1 Introduction

Exchange and correlation phenomena describe the effects of electron–electron interactions beyond the one-electron mean-field approximation; they are necessary to build correct electronic configurations based on an orbital description for 3dⁿ ions. A large part of the correlation energy, in the sense of electron–electron interactions, is included in mean-field approaches: for example, most one-electron schemes account for screening phenomena (i.e. Slater screening constant). In the Hartree–Fock framework, correlation energy is defined restrictively as the difference between the exact and the Hartree–Fock ground-state energy.

Exchange phenomena are more complex, including the symmetry of the wave functions and the Fermi hole, both giving rise to Hund’s rules [1]. In the case of partially occupied orbitals, for example 3d orbitals for transition metals, exchange will contribute to differentiate up- and down-spin populations.

In inorganic molecular chemistry, exchange and correlation are discussed on the basis of Racah A, B, C, or Slater parameters, which allow to quantify electron–electron interactions and result in the Tanabe–Sugano diagrams for octahedral complexes. What about solid-state chemistry?

Tight-binding methods as Extended Hückel (EHTB), which emphasise atomic orbital overlaps, do not include exchange and correlation effects. Moreover, in non self-consistent methods as EHTB, crystal or molecular orbital energies do not depend on the electron count.

Methods based on the Density Functional Theory (DFT) use approximate exchange-correlation functionals (so-called LDA, GGA, WDA...), based on the analytical expression for the exchange-correlation energy of an homogenous electron gas. Although closer to the exact energy than single-determinant Hartree–Fock, DFT remains at the mean-field stage. A step beyond correlation effects can be achieved through the LDA+U method, in which on-site electron–electron repulsion terms are added to the exchange-correlation functional [2].

Chemical bonding in transition metal oxides can be described on the basis of two types of atomic orbitals:

• • (i) s and p orbitals, for which ζ Slater exponents take low values (ζ_4s ∼ 1–1.5 [3]), corresponding to diffuse orbitals;
• • d orbitals, more localised (ζ_3d ∼ 2.5–4.5 [3]), describing strongly interacting electrons, for which exchange and correlation phenomena are often crucial.

As an example, a monoelectronic band description of the three following double oxides with rhombohedral perovskite-related structures would predict a metallic behaviour for LaFeO₃ (d⁵) and LaNiO₃ (d⁷), and an insulating one for LaCoO₃ (d⁶) (Fig. 1). Whereas LaNiO₃ indeed is a metal at any temperatures (within its stability range), LaFeO₃ actually is always insulating and LaCoO₃ undergoes an insulating to metal transition at high temperature. So a simple low-spin band description, neglecting exchange and correlation effects, does not correspond to experimental facts.

In LaFeO₃, trivalent iron (3d⁵) is not in the low-spin t_2g⁵e_g⁰ configuration (favoured by the crystal field), but in the high-spin t_2g³e_g² configuration (favoured by exchange and correlation). The competition between crystal field (which stabilises some energy levels) and exchange and correlation (which tend to keep electrons apart in space) is at the origin of spin equilibria and spin transition phenomena. At low temperature, the six 3d electrons of cobalt in LaCoO₃ lie in the three low-lying t_2g orbitals, giving an insulating state; at high temperature, exchange and correlation prevail and spread the six electrons in the five d orbitals. Then LaCoO₃ becomes a metal, with the cobalt ion in an intermediate spin state.

In the following, we will describe exchange and correlation effects for transition metal ions in various coordination polyhedra the symmetry of which strongly influences the energy splitting of the five 3d orbitals. We will extend the treatment of most-common octahedral (O_h) and tetrahedral (T_d) sites to the square-based pyramidal (C_4v), square planar (D_4h), dumbbell (D_∞h), cubic (O_h) and antiprismatic (D_4d) environments (Fig. 2).

We will restrict our discussion to d⁴, d⁵ and d⁶ ions, most interesting because they are to be found in three different spin states: S = 0, 1 and 2 for d⁴ and d⁶ ions; S = 1/2, 3/2 and 5/2 for the d⁵ ion. Our approach can then be applied to metal ions such as, for example, Fe⁴⁺ and Mn³⁺ (3d⁴), Fe³⁺ and Co⁴⁺ (3d⁵), Co³⁺ (3d⁶). Its extension to the 4d and 5d series should be handled cautiously because of the spin–orbit coupling.

The aim of this work is to guide inorganic chemists in the choice of appropriate crystal structures for stabilising a chosen electronic configuration for these 3d transition ions, and to help in the interpretation of their remarkable physical properties.

2 Expression of the exchange and correlation using U, U′ and J_H parameters of Kanamori and Brandow

According to Kanamori [4] and Brandow [5], Coulomb repulsion energy between two electrons belonging to the same atomic orbital (U) or to two different orbitals (U′) is balanced by an exchange energy (J_H) that stabilises parallel vs antiparallel spins. These parameters are expressed by the following integrals:

These energies are linked to Racah parameters through the following relationships [5]:

These relationships show that U, U′ and J_H are not independent and immediately lead to U – U′ = 2 J_H, which will allow us to compare the energy of the various spin states using a single parameter, the Hund’s rule exchange energy, J_H. For a 3d⁴ ion in an octahedral site and a HS configuration (t_2g³e_g¹), the energy of a singly occupied (so) e_g orbital can be written as:

Summing the crystal field energy over the four electrons and taking half the sum of exchange and correlation energies (as each contribution is shared by two electrons), the ion overall energy (E_T), including nucleus–electron and electron–electron interactions, becomes:

The so-called t_2g⁴(S = 1) low spin (LS) state actually is an intermediate spin (IS) state, but in no case the S = 0 state can be the ground state in an octahedral site of O_h symmetry owing to the Hund’s rule. Depending on whether the site is singly (so) or doubly occupied (do) and whether the spin is a majority (denoted α) or minority (denoted β) one, the electron energies respectively are:

In the same way as above, we obtain for the total energy:

It should be pointed out that in a T_d tetrahedral site where the e orbitals are stabilised by –2.67 Dq and t₂ orbitals are lifted by 1.78 Dq, the e⁴ LS (S = 0) configuration reversely becomes stable vs the S = 1 state. The energy of an electron belonging to one of the two doubly occupied degenerate e-orbitals (as there is the same number of spin up and spin down electrons) is:

Making equal the E_T energies of the HS (S = 2) and IS (S = 1) states of a d⁴ ion in O_h symmetry leads to J_H/Dq = 2. For high values of J_H, the HS state is stable and for high values of Dq the IS state on the contrary becomes the ground state.

In T_d symmetry, the condition for passing from S = 2 to S = 0 becomes J_H/Dq = 1.1125.

For the d⁵and d⁶ cations, the IS state (S = 3/2 or S = 1, respectively) is unstable in O_h symmetry from the following conditions: in the d⁵ case, the LS→IS transition requires J_H/Dq > 5/2, but the IS→HS transition requires only J_H/Dq > 5/3. Therefore, one predicts a direct LS→HS transition which is actually expected for J_H/Dq > 2. In the d⁶ case, the LS→IS transition requires J_H/Dq > 10/3, the IS→HS transition only J_H/Dq > 2. The direct LS→HS transition occurs at J_H/Dq > 2.5.

3 Influence of a structural distortion on spin transitions of a d⁴ ion

Let us first consider the case in which one of the six oxygen atoms moves away from the central ion, the coordination progressively changing from octahedral of O_h symmetry to a (5 + 1) coordination of C_4v symmetry until the limiting case of the square based pyramid is reached. In this type of coordination, a simple crystal field approach gives the following energy values in Dq units for the five d orbitals [6]:

In octahedral (O_h) coordination:

We assume that the magnitude of the distortion can be measured by a coefficient k (0 ≤ k ≤ 1) and that the energy of the orbital i is a function of k:

For instance, in the present case, with $E_i\left(0\right)={\mathrm{E}}_i^{O_h}$ and $E_i\left(1\right)={\mathrm{E}}_i^{C_{4\mathrm{v}}}$, we obtain:

Using these energy values E_i(k,Dq) to take into account the crystal field effect in the computation of the atomic orbital energies including the exchange correlation effects and then in the computation of the overall energy of the ion for each configuration, the following relations are obtained for the three possible spin equilibria:

It is now possible to delimit in a J_H/Dq-vs-k diagram the existence domains of each electronic configuration, S = 2, S = 1 and S = 0 (Fig. 3).

The three straight lines intercept at a triple point t of abscissa k_t where the three spin states are simultaneously stable (k_t = 0.923 and J_H/Dq = 1.142). As usual, if we look for the more stable state, only three half-line starting from t correspond to an actual equilibrium. As already pointed out, S = 0 cannot be stable in O_h symmetry. On the other hand, S = 1 cannot exist beyond the k value of 0.923. We can extend the calculations of the crystal field contribution to the five 3d atomic orbitals to the seven types of distortions illustrated in Fig. 2 , which encompasses most of the crystal sites usually observed in transition metal oxides. In cases (4) and (7), k represents a rotation from 0 to 60° and from 0 to 45°, respectively. Case (5) corresponds to a hypothetic case progressively passing from an octahedron (CN = 6) to a tetrahedron (CN = 4). The orbital energies are given in Table 1. The J_H/Dq-vs-k diagrams are given for each spin state in Figs. 4–6 , for 3d⁴, 3d⁵, and 3d⁶ ions, respectively. These diagrams show that, whereas in O_h symmetry no value of J_H/Dq allows to stabilise the three possible spin states of the d⁴, d⁵ and d⁶ ions – at least for narrow d levels –, most of the distortions (C_4v, D_4h, D_{∞ h}, D_{3 h}, D_4d) allow it, with one triple point t or even two triple points t₁ and t₂. For a d⁴ ion in D_4h symmetry (i.e. an octahedron distorting into a square) k_t1 = 0.465 and k_t2 = 0.735, and for a d⁵ ion in D_{3 h} symmetry (i.e. an octahedron distorting into a triangle based prism), k_t1 = 0.175 and k_t2 = 0.728. Some diagrams exhibit particular points of abscissa k_c generally corresponding to the energy crossover of some atomic orbitals as k varies. Let us finally notice that predicted spin transitions are occurring for J_H/Dq values close to 2 (±0.5). As according to literature, depending on the covalence of the bond, J_H values are ranging from 0.6 to 0.9, such spin transitions are expected for Dq values ranging from 1.2 to 1.8 eV (i.e. from 2400 to 3600 cm^–1 ± 25%).

4 Energies of atomic orbitals for dⁿ (n = 4, 5, 6) ions at the triple point taking into account exchange and correlation

In Fig. 7 we give the orbital splitting for dⁿ (n = 4, 5, 6) ions at some triple points (t) of the above diagrams (D_4d: d⁴; C_4v: d⁵ and d⁶) for each of the three spin states.

In each case, we can point out that various orbitals with majority α spin and minority β spin are degenerate. For instance, for a d⁶ ion in C_4v symmetry, the d^αdo_xy or ${\mathrm{d}}_{\mathrm{𝑥𝑦}}^{ϐ\mathrm{do}}$ orbital from S = 0 is degenerate with d^α_z² from S = 1. In the same way, ${\mathrm{d}}_{\mathrm{𝑥𝑧},\mathrm{𝑦𝑧}}^{ϐ\mathrm{do}}$ from S = 1 is degenerate, with d^α_x²-y² from S = 2.

For a band diagram, in the vicinity of the triple point, bands with different symmetry and spin directions will thus overlap.

Hence, starting from S = 1, a spin equilibrium will be described as:

• • (i) the excitation of an electron from ${\mathrm{d}}_{\mathrm{𝑥𝑧},\mathrm{𝑦𝑧}}^{ϐ\mathrm{do}}$ into (empty) d^α_x²-y² with a spin flip; all the α spin states will be stabilised by –J_H and the energy of the remaining ${\mathrm{d}}_{\mathrm{𝑥𝑧},\mathrm{𝑦𝑧}}^{ϐ}$ electron will increase by + J_H; in this way we have got the S = 2 (HS) state of Fig. 7;
• • (ii) the excitation of an electron from d^α_z² into d^αso_xy with a spin flip; it will increase the stability of the β spin states (by –J_H); we shall then obtain the S = 0 (LS) state ofFig. 7.

In the case of the d⁴ ion, we even observe a degeneracy of the HOMO – including both spin directions – of all the three (S = 0, 1 and 2) spin states simultaneously.

For S = 1, the HOMO itself contain both majority $\left({\mathrm{d}}_{\mathrm{𝑥𝑦}}^{\alpha }\right)$ and minority $\left({\mathrm{d}}_{{𝑧}^2}^{ϐ}\right)$ spin states.

5 Discussion

5.3 Exchange, correlation and insulating vs metallic character

The metallic-vs-insulating character of a transition-metal oxide depends on the relative magnitude of the transfer energy β_MO or of the bandwidth W and of the so-called Coulomb intra-atomic repulsion U_Hubbard, corresponding, in an ionic model, to the charge transfer Mⁿ⁺ + Mⁿ⁺ → M^(n–1)+ + M⁽ⁿ⁺¹⁾⁺. An upper limit of U _Hubbard is then given by the difference between the (n + 1)th and nth ionisation energies. Actually U_Hubbard and W are not independent and U_Hubbard decreases as W increases. The Mott–Hubbard metal–insulator transition is thus governed by the magnitude of the U _Hubbard/W ratio.

For a given dⁿ configuration, it is then important to evaluate the orbital energy of the d ^{n +1} configuration for the various possible spin states as a function of U or U′, and the type of orbital overlap that controls the value of W and thus the stabilisation due to the transfer. For instance, one sees that for a HS (S = 5/2) d⁵ ion in C_4v symmetry only the HS d⁶ configuration involving the π-type d_xz,yz orbitals is concerned (Fig. 7), whereas a IS (S = 3/2) ion can lead to two possible states S = 2 or S = 1 of the d⁶ configuration. Thus, the insulating-vs-metallic character of a dⁿ transition metal oxide results from the magnitude of the energy gap separating the dⁿ from the dⁿ⁺¹ configurations.

Confining us to the O_h symmetry and to the above d⁵/d⁶ case, the following electronic transitions can be considered, in which the type of bonding involved in the charge transfer is given between parentheses:

• • (i) d⁵_HS+ d⁵_HS → d⁴_HS + d⁶_HS (σ, π)
• • (ii) d⁵_IS+ d⁵_IS → d⁴_IS + d⁶_HS (σ)
• • (iii) d⁵_IS+ d⁵_IS → d⁴_HS + d⁶_IS (π)
• • (iv) d⁵_LS+ d⁵_LS → d⁴_IS + d⁶_LS (π)

for each of them we can calculate the energy gaps using the total energies including exchange, correlation and crystal field contributions for all the valence electrons (Table 2).

Table 2

Exchange and correlation energies for valence electrons of various d⁴, d⁵ and d⁶ configurations.

	HS	IS	LS
d4	6 U' – 6 JH – 6 Dq	U + 5 U' – 3 JH – 16 Dq
d5	10 U' – 10 JH	U + 9 U' – 6 JH – 10 Dq	2 U + 8 U' – 4 JH – 20 Dq
d6	U + 14 U' – 10 JH – 4 Dq	2 U + 13 U' – 7 JH – 14 Dq	3 U + 12 U' – 6 JH – 24 Dq

With the exception of (i), all the transfers correspond to a minimal energy of U′ – J_H. For the first case, the gap (U′ + J_H, assuming J_H/Dq = 2) is less favourable to such a transfer.

However, only (ii) involves σ-type bands that are broader than π-type ones. The ratio (U′ – J_H)/W_σ is then the most favourable for the formation of a metallic state from an IS state.

In the case of a lower symmetry favouring the HS↔IS spin equilibrium (Fig. 5), it should be noticed that case (i) can be split into two steps:

• • (i′) d⁵_HS + d⁵_HS → d⁵_IS + d⁵_HS
• • (i′′) d⁵_IS + d⁵_HS → d⁴_HS + d⁶_HS (σ)

the gap corresponding to (i′′) is equal to U′.

This HS/IS configuration mixing could also lead to a metallic state, as it implies a σ-type transfer.

This simple ionic-type approach of exchange and correlation explains the insulating behaviour of LaFeO₃and of LaCoO₃ at low temperature as well as the transition to a metallic behaviour at high temperature for the latter.

Indeed, for Fe³⁺ in LaFeO₃, the J_H/Dq ratio, much larger than 2.5, favours the exchange energy gain with respect to the crystal field and leads to a HS $\left({\mathrm{t}}_{2\mathrm{g}}^{\alpha 3}{\mathrm{e}}_{\mathrm{g}}^{\alpha 2}{\mathrm{t}}_{2\mathrm{g}}^{ϐ1}\right)$ state. The upper Hubbard band corresponding to the HS $\left({\mathrm{t}}_{2\mathrm{g}}^6{\mathrm{e}}_{\mathrm{g}}^0\right)$ d⁶ configuration lies at a much higher energy than the lower Hubbard band and is well separated from it, which accounts for the insulating character.

On the other hand, in LaCoO₃, Co³⁺ is in a LS

$${\mathrm{M}}^{5+}+{\mathrm{Fe}}^{3+}\to {\mathrm{M}}^{6+}+{\mathrm{Fe}}^{2+}$$

state at low temperature and thus insulating, because the crystal field energy overcomes the exchange contribution. As the temperature increases, the Co–O distances become longer, leading to a decrease of the covalence and of Dq, and the exchange and correlation parameters slightly increase, which leads to a change in the electronic configuration. LaCoO₃ was first described as undergoing a LS(S = 0)↔HS(S = 2) transition [9]. Recent works now describe the transition as LS(S = 0)↔IS(S = 1), which better agrees with our description of the insulator to metal transition given above [10, 11]. Double perovskites synthesised several decades ago such as Sr₂FeMoO₆ (M = Mo, W, Re...) and characterised by very different physical properties (either ferromagnetic metal or antiferromagnetic insulator) depending on the nature of M, are now revisited, since giant magnetoresistance has been observed for these materials [12]. If we consider the possibility of a charge transfer between M⁵⁺ and Fe³⁺ ions such as:

${\mathrm{t}}_{2\mathrm{g}}^{ϐ\mathrm{do}}$we have to compare the energy of iron d orbitals (E_Fe:3d = –16.5 eV) modified by crystal field, exchange and correlation effects (t^αso_2g of Fe²⁺ is shifted by 6 J_H above $\pi *\left({\mathrm{t}}_{2\mathrm{g}}^{ϐ}\right)$) with that of M⁵⁺ ions orbitals (Mo⁵⁺: 4d¹, W⁵⁺: 5d¹, Re⁵⁺: 5d²), the energies of which depend on the main quantum number n and atomic number Z (E_Mo:4d = –11.5 eV, E_W:5d = –11.0 eV, E_Re:5d = –12.3 eV) (Fig. 8).

We now understand why a partial transfer is possible in the molybdenum and rhenium cases: a ${ϐ}_{\mathrm{MO}}^2/\left(H_{\mathrm{ii}}\left(\mathrm{M}\right)-{\mathrm{H}}_{\mathrm{ii}}\left(\mathrm{O}\right)\right),$ band of the Fe–Mo(Re)–O type involving mixed valences Fe^III/Fe^II and Mo^V/Mo^VI gives rise to a metallic character and strong ferromagnetic interactions between localised iron moments via the itinerant π*^β electrons accounting for a high T_c value.

The highest energy of tungsten 5d orbitals leads to a complete charge transfer, giving single valence Fe²⁺(3d⁶) and W⁶⁺ (5d⁰) ions, resulting in an insulating character and weak antiferromagnetic interactions (T_N ≈ 20 K) between Fe²⁺ ions.

6 Conclusions

In order to clarify the competition of exchange and correlation energies vs crystal field stabilisation energy, a simple approach is proposed. The exchange and correlation contributions have been described on the base of Brandow and Kanamori’s U, U′ and J_H parameters. The dependence of the crystal field on site distortion has simply been modelled using a linear interpolation between undistorted and fully distorted site. This approach led to establish phase diagrams for d⁴, d⁵ and d⁶ cations, allowing us to predict the spin-state stability range depending on exchange (J_H), crystal field (Dq) and distortion (k) parameters. It can be used to complement the Extended Hückel Tight Binding calculations of the electronic structure of materials with a noticeable ionic character such as 3d transition-metal oxides, in order to interpret their electronic behaviour and, particularly, discuss their insulating-vs-metallic character, as we recently did [13] .