observe this unusual state, it still appears highly controversial. The approach of Pouchard is based on the use of the point charge model taking into account the mono-and bielectronic terms; the latter include both the interelectronic Coulomb repulsion energies (U, U′) and the intra-atomic exchange energy (J H ≈ U -U′), as defined by Kanamori and Brandow what allows building (k, J H /Dq) spin state diagrams where k stands for the distortion parameter and Dq is the cubic field parameter. Three kind of domains are observed in such diagrams with either low, intermediate, or high spin state (respectively, LS, IS, and HS). The crystal field model has largely proved its efficiency to describe many aspects of the solid, and it generally offers a strong basis for discussion. The size of the IS state domain that Pouchard reports is then sufficiently large that one can wonder why such state has so few been reported until now. Recently, Lamonova et al. [START_REF] Lamonova | Intermediate-Spin State of a 3d Ion in the Octahedral Environment and Generalization of the TanabeSugano Diagrams[END_REF] proposed an alternative scheme using a modified crystal field approach and an effective nuclear charge at the transition metal cation to draft in (Z eff , k′) planes the spin state diagrams of 3d ions in distorted octahedral complexes (k′ representing the most typical octahedral distortions). Though interesting, their approach remains in essence very close to the one of Pouchard, and their results nicely compare with similar extension of the domains. In this article, we follow the approach of Pouchard, but we improve their description of the crystal field parameters and extend the results to other tetragonal distortions using the formalism of phase transitions for the distortion parameter.

THEORETICAL BASIS AND DESCRIPTION OF THE EFFECTIVE PARAMETERS

As it is well-known, the crystal field interaction corresponds to the electrostatic coupling between the shell (here 3d) and its surroundings. Its treatment is only shortly reminded hereafter [START_REF] Ballhausen | Introduction to Ligand Field Theory[END_REF] and a more detailed calculation is proposed in Appendix I. The potential at the transition metal cation due to the ligands is proportional to the Green's function for the Laplace operator and is given by V = ∑ i eZ i r ij -1 where i runs over the nearest ligands. Dealing with orbitals, a more convenient form is obtained using the Laplace expansion r ij -1 = ∑ l ∑ m=-l l ((4π)/(2l +1))(r < l /r > l+1 )Y lj m Y li * m ; r </> are min/max(r i , r j ), which reduce in the crystal field model to the distance from a point j to the transition metal cation and to the metal-ligand distance; Y l m is a normalized spherical harmonic. Handling d orbitals, all terms with n > 4 vanish and focusing on energy levels (E ∝ ∫ Ω Y 2 m ′VY 2 m dΩ with Ω the solid angle), only even terms remain. Rewriting the previous expansion as V = ∑ l ∑ m V l m and taking further into account the present tetragonal symmetry constraints, the only nonzero terms of the potential acting on the crystal field are diag ( 1, 4, 6, 4, 1) antidiag(1, 0, 0, 0,

a, b, and c, expressed in Dq unit (10Dq = 5/3•Ze⟨r 4 ⟩/r 0 [START_REF] Watson | Iron Series Hartree-Fock Calculation II[END_REF] , where r 0 is the metal-ligand distance in octahedral field), depend on the details of the symmetry: where the sum runs over the 6 ligands. Here, b and c depend only on the distortion parameter (γ D 4h , α D 2d ... described in next section), while a depends in addition on the ratio of the second to the fourth moments of the radial distribution of the 3d electrons with p = r 0 2 ⟨r 2 ⟩/⟨r 4 ⟩.

∑ ∑ ∑ θ φ θ φ θ φ = - = = ⎡ ⎣ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ ⎡ ⎣ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ ⎡ ⎣ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ a p r d Y D q b r d Y D q c r d Y D q 6 7 ( , ) 2 7 ( ,
The orbital energies are given by In the case of O h symmetry, a = 0, b = 1, and c = 5 leading to the expected t 2g triplet (E t 2g = -4Dq) and e g doublet (E e g = 6Dq).

= =-- = = + - = = + + = = -+ - E E a b E E a b c E E a b c E E a b
The bielectronic contribution is calculated by considering all the possible electron pairs' interactions and assuming that the interelectronic Coulomb repulsion energies (U, U′) and the intra-atomic exchange energy (J H ≈ U -U′) are independent of the occupied orbitals.

Considering for instance an e g 4 b 2g 2 a 1g 0 b 1g 0 LS state, the monoelectronic contribution is given by summing the individual energy over all the occupied orbitals E mono-e

LS = 4E e g + 2E b 2g = 4(-a -4b) + 2(2a + b -c) = -14b -2c.
To determine the bielectronic contribution (see Appendix II), one can see that all 6 electrons interact with one electron in the same orbital (U) and 4 electrons in different orbitals (4U′); in addition, each electron interacts with two electrons with parallel spin. This results in

E bi-e LS = 6(1/2)(4U′ + U -2J H ) = 3(4(U - 2J H ) + U -2J H ) = 15U -30J H (the (1/2)
term is to prevent the double counting of the interaction energies). The total energy is finally E LS = E core -14b -2c + 15U -30J H where the core energy E core (interaction of the 3d electrons with the nucleus and the [Ar] core electrons) is assumed constant. Repeating such treatment for the IS and the HS states, we obtain E IS = E core -4a -9bc + 15U -33J H and E HS = E core a -4b + 15U -38J H . Generally, the total energies of each spin state, and further each possible orbital configuration, are then compared to determine the stability domain of each state. For instance, in the case of the e g b 2g a 1g b 1g configuration, the LS state is expected to be more stable than both the IS and HS states if {E LS < E IS and E LS < E HS }, i.e., {J H < (1/3)(-4a + 5b + c) and J H < (1/8)(-a + 10b + 2c)}.

In these expressions, several points have to be highlighted: (i) The spin state of the cation depends on the crystal field strength, and the cubic field strength parameter Dq is thus different in the several spin states. Experimental data for Dq in the several spin states are rare or even absent for the IS state. The cubic field for a given 3d ion in HS state is typically ranging between 50 and 80% of its value in LS state. Speculating on the IS cubic field value, several points should be taken into account: (i) the IS state ion size is expected to range between the LS state and HS state ion sizes; [START_REF] Radaelli | Structural Phenomena Associated with the Spin-State Transition in LaCoO 3[END_REF] (ii) for a d 6 ion in O h symmetry, e g orbitals are empty only in LS state; (iii) the nephelauxetic effect is stronger for e g orbitals. [START_REF] Jorgensen | Orbitales Molećulaires Dans les Chromophores Mineŕaux et Parametres Angulaires e de Liaison Covalente[END_REF] These observations tend to show that Dq IS should have a midvalue or even be closer to Dq HS . We define two effective parameters ρ I and ρ H , which represent the ratios of the cubic field strength for intermediate or high spin state cations to the cubic field strength for the low spin state cation. The stability conditions of the LS state in e g b 2g a 1g b 1g configuration for instance thus rewrites {J H < (1/3)[(-4a -9bc)ρ I + (14b + 2c)] and J H < (1/8)[(-a -4b)ρ H + (14b + 2c)]} in Dq LS units.

(ii) The moments of the radial distribution of the 3d electrons are generally taken from calculations for the free ions using Hartree-Fock radial wave functions [START_REF] Watson | Iron Series Hartree-Fock Calculation II[END_REF] or double ζ Slater type orbitals, [START_REF] Richardson | Approximate Radial Functions for First-Row Transition-Metal Atoms and Ions. I. Inner-Shell, 3d and 4s Atomic Orbitals[END_REF] which are comparable within ∼10%; however, several authors argued that the Hartree-Fock values are not appropriate to solid state and determined experimental ratios ⟨r 2 ⟩/⟨r 4 ⟩ 3-5 times lower than the calculated ones. [START_REF] Zhao | Orbital Theory for Mn 2+ -Complex Ions in Crystals[END_REF] For the lanthanide series, it was earlier shown that Hartree-Fock calculations largely overestimate ⟨r 2 ⟩, while they underestimate both ⟨r 4 ⟩ and ⟨r 6 ⟩; 9 this was attributed to the 4f orbital expansion in crystals and the screening effect by 5s 2 p 6 electrons (see Burns for more details [START_REF] Burns | Shielding and Crystal Fields at Rare Earth Ions[END_REF] ). The first argument, which is closely related to the nephelauxetic effect, [START_REF] Jorgensen | Orbitales Molećulaires Dans les Chromophores Mineŕaux et Parametres Angulaires e de Liaison Covalente[END_REF] holds for 3d electrons. Leavitt et al. [START_REF] Leavitt | Rare Earth Ion-Host Crystal Interactions 3. Three-Parameter Theory of Crystal Fields[END_REF] have proposed to use a factor τ to account for the expansion of the radial wave function in crystals: R cryst (r) = NR HF (τr) (N is the normalization coefficient); the Slater integrals then become F k = τF HF k , and the moments of the radial distribution of the 3d electrons become ⟨r k ⟩ = ⟨r k ⟩ HF /τ k . With such definition, p = τ 2 p HF and takes Z eff into account. [START_REF] Jorgensen | Orbitales Molećulaires Dans les Chromophores Mineŕaux et Parametres Angulaires e de Liaison Covalente[END_REF] Using for the strong field case 10Dq = 2 eV, one finds ⟨r 4 ⟩ ≈ 17(a.u.) and hence an average τ ̅ ≈ 0.6 (with Z = 2; d MO = 2 Å ́); similarly, using the experimental reduced Racah parameter 13 B (= F 2 -5F 4 ), τ ̅ ≈ 0.63. From these results, we define an effective p ̅ ≈ p HF /3 ≈ 1.6 (d MO = 2 Å ́) for 3d [START_REF] Ballhausen | Introduction to Ligand Field Theory[END_REF][START_REF] Watson | Iron Series Hartree-Fock Calculation II[END_REF][START_REF] Richardson | Approximate Radial Functions for First-Row Transition-Metal Atoms and Ions. I. Inner-Shell, 3d and 4s Atomic Orbitals[END_REF] ions. (Note that using the experimental diamagnetic susceptibilities for ions [START_REF] Konig ; Hellwege | Magnetic Properties of Coordination and Organometallic Transition Metal Compounds[END_REF] (with χ dia = -⟨r 2 ⟩Ne 2 Z i /6mc 2 ), τ ̅ ≈ √2, and estimations from p ≈ -3/ 2Z/d MO(Å ́)3 χ (10 -6 emu/mole) dia /Z i Dq (eV) ) would give p values 10-30 times smaller than the Hartree-Fock values.) (iii) Generally speaking, on building spin state diagrams, the possibility of combination/mixture of states is not taken into account. Here, we consider the opportunity of stabilizing a regular states (LS, HS) mixture rather than an IS state (though theoretically possible, the opportunity to mix regular and IS states is not considered in this article). In the treatment of the mixed state domains, we restrict the analysis to the simplest case of 1:1 mixtures. Such case can be treated by comparing for each possible distortion k and at constant J H /Dq the total energy for 2 transition metal cations with IS state E t IS =2E IS (k) to the total energy of a couple of transition metal cations with regular (LS or HS) states authorized to be more or less distorted and having a total energy E t mix = E LS/HS (k ∓ δk)+ E LS/HS (k ± δk). This can be understood as a strain compensation: for instance, while one of the initial IS state cations inflates more than expected on evolving from IS to HS state, the second one releases the excess strain by deflating on evolving from IS to LS state; because this occurs at constant J H / Dq, this release cannot occur at constant distortion (meaning otherwise that the IS state is initially not the most stable). Another possibility of stabilization of a regular states (LS, HS) mixture arises from the partial transfer of the coupling energy from an IS state ion to another one leading to a mixture of LS(J H /Dq(1 ∓ δj)) and HS(J H /Dq(1 ± δj)) states. Such situation at constant distortion can be understood as a balanced action of Γ 1 + breathing mode: on one side, an IS state cation environment is relaxed, and on the other side, a second IS state cation environment is constrained.

DESCRIPTION OF THE ORDER PARAMETERS

We are focusing on the subgroups of O h resulting from a strain induced or displacive transition leading to a tetragonal distortion of a 6-fold coordinated transition metal (respectively, Wyckoff positions 3c and 1b of Pm3̅ m).

Dealing with point groups, we are only concerned with irreducible representations of Pm3̅ m at the Γ point. The point groups of interest were determined using the computer program ISOTROPY. 16 With such constraints, only three tetragonal subgroups have to be considered: one strain induced, D 4h ,-and two displacive induced, D 2d and C 4v ; Figure 3 illustrates these descents in symmetry. The irreducible representations involved in the transitions from O h are Γ 3 + , which is a pure strain mode, and Γ 5

-and Γ 4 -, which are displacive modes.

The first mode, Γ 3 + , corresponds to a compression (dilatation) along the 4-fold axis coupled with an isotropic dilatation (compression) in the m z plane (Figure 1) leading to the D 4h symmetry. One single parameter γ acting only on the metal-ligand distance is sufficient to describe the change in the environment (Table 2).

The second mode, Γ 5 -, corresponds to the activation of the E u mode with an antiphase shift of the adjacent ligands (alternatively up and down) in the square base (Figure 1) leading to the D 2d symmetry. One single parameter α acting only on the elevation angle is sufficient to describe the change in the environment (Table 2).

The last mode, Γ 4 -, leads to the C 4v symmetry. It implies several modes and two distortion parameters α and γ acting on the metal-ligand distance and the elevation angle, respectively, have to be considered depending on the active modes T 1u , which triggers the transition metal cation (Z t ), while E u and A 2u trigger, respectively, the equatorial (Z e ) and apical (Z a ) ligands (Figures 1 and9). By fixing the reference at the transition metal cation position one can define two distortion parameters α and γ expressed as t g α = (z ez t )/r 0 and γ = (z az t )/r 0 (Table 2). T 1u acts on both α and γ, while E u acts only on α and A 2u acts only on γ.

Note that Γ 3 + can superimpose to the displacive modes while keeping the overall symmetry (either D 2d or C 4ν ); the breathing mode Γ 1 + can also superimpose in all cases.

EFFECT OF THE D 4h DISTORTION ON THE SPIN STATE OF 3d n IONS

The D 4h distortion is due to the Γ 3 + strain mode; it implies both an elongation (compression) of the octahedron along a 4-fold axis and a compression (elongation) in the perpendicular square base.

With the definitions in eqs 1a, 1b, and 1c and Table 2, the crystal field parameters are given by

γ γ γ γ γ = - + - - = + + - = - - - - - - a p b c 12 7 [(1 2 ) (1 ) ] 1 7 [4(1 2 ) 3(1 ) ] 5 (1 ) 3 3 5 5 5
(2)

In the following, we constrain γ to [-1/8;1/7], i.e., the ratio of the out-of-plane to the in-plane metal-ligand distances ranges from 2/3 to 3/2 (these values are chosen to restrict the analysis to reasonable parameters, ;however, selecting any other value in a physically meaningful range hardly impacts on the result). The orbital energies are given by eq 1d and are plotted in Figure 2a. In the case of d 6 ions, the reversal of the e g doublet (d xz , d yz ) and the b 1g singlet n (d x 2 -y 2 ) for strong negative strain also prevents the existence of the IS state and the S = 1 state is here also the actual LS state with configuration b 2g 2 b 1g 2 e g 2 . Note that within the limits chosen for γ, the (e g /b 1g ) reversal only occurs for p ≳ 1.866 (Figure 2b).

The energy of the mixtures is calculated at constant total distortion:

2(1 + 2γ) = (1 + 2γ -) + (1 + 2γ + ) and equivalently 2(1 -γ) = (1 -γ -) + (1 -γ + ) (see eq 2); setting γ ± = (2 + κ)γ then γ ∓ = -kγ.
Looking at the mixture stability at, e.g., κ = 0, then simply corresponds to comparing the energy of a given state at γ distortion to the energy of a mixture of regular states at, respectively, 2γ and γ = 0; such situation corresponds for instance to a chessboard of (almost) regular and overdistorted octahedrons. If 2γ ∉ [γ min ; γ max ], the overdistortion is set at the limit and the under-distortion is adapted to maintain the overall distortion constant. The value of κ was checked to only affect the details of the domains' contours by increase of the mixed state domains' size, and only the results at κ = 0 are presented.

Figure 3 shows the diagrams for d [START_REF] Ballhausen | Introduction to Ligand Field Theory[END_REF][START_REF] Watson | Iron Series Hartree-Fock Calculation II[END_REF][START_REF] Richardson | Approximate Radial Functions for First-Row Transition-Metal Atoms and Ions. I. Inner-Shell, 3d and 4s Atomic Orbitals[END_REF] ions. The results account for the use of the effective parameters described: (i) p = p HF /3 = 1.6 (average value based on 10Dq = 2 eV to determine τ and on the ionic radius of d [START_REF] Ballhausen | Introduction to Ligand Field Theory[END_REF][START_REF] Watson | Iron Series Hartree-Fock Calculation II[END_REF][START_REF] Richardson | Approximate Radial Functions for First-Row Transition-Metal Atoms and Ions. I. Inner-Shell, 3d and 4s Atomic Orbitals[END_REF] ions in LS state to determine the transition metal cation-oxygen bond length); (ii) ϱ H = 3/4 and ϱ I = 7/8 (i.e., without loss of generality in the results, Dq IS = (1/2)(Dq LS + Dq HS ); and (iii) possibility of regular states mixtures (γ min = -1/8, γ max = 1/7, and δj = 10%).

In the insets, the results are based on the approach of Pouchard (p = 2, 7 ϱ H = 1, and ϱ I = 1); our results are almost similar to theirs except that the borders are not linear and that the diagrams are extended on the left side to account for negative strain (compression of the octahedron along the 4-fold axis). One can remark in particular that the IS state domain is very broad for d [START_REF] Watson | Iron Series Hartree-Fock Calculation II[END_REF][START_REF] Richardson | Approximate Radial Functions for First-Row Transition-Metal Atoms and Ions. I. Inner-Shell, 3d and 4s Atomic Orbitals[END_REF] ions within this approximation and exists as soon as the metal-ligand distance difference (from the apex and the equator) exceeds ca. 8%; in contrast, for d 4 ions, the IS state domain exists from zero distortion to ca. 28% difference in the metal-ligand distances (γ > 0).

The use of the effective parameters fully changes the diagrams with a strong reduction (about a half) of the initial IS state domain size (delimited with a dashed line). Except at zero distortion, the LS state domain is fully replaced with a more stable mixture of LS(γ ± ) and LS(γ ∓ ) states. The single HS state is stable only for d [START_REF] Watson | Iron Series Hartree-Fock Calculation II[END_REF] ions (for which the HS state energy is independent of γ), while it is replaced with HS states mixtures for d [START_REF] Ballhausen | Introduction to Ligand Field Theory[END_REF] and d [START_REF] Richardson | Approximate Radial Functions for First-Row Transition-Metal Atoms and Ions. I. Inner-Shell, 3d and 4s Atomic Orbitals[END_REF] ions (except at zero distortion). For the IS state domain, the behavior is quite similar for d 5 and d 6 cations with some remainder for strong distortion (as the difference in the magnitude of the distortion between the under-and overdistorted octahedrons in the mixture decreases). The consideration of a partial transfer of the coupling energy can further decrease the size of the IS state domain; however, within the transfer limit of 10% used in the calculations, some IS state domains still linger for the strongest positive strain (γ > 0, elongation) and for negative strain (γ < 0, compression). For d 4 cations, a small IS state domain remains only around zero distortion for a positive strain. (3)

D 4h D 2d C 4v r θ φ r θ φ r θ φ O 1 r 0 (1 + 2γ) 0 0 r 0 0 0 r 0 (1 + γ) 0 0 O 2 r 0 (1 + 2γ) π 0 r 0 π 0 r 0 (1 -γ) π 0 O 3 r 0 (1 -γ) π/2 0 r 0 /cos α π /2 + α 0 r 0 /cos α π /2 + α 0 O 4 r 0 (1 -γ) π/2 π/2 r 0 /cos α π /2 -α π /2 r 0 /cos α π /2 + α π /2 O 5 r 0 (1 -γ) π/2 2π/2 r 0 /cos α π /2 + α 2π/2 r 0 /cos α π /2 + α 2π/2 O 6 r 0 (1 -γ) π/2 3π/2 r 0 /cos α π /2 -α 3π/2 r 0 /cos α π /2 + α 3π/2

EFFECT OF THE

The crystal field parameters in eq 3 are even functions of α, and it is sufficient to analyze the results for positive distortion. The orbital energies are given by eq 1d and are plotted for p = 1.6 in Figure 4a; changing the p value impacts the value of α at the levels reversal (Figure 4b) but the qualitative features of the plot are maintained.

In the case of d [START_REF] Ballhausen | Introduction to Ligand Field Theory[END_REF] ) is the IS state. The energy of the mixtures is calculated at constant total distortion, i.e., for 2t g α = t g α -+ t g α + (see Table 2); imposing α - = 0 (octahedral case) then α + = arc t g (2t g α); if α + ∉ [α min ; α max ], the overdistortion is set at the limit (α max ) and the underdistortion is adapted to maintain the overall distortion constant. Note that because the crystal field parameters are even functions of α, the mixtures are also even; however, a mixture S 1 (α -)/S 2 (α + ) with positive distortion is reversed for negative distortion and becomes S 2 (α -)/S 1 (α + ); this only impacts the mixtures combining different states (LS + HS).

As in the case of the D 4h distortion, Figure 5 shows the diagrams for d [START_REF] Ballhausen | Introduction to Ligand Field Theory[END_REF][START_REF] Watson | Iron Series Hartree-Fock Calculation II[END_REF][START_REF] Richardson | Approximate Radial Functions for First-Row Transition-Metal Atoms and Ions. I. Inner-Shell, 3d and 4s Atomic Orbitals[END_REF] ions resulting from the use of the effective parameters (p = p HF /3 = 1.6, ϱ H = 3/4, and ϱ I = 7/8) and the possibility of regular states mixtures (with α min = 0°, α max = 30°, and δj = 10%). In the insets, the results are based on the approach of Pouchard (p = 2, ϱ H = 1, and ϱ I = 1).

Preliminary observations of the insets in Figure 5 show that, in the selected α range, the regular LS and HS states are stable over a large domain. The IS state appears from α ≈ 21, 11, and 14°for, respectively, d 4 , d [START_REF] Watson | Iron Series Hartree-Fock Calculation II[END_REF] , and d 6 ions, i.e., it is pushed further as the d-shell occupancy moves away from 5; these values of α correspond to metal-ligand distance elongations of, respectively, ca. 7%, 2%, and 3%, i.e., much lower than in the case of the D 4h distortion (except for d [START_REF] Ballhausen | Introduction to Ligand Field Theory[END_REF] ions for which the IS state was stable from zero distortion). This suggests, by comparison only of the metal-ligand distance elongation, that Γ 5

-is more likely to favor the emergence of the IS state.

The use of the effective parameters here also strongly reduces (by about 2/3) the initial IS state domain size (delimited with a dashed line). Its existence is, however, quite fragile, and it is mostly replaced with more stable mixtures of regular states. In the case of d 4 and d [START_REF] Richardson | Approximate Radial Functions for First-Row Transition-Metal Atoms and Ions. I. Inner-Shell, 3d and 4s Atomic Orbitals[END_REF] ions, the IS state domain fully vanishes, while for d [START_REF] Watson | Iron Series Hartree-Fock Calculation II[END_REF] ions some IS state domain traces still remain for stronger distortion.

As in the case of D 4h distortion, the partial transfer of the coupling energy (with δj = 10%) further decreases the remaining IS state domain; it also reduces the regular states extension for low distortion.

Interestingly, the D 2d distortion hardly modifies the J H /Dq limit to the HS state (or HS mixtures) for smaller α value, and this limit is constant until the orbital doublet inflects. Just as for the D 4h distortion, this HS state is fully stable only for d 5 ions (its energy is independent of α); otherwise, it exists only for weak (d 4 ) or strong (d 6 ) distortion. Elsewhere, it is replaced with HS(α -)/HS(α + ) mixtures.

In contrast, the LS state domain size appears more extended than in the case of D 4h , and it is only replaced with a more stable mixture of LS(α -)/LS(α + ) in the α range ca. [13°, 27°]; this corresponds to metal-ligand distance elongation com-prised between ca. 3 and 11%. The lower limit is twice the one calculated for D 4h (γ ≈ 0.005; elongation ≈1.5%). Note that interestingly, in the case of d 4 and d [START_REF] Richardson | Approximate Radial Functions for First-Row Transition-Metal Atoms and Ions. I. Inner-Shell, 3d and 4s Atomic Orbitals[END_REF] ions, owing to the definitions of α ± (see above) and the α range for the mixtures stability, the LS(α -) state is always associated with S = 1 and S = 0 for, respectively, d 4 and d 6 ions, while the LS(α + ) is always associated with S = 0 and S = 1 for, respectively, d 4 and d [START_REF] Richardson | Approximate Radial Functions for First-Row Transition-Metal Atoms and Ions. I. Inner-Shell, 3d and 4s Atomic Orbitals[END_REF] ions. This means for these ions that the LS(α -)/LS(α + ) mixture is actually a mixture of S = 0 and S = 1 states (though both are LS states). In the same way, the LS(α -)/HS(α + ) domain (for weaker distortion before the reversal of the e doublet and the b 1 singlet) actually mixes either S = 1 and S = 2 states for d [START_REF] Ballhausen | Introduction to Ligand Field Theory[END_REF] ions or S = 0 and S = 2 states for d 6 ions. Finally, the HS(α -)/ LS(α + ) domain (for stronger distortion after the reversal of the e doublet and the b 1 singlet) actually mixes either S = 0 and S = 2 states for d [START_REF] Ballhausen | Introduction to Ligand Field Theory[END_REF] ions or S = 1 and S = 2 states for d 6 ions.

EFFECT OF THE C 4ν DISTORTION ON THE SPIN

STATE OF 3d n IONS The C 4ν distortion is due to the Γ 4 -displacive mode. This case is more complex because up to 3 modes act on the z coordinates of the transition metal cation and/or the ligands to modulate the distortion (Figure 1).

With the definitions in eqs 1a, 1b, and 1c and Table 2, the crystal field parameters are given by The orbital energies are still given by eq 1d and are plotted in Figure 6a,b.

The orbital energies as well as the crystal field parameters are even functions of α and γ (Table 2 and Figure 6a,b). The compensation scheme for mixtures is also an even function of the distortion parameters, and it is sufficient to analyze the results in one quadrant, e.g., for positive distortions. Just as in the case of the D 2d distortion, one should care that a mixture S 1 (α/γ -)/S 2 (α/γ + ) is reversed for negative distortion and becomes S 2 (α/γ -)/S 1 (α/γ + ); this impacts only the mixtures combining different states (LS + HS).

Figure 6c shows that only for strong γ distortion there are no restrictions for the existence of an IS state for both d [START_REF] Ballhausen | Introduction to Ligand Field Theory[END_REF][START_REF] Watson | Iron Series Hartree-Fock Calculation II[END_REF][START_REF] Richardson | Approximate Radial Functions for First-Row Transition-Metal Atoms and Ions. I. Inner-Shell, 3d and 4s Atomic Orbitals[END_REF] As in the case of D 4h and D 2d distortions, the energy of mixtures is calculated at constant total distortion. Having two distortion parameters, several mixing scenarios can be used either based on a single parameter or combining them. For the sake of simplicity, we only consider the solutions for which either α or γ is used to stabilize a mixture. Such situations are sketched in Figure 7: in Figure 7a, only the apical ligands move relatively to the transition metal cation, and α is constant (t g α = (z ez t )/r 0 ), while γ evolves around the initial value (γ = (z az t )/r 0 ); in Figure 7b, only the equatorial ligands move relatively to the transition metal cation and γ is constant, while α evolves around the initial value.

By following the definitions of the distortion parameters, the present γ mixing scenario is equivalent to the one used with the D 4h distortion (2(1 -γ) = (1 -γ -) + (1 -γ + )), and the α mixing scenario is equivalent to the one used with the D 2d distortion (2t g α = t g α -+ t g α + ); the maximum values are set to γ max = 1/4 and α max =30°. Figure 8 shows the diagrams for d [START_REF] Ballhausen | Introduction to Ligand Field Theory[END_REF][START_REF] Watson | Iron Series Hartree-Fock Calculation II[END_REF][START_REF] Richardson | Approximate Radial Functions for First-Row Transition-Metal Atoms and Ions. I. Inner-Shell, 3d and 4s Atomic Orbitals[END_REF] ions as a function of γ with α = 0, 10, or 20°(p = p HF /3 = 1.6, ϱ H = 3/4, and ϱ I = 7/8).

As aforementioned, the IS state cannot exist for d [START_REF] Ballhausen | Introduction to Ligand Field Theory[END_REF] ions when the e doublet is the second lowest level. It only appears from the reversal of e and b 1 levels at α ≈ 23°(Figure 6b), but it decreases as α further increases (for ϱ H = ϱ I = 1 only the IS and HS state domains coexist at α ≈ 23.21°and then it decreases at the benefit of both LS and HS states). For d [START_REF] Richardson | Approximate Radial Functions for First-Row Transition-Metal Atoms and Ions. I. Inner-Shell, 3d and 4s Atomic Orbitals[END_REF] ions, the IS state cannot exist when the e doublet is the second highest level, i.e., for strong α distortion (Figure 6c); the IS state domain is anyway either absent or very small (depending on the magnitude of the effective parameters) for small α distortion, and it grows as α increases up to the limit where e and b 1 reverse and where nearly only the IS and HS state domains coexist; after, it disappears. In the same way, for d 5 ions, the IS state domain is either absent or very small for small α distortion, but then it regularly increases with α. Actually, with the values selected for the effective parameters in this study, the IS state domain can only exist for d [START_REF] Watson | Iron Series Hartree-Fock Calculation II[END_REF][START_REF] Richardson | Approximate Radial Functions for First-Row Transition-Metal Atoms and Ions. I. Inner-Shell, 3d and 4s Atomic Orbitals[END_REF] ions for strong alpha distortion (dashed contour for α = 20°in Figure 8). The effect of mixing the regular states is to erase the last traces of the IS state domain (except for the d 5 ion at α = 20°). This is evidence that the IS state domain can hardly survive a C 4v distortion.

Figure 8 highlights that, except the HS state for d 5 ions (energy independent of both α and γ), the regular states are not stable and are mostly replaced with mixtures either α or γ stabilized (Figure 9); furthermore, depending on the relative position of the e doublet, the LS state of d 4 and d 6 ions in the mixtures has either a total electron spin S = 0 or 1.

Note that the sharpness of the domain borders strongly (but not only) depends on the selected mixing scenarios (Figure 8) and that they might be much smoother in average systems. ). The dashed lines mark the borders of the initial IS state domain before considering the higher stability of a mixture of regular states. Exponents α and γ nearby a given mixture marks the most stable mixing scenario for the mixture (see Figure 7). The LS ∓δj /HS ∓δj 1:1 mixture of LS(γ, J H /Dq(1 ∓ δ j)) and HS(γ, J H /Dq(1 ± δj)) states with δj = 10% is highlighted with a darkened area.

DISCUSSION

Skipping the details due to each specific tetragonal distortion, these results globally show that the IS state domain size is strongly sensitive to the ion nature and its shell (p, ϱ H , and ϱ I ), that it is basically mostly stable for strong distortions as compared to regular states (though it appears globally more robust under the action of the strain mode as compared to the action of the displacive modes), and that mixtures of regular states are generally more stable than both the IS state and the regular states over a large part of the diagrams.

It was checked varying the magnitude of the parameters p, ϱ H , and ϱ I that, although the borders of the domains are obviously more or less shifted, the global features of the diagrams are maintained: i.e., without loss of generality, the size of the IS state domain, if such state exits, remains small and concentrates at the limits of the diagrams, and the mixed state domains are always present whatever the representation involved (Γ 3 + , Γ 4 -, or Γ 5 -

). In the treatment of the mixed state domains, we have restricted the analysis to the simplest case of 1:1 mixtures; though rough, this approximation shows an effective mixing over the majority of the diagrams, independently of the nature of the ion and its shell; pushing further the investigation with higher order mixtures only reinforces this results; i.e., a given state looks more likely represented by a superimposition of several states at more or less distorted sites rather than a unique one.

The present mechanism for mixture stabilization supposes the possibility of exchange between two near IS state ions. Such situation can hardly occur in the case of diluted solution. However, in such case, we can still consider the opportunity to stabilize a regular state mixture by propagation of the distortion to the closer environment. For solutions that lead to LS -/LS + or HS -/HS + , the global spin state is not modified (but the total electron spin is eventually modified), and only some distribution in the magnitude of the distortion is expected. More interesting are those mixtures combining different states, which are responsible for the narrowing or verify the full disappearance of the IS state domain. Such result actually shows that, under tetragonal distortion, the IS state is very unlikely and more realistically replaced by combinations of under/ overdistorted LS and HS state. Practically, a transition sequence implying an evolution from an LS ground state to some HS exited state will hardly go through an IS state but rather through a mixture of the ground and exited states. Experimentally, dealing with spins and orbital occupancy, accurate information to probe such scenario can be obtained from electron paramagnetic resonance and possibly polarized X-ray absorption spectroscopy experiments; complementary information can be given by techniques like extended X-ray absorption fine structure to obtain precise bond length information or resonance methods (electron paramagnetic resonance, nuclear magnetic resonance, and Mossbauer) thanks to their short length scale.

The stability of the mixing scenarios as well as their extension over a majority of the diagrams obviously raise the question of the several states organization/distribution in the compound, i.e., the possibility of clusterization and phase separation or the possibility of spin state induced nuclear superstructures. A full detailed discussion is out of the scope of this article, and this point is here only shortly addressed for the D 4h distortion with a simple approach based on the calculation [START_REF] De Wette | On the Calculation of Lattice Sums[END_REF] of the difference in the Madelung potential between a superstructure as sketched in Figure 9 and the sum of the individual Madelung potentials resulting from a phase separation (ignoring interface/surface or any other effect). The results are given in Figure 9 for a D 4h distorted ABO 3 perovskite-like structure, where B is a transition metal cation with a valence between 2 and 4, and charge compensation is achieved by the A site cations in a purely ionic fashion. For a tetravalent 3d transition metal cation, such as Fe 4+ , Co 4+ , and Ni 4+ , the superstructure scenario seems to be favored for any value of the distortion parameter. In contrast, for a divalent, Cr 2+ , Mn 2+ , and Fe 2+ , or a trivalent, Mn 3+ , Fe 3+ , and Co 3+ , 3d transition metal cation, the superstructure scenario is only favored in the case of a compression; phase separation is favored in the case of an elongation. For small distortions, the difference in the Madelung potential between the superstructure and the phase separation scenarios is rather small, and either intermediate solutions occur or other parameters, for instance, the surface, interface, and magnetic interactions, may determine the resulting phase. The analysis is here restricted to the simplest case of corner shared octahedrons under the action of Γ 3 + , but similar calculations could be done for any polyhedron ordering schemes (corner, edge, face shared, or combinations); depending on the active distortion (Γ 3 + , Γ

) and the structure details, it might, however, be necessary to add further contributions of Γ 3 + and Γ 1 + to sketch realistic/actual structures. These results show that a global tetragonal distortion can result not only in a spin state disproportionation, what further implies some distribution in the magnitude of the distortion, but also can induce a distortive ordering to further minimize the lattice energy; in such case, superstructures like the one sketched in Figure 9 can be expected. From an experimental point of view, their diffraction hallmarks may be hardly visible depending on the distortion magnitude. For instance, mixtures based on a partial transfer of the coupling energy should, generally speaking, display only a very light modulation of the oxygen positions. In contrast, the LS/HS states combinations at the border between LS and HS states (corresponding to an IS state disproportionation in LS and HS states) could be easier to identify from diffraction studies. Indeed, they involve stronger structural change with mixtures of small (big) under-distorted octahedral environments containing LS (HS) state ions and big (small) overdistorted octahedrons containing HS (LS) state ions; with such configuration, the ligand position can be sufficiently highly modulated to visibly impact the structure factor (particularly in the extreme case of a full transfer of the distortion to one site (e.g., LS(0)/HS(2k)), the modulation amplitude reaches 2k). Specific techniques can then provide additional useful information like pair distribution function analysis to probe slight bond length differences, small-angle Xray/neutron scattering, and Brillouin scattering that can probe the mesoscopic arrangement and evidence/discard a phase separation or vibrational techniques (infrared and Raman) that can give some information about the bonds and some short (medium) range interactions.

SOME RESTRICTIONS TO THE MODEL AND RESULTS

Within the model proposed to define the spin state borders, a set of minimum parameters was used discarding most of the interactions or perturbations. Hereafter, the possible effects of some of these interactions are shortly addressed.

(i) The effect of the spin-orbit coupling is fully ignored. Comparison of the results by Pouchard (without spin-orbit coupling) and Lamonova et al. [START_REF] Lamonova | Intermediate-Spin State of a 3d Ion in the Octahedral Environment and Generalization of the TanabeSugano Diagrams[END_REF] (with spin-orbit coupling) that turn out to be quite similar tend, however, to show that in first approximation this effect can be neglected to build the diagrams and draft global trends. The effective spin value might anyway depend on the magnitude of the spin-orbit coupling as shown in the case of LaCoO 3 on the basis of electron paramagnetic resonance experiments. [START_REF] Noguchi | Evidence for the Excited Triplet of Co 3+ in LaCoO 3[END_REF][START_REF] Ropka | 5 D Term Origin of the Excited Triplet in LaCoO 3[END_REF][START_REF] Hoch | Diamagnetic to Paramagnetic Transition in LaCoO 3[END_REF] (ii) The point charge model strictly applies to ionic structure only. However, replacing the point charge model parameters by some effective values, it becomes a reformulation of the superposition model. [START_REF] Newman | The Superposition Model of Crystal Fields[END_REF] The physics behind this model was in particular discussed in refs 22 and 23. It was shown that the dlevel splitting of the transition metal cation comes from the hybridization with the highest occupied orbitals of the surrounding ligands and that in the second order of perturbation theory the contributions from ligands are summed up independently and the symmetry properties of the contributions are the same for both the point charge and the superposition models. The same theory also explains the origin of the reduction of spin-orbit constant and of the Racah parameters in solids as compared to their free-ion values. A certain degree of covalence can thus easily be accounted for by tuning the parameters p and ρ H/I . As discussed earlier, the global features of the diagrams will remain, but the borders of the domains will drift.

(iii) Magnetic interactions are neglected. Depending on the spin-spin interaction magnitude, the diagrams should either remain unchanged, or their borders could drift; however, in agreement with available experiments (no interacting IS have been yet confirmed), the probability of the IS state domain reappearance, growth, or stabilization remains slight.

(iv) The possibility of charge disproportionation is not considered. Depending on the ion and the flexibility of the structure, such situations (e.g., 2Co LS 3+ → Co LS 4+ + Co LS 2+ as proposed for TlSr 2 CoO 5 ) [START_REF] Doumerc | Crystal Structure of the Thallium Strontium Cobaltite TlSr 2 CoO 5 and Its Relationship to the Electronic Properties[END_REF] could be more or less favored. In such case anyway, the spin state details for each resulting ion shell can be obtained from the diagrams calculated in this work.

(v) The present model is static (or frozen at low temperature). It cannot be excluded that dynamical effects will smooth or average some effects. For instance, we cannot discard a narrowing (or zeroing) of the distribution of the distortion parameter in LS -/LS + or HS -/HS + mixtures or that some LS/HS mixtures, upon, e.g., phonon dynamics or electron-phonon interactions, can result in an average IS solution; only ultrafast techniques should then be able to settle. Generally speaking, this last point raises the question of the actual definition of the intermediate spin and its time dependence; defining the IS state in a purely static model, the present results highlight its limited existence; considering however an extended definition in a dynamic scheme with (ultra)fast hopping between, e.g., LS and HS states (some kind of spin state pairing mediated, e.g., by phonons, orbital excitation, etc.) results in some orbital occupancy distribution, what can largely extend the dynamic-IS state existence domain depending on the selected time threshold.

(vi) Mono-and bielectronic terms are considered in the model, but further electron-electron interactions in the 3dmanifold are ignored. Adding such effects could destabilize the LS state resulting, for instance, in a global offset or a shrinkage of the diagrams along J H axis; the opportunity to stabilize an IS state or to simply grow an already existing IS state domain cannot be discarded at this stage.

Finally, in addition to this description and as aforementioned, any extra interaction (spin-orbit, charge, time, phonons, and delectron manifold), depending on its magnitude, can influence the stability of the (super)structures or mesostructures (e.g., phase separation) discussed earlier and drive their details.

CONCLUSIONS

Spin states diagrams of 3d [START_REF] Ballhausen | Introduction to Ligand Field Theory[END_REF][START_REF] Watson | Iron Series Hartree-Fock Calculation II[END_REF][START_REF] Richardson | Approximate Radial Functions for First-Row Transition-Metal Atoms and Ions. I. Inner-Shell, 3d and 4s Atomic Orbitals[END_REF] were calculated for tetragonally distorted 6-fold oxygen environment (D 4h , D 2d , and C 4v ). The calculations were made within the point charge model approximation using a restricted set of effective parameters to account for the cubic field drift on spin state change, and the generally observed lowering in the expectation values of the 3dradial wave functions and using the possibility of mixed states. With such treatment, it is shown, in agreement with experiments, that the IS state is quite rare under tetragonal distortion. The narrowing of the calculated IS state domain as compared to other reports is due to the use of more realistic parameters for the crystal field but also largely due to the possible stabilization of mixed (LS, HS) domains. These later impact not only the IS state but also the extension of regular LS and HS states' domains with the appearance of stable mixtures combining cations in over-and under-distorted environments, with either LS or HS state. These mixtures were further discussed on the basis of simple Madelung potential calculation in the case of the D 4h distortion, and it was shown that they are likely to induce spin state ordering associated with a distortive ordering.

■ APPENDIX AI. Calculation of the Crystal Field Hamiltonian

AI.1. Crystal Field Potential. The potential at the transition metal cation due to the ligands is given by

∑ ∑ ∑ π = + * =- + V eZ l r d Y Y 4 2 1 i i l m l l l ML l lj m li m 1
where i runs over the nearest ligands, i is even (any matrix representation of an odd-order potential in a basis of orbitals with the same parity vanishes identically [START_REF] Ballhausen | Introduction to Ligand Field Theory[END_REF] ) and up to the fourth order only (higher orders exert no influence because the direct product of two d-orbital sets spans no representations of the rotation group of order higher than four [START_REF] Ballhausen | Introduction to Ligand Field Theory[END_REF] ). The potential has the same symmetry as the cationic site, and it remains invariant under all operations of the group, i.e., it transforms as a Γ 1 (+) (=A (1)(g) ), the identity representation of the symmetry group.

For instance, in the case of O h point group, only the set Y 4 m has a Γ 1 (+) (= A 1g ) irreducible component (see Table A1), ). Dealing with tetragonal distortion, both {Y 2 m } and {Y 4 m } sets have a Γ 1 (+) (= A (1)(g) ) irreducible component (see Table A1). We also take the 4-fold axis to be the axis of quantization. In particular, in this case, the potential is invariant upon the action of C ̂4 and/or S ̂4 (= C ̂4 ⊗ i ̂i ̂, the inversion operator); . Looking at the definition of the potential given at the beginning of the paragraph, we see that the coefficients t,u,v,w depend on the details of the ligands' position with t = ∑eZ i (4π/5) 1/2 (r 2 /d

ML i 3 ) Y 2 0 (θ i , φ i ), u = ∑ i eZ i (4π/9) 1/2 (r 4 (d ML i 5 )Y 4 0 (θ i , φ i ), and v = w ̅ = ∑ i eZ i (4π/9) 1/2 (r 4 /d ML i 5 )Y 4 -4 (θ i , φ i ) (using Y l -m = (-1) m Y l * m );
the potential at the transition metal cation is real, and this expression simplifies to

V = tY 2 0 + uY 4 0 + v(Y 4 -4 + Y 4 4 ) or simply V = V 2 0 + V 4 0 + V 4 ±4
. AI.2. Crystal Field Hamiltonian. The effect of the crystal field is to lift up the orbitals' degeneracy; we use the perturbation theory for degenerate systems to evaluate the d orbitals splitting, i.e., we need to solve for Vd m = Ed m where d m = R n=3,l=2 Y l = 2 m for 3d ions. [START_REF] Ballhausen | Introduction to Ligand Field Theory[END_REF] Each individual element of the secular determinant is given by

H mm′ = e∫ τ d m *Vd m' dτ = e∫ r R 3,2 * R 3,2 r 2 dr∫ Ω Y 2 m *VY 2 m'
dΩ where τ and Ω represent the volume and the solid angle, respectively (dτ = r 2 drdΩ = r 2 sin(θ)dθdφdr). Using the potential obtained above, one gets

∫ ∫ ∫ ∫ = ′⟨ ⟩ Ω + ′ Ω + ′ Ω + Ω ′ Ω * ′ Ω * ′ Ω * - ′ Ω * ′ ⎜ ⎟ ⎡ ⎣ ⎢ ⎛ ⎝ ⎞ ⎠ ⎤ ⎦ ⎥ H t r Y Y Y r u Y Y Y v Y Y Y Y Y Y
CF By applying the definitions given in Table 2 for the ligands' position, the crystal field parameters given in 2-4 are obtained.

AII. Total energy of the transition metal cation This section gives an example of the calculation of the total energy of a d 6 cation with an e g b 2g a 1g b 1g orbital configuration; 2 the energy for the three possible spin state configurations are determined by taking into account both the mono-and bielectronic terms (Table A2). The core energy E core (interaction of the 3d electrons with the nucleus and the [Ar] core electrons) is assumed constant. The bielectronic contribution is calculated by considering all the possible electron pairs' interactions and assuming that the interelectronic Coulomb repulsion energies (U, U′) and the intra-atomic exchange energy (J H ≈ U -U′) are independent of m in {d m } set. In LS state, all 6 electrons interact with one electron in the same orbital (U) and 4 electrons in different orbitals (4U′); in addition, each electron interacts with two electrons with parallel spin. This results in

= ′ + - = - + - = - - E U U J U J U J U J 6 1 2 ( 4
2 ) 3(4( 2 )

2 ) 15 30

Figure 1 .

 1 Figure 1. Tetragonal distortion path with the three types of polyhedra resulting from a tetragonal distortion of an O h polyhedron. Each figure is designated with its point group symmetry in Schoenflies notation as well as its full international symbol equivalent. The black arrows and legends between figures show the path between point groups and the lost symmetry elements. The blue double lines mark the irreducible representations involved in the transitions from O h (Miller-Love notation and point group notation into brackets). When the transition is displacive (O h → D 2d and O h → C 4v ), the responsible modes are also pictured.

  Figure 2b shows the orbital configuration as a function of the distortion parameter and p. It highlights several domains where the IS state is forbidden by Hund's rule for d 4 and d 6 cations, namely, where the e g doublet (d xz , d yz ) is the second or third lowest level, respectively. In the case of d 4 ions, the IS state can generally not exist for negative strain, γ < 0, except when the e g doublet (d xz , d yz ) and the b 1g singlet (d x 2 -y 2 ) reverse for strong distortion; within these limits, the S = 1 state is thus the actual LS state. This can be seen for instance from the orbitals' energy diagram in Figure 2a (p = 2), which shows the reversal of the e g doublet (d xz , d yz ) and the b 2g singlet (d xy ) at γ = 0 and the reversal of the e g doublet (d xz , d yz ) and the b 1g singlet (d x 2 -y 2 ) at γ ≈ -0.11. This leads for S = 1 to an e g 3 b 2g 1 IS state for γ > 0 and either a b 2g 2 e g 2 LS state or a b 2g 2 b 1g 1 e g 1 IS state for γ < 0.

Figure 2 .

 2 Figure 2. Crystal field splitting at a D 4h site for d 4-6 ions. (a) Moments ratio p = 2; (b) highlighting of domains splitting in the (p, γ) plane; only the relative positions of the levels are sketched with the orbital configuration written nearby; the plot also mentions where restrictions occur for the existence of the IS state.

b 1 2 e 2 )

 22 ions, the IS state cannot exist before the distortion reverses the e doublet (d xz , d yz ) and the b 1 singlet (d x 2 -y 2 ) levels (Figure 4); in the low distortion region, the S = 1 state (b 2 2 e 2 ) is thus the actual LS state, and for stronger distortion, the S = 1 state (b 2 2 b 1 1 e 1 ) is the IS state. In contrast, in the case of d 6 ions, the IS state cannot exist after the distortion reverses the e doublet (d xz , d yz ) and the b 1 singlet (d x 2 -y 2 ) levels; in the high distortion region, the S = 1 state (b 2 2 is thus the actual LS state, and for weaker distortion, the S = 1 state (b 2 2 e 3 b 1 1

Figure 3 .

 3 Figure 3. Spin state diagrams at a D 4h site for d 4-6 ions under the action of Γ 3 + (p = 1.6, ϱ H = 3/4, and ϱ I = 7/8) and considering the higher stability of a mixture of regular states (κ = 0); the light dashed lines mark the borders of the initial IS state domain before considering the possible higher stability of a mixture of regular states. Where the number referring to a domain is red, i.e., 1 and 4, the total electron spin is S = 1 though the spin state is LS. The bold vertical dashed line marks the reversal of the e g doublet (d xz , d yz ) and the b 2g singlet (d xy ) at γ = 0. Inset: only the action of Γ 3 + is considered and the moments ratio p = 2 (the second vertical dashed at line at γ ≈ -0.11 mark the reversal of the e g doublet (d xz , d yz ) and the b 1g singlet (d x 2 -y 2 )). The LS ∓δj / HS ∓δj 1:1 mixture of LS(γ, J H /Dq(1 ∓ δj)) and HS(γ, J H /Dq(1 ± δj)) states with δj = 10% is highlighted with a darkened area.

Figure 4 .

 4 Figure 4. Crystal field splitting at a D 2d site for d 4-6 ions. (a) Moments ratio p = 1.6; (b) highlighting of domains splitting in the (p, α) plane; only the relative positions of the levels are sketched with the orbital configuration written nearby; the plot also mentions where restrictions occur for the existence of the IS state.

Figure 5 .

 5 Figure 5. Spin state diagrams at a D 2d site for d 4-6 ions under the action of Γ 5 -(p = 1 0.6, ϱ H = 3/4, and ϱ I = 7/8); inset, action of Γ 5 -(p = 2 and ϱ H = ϱ I = 1). The light dashed lines mark the borders of the initial IS state domain before considering the higher stability of a mixture of regular states. The vertical bold dashed line marks the reversal of the e doublet (d xz , d yz ) and the b 1 singlet (d x 2 -y 2 ). Where the number referring to a domain is red, i.e., 1 and 4, the total electron spin is S = 1 though the spin state LS ∓δj /HS ∓δj 1:1 mixture of LS(γ, J H /Dq(1 ∓ δj)) and HS(γ, J H /Dq(1 ± δj)) states with δj = 10% highlighted with a darkened area.

Figure 6 .

 6 Figure 6. Effect of the distortion parameters on the levels splitting with p = 1.6. (a) Effect of γ with α = 0°(e.g., action of A 2u mode only); (b) effect of α with γ = 0 (e.g., action of E u mode only); (c) effect of both α and γ with highlighting of splitting domains for 0 ≤ p ≤ 4; only the relative positions of the levels are sketched with the orbital configuration written nearby; the plot also mentions where restrictions occur for the existence of the IS state.

  ions (light green domain eb 2 b 1 a 1 . However, as γ decreases, large domains where the IS state is not possible open depending on the relative position of the e doublet (d xz , d yz ) and the b 1 singlet (d x 2 -y 2 ), i.e., for weak α distortion in the case of d 4 ions (b 2 eb 1 a 1 ) and for strong α distortion in the case of d 6 ions (b 2 b 1 ea 1 ). When p decreases, the eb 2 b 1 a 1 domain stretches and the e/b 1 reversal occurs for higher value of α; when p increases the eb 2 b 1 a 1 domain vanishes and the e/b 1 reversal occurs for lower value of α.

Figure 7 .

 7 Figure 7. Sketch of the action of the several modes activated in Γ 4 representation.T 1u triggers the transition metal cation (z t ); E u and A 2u trigger, respectively, the equatorial (z e ) and apical (z a ) ligands. The drawings also sketch the mixing scenarios used for the calculation of the diagrams with, in blue, the original distortion at (α, γ), in green, the underdistorted case, and in red, the overdistorted case: (a) the mixture is based on couple (α, γ -)/(α, γ + ); (b) the mixture is based on couple (α -, γ)/(α + , γ).

Figure 8 .

 8 Figure 8. Spin state diagrams at a C 4v site for d 4-6 ions under the action of Γ 4 -(p = 1.6, ϱ H = 3/4, and ϱ I = 7/8). The dashed lines mark the borders of the initial IS state domain before considering the higher stability of a mixture of regular states. Exponents α and γ nearby a given mixture marks the most stable mixing scenario for the mixture (see Figure7). The LS ∓δj /HS ∓δj 1:1 mixture of LS(γ, J H /Dq(1 ∓ δ j)) and HS(γ, J H /Dq(1 ± δj)) states with δj = 10% is highlighted with a darkened area.

Figure 9 .

 9 Figure 9. Difference in the Madelung potential between the 1:1 superstructure scenario (SS) and the phase separation scenario (PS) for a D 4h distorted ABO 3 perovskite-like structure and a transition metal with valence +4 (red solid line), +3 (blue dashed line), and +2 (green dot line). Calculation details: a cubic = 4 Å ́; 2γ = γ -+ γ + (see text); full charge compensation on A site only in a purely ionic scheme; interface/surface and any other effects are ignored.

4 .

 4 meaning that only linear combinations of Y 4 m can transform as a Γ 1 (+) representation in O h (to a first approximation, Y 0 0 only contributes to a uniform shift of all levels and is neglected). If we take the 4-fold axis to be the axis of quantization and operate C ̂4 (operator for the 4-fold rotation around the axis of quantization) on the set {Y 4 m }, we must have C ̂4 Y l m = Y l m e im(π/2) = Y l m , which is true only for m = 0 or ±4. In this case, the potential is, as well-known, simply given by V = uY 4 0 + vY 4 -4 + wY 4 Using further the invariance condition upon the action of C ̂2 and C ̂3 (operators for the 2-fold and 3-fold rotations around [110] and [111] axis, respectively) and normalizing, one finally gets V = Y 4 0 + (5/14) 1/2 (Y 4 -4 + Y 4 4

  we have C ̂4 Y l m = Y l m e im(π/2) = Y l m and/or (C ̂4 ⊗ i ̂) Y l m = (-1) l Y l m e im(π/2) = Y l m, which is true only for l even and m = 0 or ± 4. Consequently, one gets V = tY 2 0 + μY 4 0 + vY 4 -4 + wY 4 4

where

  et = t'r 2 , eu = u′r 4 , ev = v′r 4 , and the nth moments of the radial distribution of the 3d electrons are ⟨r n ⟩ = ∫ r R 3,2 * r n R 3,2 r 2 dr. The evaluation of the remaining integrals are considerably simplified noting that ∫ φ Y 2 m *Y L M Y 2 m ′dφ ≠ 0 only if M = m -m′. The results of these integrations are given below in matrix form C L and the value of M along the diagonals are explicitly mentioned: The resulting secular determinant writes where p′ = ⟨r 2 ⟩/⟨r 4 ⟩ or highlighting the contributions of the different terms in the potential gives Introducing 10Dq = 5/3Ze 2 ⟨r 4 ⟩/r 0 5 and p = r 0 2 ⟨r 2 ⟩/⟨r 4 ⟩ where r 0 is the metal-ligand distance in octahedral field, and the crystal field parameters simplify to

LS state configuration e g 4 b 2g 4 a 1g 4 b 1g 0 :

 0 The monoelectronic contribution is given by summing the individual energy over all the occupied orbitals:

The ( 1 / 2 )

 12 term is to prevent the double counting of the interaction energies. The total energy isE LS = E core -14b -2c + 15U -30J H .IS°state configuration:

Table 1 .

 1 Definition of the Symbols Used in the Article

	symbol	definition
	a,b,c	symmetry dependent crystal field parameters
	D q	cubic field parameter
	d m	normalized spherical harmonic Y l m with l = 2
	k	distortion parameter
	p	p= r 0 2 ⟨r 2 ⟩/⟨r 4 ⟩
	⟨r n ⟩	nth moment of the radial distribution of the 3d electrons
	r 0	metal-ligand distance in octahedral field
	U,U′	interelectronic Coulomb repulsion energies
	J H	hund coupling energy (intra-atomic exchange energy)
	LS,IS,HS low, intermediate, and high spin
	Y l m	normalized spherical harmonic
	Z eff	effective charge of the transition metal
	α	distortion parameter (D 2d , C 4v )
	γ	distortion parameter (D 4h , C 4v )
	ρ H ,ρ I	ratio of the cubic field for high or intermediate spin state ions to the cubic field for low spin state ion
	Γ i	irreducible representation

Table 2 .

 2 Oxygen Positions Relative to the Transition Metal Cation in Spherical Coordinates (r,θ,φ)

  D 2d DISTORTION ON THE SPIN STATE OF 3d n IONS The D 2d distortion is due to the Γ 5

-displacive mode. It corresponds to the activation of the E u mode with an antiphase shift of the adjacent ligands in the square base. As mentioned in the introduction, the Γ 3 + strain mode can superimpose to Γ 5 while leaving the symmetry unchanged (D 2d ); however, such situation is not treated in this article. With the definitions in eqs 1a, 1b, 1c and Table

2

, the crystal field parameters are given by

Table A1 .

 A1 Rotation Groups (Miller-Love Notation and Point Group Notation into Brackets)A2g + B 2g + 2E g ) D 2d Γ 1 + Γ 2 + Γ 4 + Γ 5 (A 1 + B 2 + B 1 + E) 2Γ 1 + Γ 2 + Γ 3 + Γ 4 + 2Γ 5 (2A 1 + B 2 + A2 + B 1 + 2E) C 4v Γ 1 + Γ 2 + Γ 3 + Γ 5 (A 1 + B 1 + B 2 + E) 2Γ 1 + Γ 2 + Γ 3 + Γ 4 + 2Γ 5 (2A 1 + B 1 + B 2 + A2 + 2E) D 4 Γ 1 + Γ 2 + Γ 4 + Γ 5 (A 1 + B 1 + B 2 + E) 2Γ 1 + Γ 2 + Γ 3 + Γ 4 + 2Γ 5 (2A 1 + B 1 + A 2 + B 2 + 2E) E g + 2 2 E g ) S 4 Γ 1 + 2Γ 2 + Γ 3 + Γ 4 (A + 2B + 1 E + 2 E) 3Γ 1 + 2Γ 2 + 2Γ 3 + 2Γ 4 (3A + 2B + 2 1 E + 2 2 E) C 4 Γ 1 + 2Γ 2 + Γ 3 + Γ 4 (A + 2B + 1 E + 2 E) 3Γ 1 + 2Γ 2 + 2Γ 3 + 2Γ 4 (3A + 2B + 2Finally, changing the basis from {d m=-2.2 } to {d xy , d xz , d yz , d x 2 -y 2 , d z 2 }, one obtains the results given in 1d:

	point group	Γ l=2	Γ l=4
	O h	Γ 3 + + Γ 5 + (E g + T 2g )	Γ 1 + + Γ 3 + + Γ 4 + + Γ 5 + (A 1g + E g + T 1g + T 2g )
	D 4h + B 1g + C 4h Γ 1 + + Γ 2 + + Γ 4 + + Γ 5 + (A 1g + B 1g + B 2g + E g ) 2Γ 1 + + 2Γ 2 + + 2Γ 3 + + 2Γ 4 + + 2Γ 5 + (2A 1g Γ 1 + + 2Γ 2 + + Γ 3 + + Γ 4 + (Ag + 2B g + 1 E g + 2 E g ) 3Γ 1 + + 2Γ 2 + + 2Γ 3 + + 2Γ 4 + (3Ag + 2B g + 2 1

1 E + 2 2 E)

Table A2 .

 A2 Bielectronic Contribution to the Energy for d[START_REF] Ballhausen | Introduction to Ligand Field Theory[END_REF][START_REF] Watson | Iron Series Hartree-Fock Calculation II[END_REF][START_REF] Richardson | Approximate Radial Functions for First-Row Transition-Metal Atoms and Ions. I. Inner-Shell, 3d and 4s Atomic Orbitals[END_REF] Ions in Different Spin State for Any J H and When U -U′ Is Approximated to 2J H

						mono e HS	3(	4 ) (2	) ( 2	6 )
						(2		)	4
					E	HS bi e 4	1 2	U (4 U (5 1 2	U 4 ) 15 J J U 4 ) H H	U (4 J 38 1 2 H	U	)
					E	HS	E core	a	b 4	U 15	J 38 H
	ion shell d 4	spin state LS IS	2J H ≠ U -U′ 2U + 4U′ -2J H 1U + 5U′ -3J H	2J H = U -U′ 6U -10J H 6U -13J H	■ AUTHOR INFORMATION Corresponding Author
		HS	0U + 6U′ -6J H	6U -18J H
	d 5	LS	2U + 8U′ -4J H	10U -20J H
		IS	1U + 9U′ -6J H	10U -24J H
		HS	0U + 10U -10J H	10U -30J H
	d 6	LS	3U + 12U′ -6J H	15U -30J H
		IS	2U + 13U′ -7J H	15U -33J H
		HS	1U + 14U -10J H	15U -38J H

*(M.P.) E-mail: pollet@icmcb-bordeaux.cnrs.fr.

Notes

The authors declare no competing financial interest.

■ ACKNOWLEDGMENTS

We acknowledge M. Pouchard, A. Villesuzanne, and J-P. Doumerc for fruitful discussions and for their critical reading of the manuscript. M.P. acknowledges the financial support by the European project ''SOPRANO'' under Marie Curie actions (Grant no. PITN-GA-2008-214040). A.A. acknowledges the financial support by the European project ''EPREXINA'' under Marie Curie actions (Grant no. PIIF-GA-2009-255662).