# The Microscopic Mechanisms Involved in Superexchange

^{†}

## Abstract

**:**

^{1}ions and X is a closed-shell diamagnetic ligand. In this article, we recall this formalism and give a physical interpretation: we may rigorously predict the ferromagnetic (J < 0) or antiferromagnetic (J > 0) character of the isotropic (Heisenberg) spin-spin exchange coupling. We generalize our results to nd

^{m}ions (3 ≤ n ≤ 5, 1 ≤ m ≤ 10). By introducing a crystal field we show that, starting from an isotropic (Heisenberg) exchange coupling when there is no crystal field, the appearance of a crystal field induces an anisotropy of exchange coupling, thus leading to a z-z (Ising-like) coupling or a x-y one. Finally, we discuss the effects of a weak crystal field magnitude (3d ions) compared to a stronger (4d ions) and even stronger one (5d ions). In the last step, we are then able to write the corresponding Hamiltonian exchange as a spin-spin one.

## 1. Introduction

^{m}electronic configuration without orbital degeneracy (m = 1) [1,2], later generalized to m > 1. In this case, the exchange Hamiltonian is of the Heisenberg–Dirac type: J

**s**

_{1}.

**s**

_{2}. Anderson’s publication has become the starting point for generalizations, notably with the introduction of orbital degeneracy [2,3,4,5,6,7,8,9].

^{1}ions: the exchange energy constant J has been expressed vs. fundamental molecular integrals, characterizing each of the σ–type bonds created by the presence of a diamagnetic ligand and a similar (or different) magnetic ion [10].

^{m}(3 ≤ n ≤ 5, 1 ≤ m ≤ 10). In addition, we rigorously show for the first time that the Hamiltonian is given by J

**s**

_{i}.

**s**

_{j}, where J is the “exchange constant”. The key physical points are as follows:

- J is expressed vs. fundamental molecular integrals in the absence of a crystal field, uniquely, for the sake of simplicity; we show that the introduction of a crystal field may be achieved very easily, thus allowing us to discuss further the notion of anisotropic couplings;
- For the first time, we may rigorously predict the ferromagnetic (J < 0) or antiferromagnetic (J > 0) character of spin-spin couplings whereas, so far, we have dealt with empirical rules, i.e., the Goodenough–Kanamori rules published between the middle of the 1950s and the beginning of the 1960s [11,12,13,14].

- Here, without a crystal field, we deal with a theoretical model, from which we derive the conclusion that when coulombic interactions are dominant, our model follows Hund’s rule and we explain why couplings are automatically ferromagnetic; when coulombic interactions are no longer dominant, our model is equivalent to the molecular orbital one and couplings are always antiferromagnetic (except in a particular case, where couplings are ferromagnetic but present a small absolute value of J);
- By introducing the notion of a crystal field, we discuss how passing from an isotropic (Heisenberg) coupling to an anisotropic one (z-z, i.e., an Ising-like coupling or an x-y one); in addition, from the theoretical expression of J, we may also predict the ferromagnetic (J < 0) or antiferromagnetic (J > 0) character of spin-spin couplings as in the absence of a crystal field, which is the key finding of this article.

^{1}ions, A and B, characterized by σ–type bonds on each side of the diamagnetic bridge, X. The cationic orbitals are of a d-type for A and B, whereas that of the diamagnetic ligand X is of the s- or p-type. We notably define and justify the general assumptions used for developing our theoretical model.

## 2. Microscopic Mechanisms Involved in Superexchange

#### 2.1. Basic Physical Considerations

#### 2.1.1. Generalities and Hund’s Rules

_{max}, L = L

_{max}(for S = S

_{max}) compatible with the exclusion principle are the respective values of the spin and orbit momenta along a z-axis of reference, the total momentum is

**$\mathcal{J}$**=

**L**+

**S**. If the external electronic shell is at most half-filled (

**L**and

**S**are antiparallel) $\mathcal{J}$ = |L − S|. If the shell is more than half-filled (

**L**and

**S**parallel)$\mathcal{J}$ = L + S (the unoccupied orbitals being considered as holes).

^{1}(ions V

^{4+}and Ti

^{3+}), where a single orbital is filled. Thus, according to Hund’s rules, the orbital momentum vanishes (L = 0, S = 5/2) at mid-filling of the 3d shell (case, 3d

^{5}, the ions, Fe

^{3+}and Mn

^{2+}). As a result, the configuration 3d

^{6}(the ion, Fe

^{2+}, L = 2, S = 2) is equivalent to the case 3d

^{4}(the ions Mn

^{3+}and Cr

^{2+}). Similarly, the configuration 3d

^{1}(ions V

^{4+}and Ti

^{3+}, L = 2, S = 1/2) is equivalent to the case 3d

^{9}(ion Cu

^{2+}), and so on.

^{1}, characterized by a single spin, ½, and coupled through a diamagnetic ligand X without a crystal field. Then we introduce a crystal field contribution. As it is one of the main sources of anisotropy for exchange, we shall discuss its influence on the nature of anisotropic couplings (z-z – Ising-type – or x-y couplings).

^{m}ions (1 < m ≤ 10), i.e., ions such as S > ½. In both cases, we consider a situation in which there is no crystal field and one for which a crystal field is introduced.

#### 2.1.2. The First “Historical” Model Proposed by Anderson for Superexchange

- The direct overlap of the involved wave functions characterizing the pair of magnetic sites A and B separated by the non-magnetic ligand X vanishes;
- The ligand wave function is weakly modified by the presence of magnetic ions;
- This modification confers a magnetic character that is the origin of the exchange interactions between the pair of magnetic ions through the non-magnetic ligand.

- Experimental measurements confirmed the transfer while examining the hyperfine interaction between the ligand nuclear spin and that of the magnetic ion;
- It has been graphically demonstrated that the ligand wave function is partially magnetic with the expected degrees;
- The electronic transfer of the up (or down) spin of ligand X to the empty left (or right) d orbital must remain ballistic, i.e., it conserves the spin so that it leads to an antiferromagnetic coupling;

_{ionic}(S

_{tot}) and ψ

_{excited}(S

_{tot}) are the respective wave functions of the starting ionic configuration and the excited one, the final state of the entity A–X–A is described via the global wave function ψ(S

_{tot}) = aψ

_{ionic}(S

_{tot}) + bψ

_{excited}(S

_{tot}), where a and b are small.

**r**and

**n**allow us to define the position of the concerned electron(s) and the

**τ**’s represent the fundamental translations of the lattice. Due to lattice periodicity, the function φ

_{m}is a Wannier function. This contribution has been called kinetic exchange because, during the formation of the weak chemical bond between A and X, the antiferromagnetic coupling of electronic spins is characterized by a gain in kinetic energy. If U is the coulombic repulsion energy, Anderson has defined the corresponding exchange energy as J

_{m}

_{,m’}(kinetic) = −2b

_{m}

_{,m’}(

**τ**)

^{2}/U.

_{mm}

_{’}= J

_{mm}

_{’}(potential) + J

_{mm}

_{’}(kinetic), with the conventional writing of an exchange Hamiltonian, $-2{J}_{m,{m}^{\prime}}{s}_{A}^{m}.{s}_{\mathrm{B}}^{m\prime}$ for a couple of d bands (m,m’), with one d band per ion [2,3,24].

#### 2.2. Starting Assumptions

#### 2.2.1. Assumption 1

^{1}in a first step, then 3d

^{m}ions, 1 < m ≤ 10). Regarding the compounds whose magnetic properties may be described, we have “natural” compounds, such as oxides, fluorides, and garnets (notably ferrites), that we call “Class I compounds”. In these compounds, the average size of the ligand orbital (p orbital) is close to that of the magnetic cation (d orbital). With the appearance of new “synthetized” magnetic compounds at the beginning of the 1990s, the magnetic cations are well separated by more or less long organic ligands [15,16,17,18]. In other words, the average size of a cation orbital is plainly lower than that of a ligand. We call these compounds “Class II compounds”.

_{e}is the electron mass. The potential operators V

_{1}= V(

**r**

_{1}) and V

_{2}= V(

**r**

_{2}) include all the nucleus and extra electron contributions to the Coulomb field acting on electrons 1 and 2, involved in the σ bond of the fragment A–X. The unique electron 1 comes from the 3d

^{1}external shell of the magnetic ion A, and electron 2 comes from the full external electronic shell of ligand X. As a result, we work within the framework of the Hartree–Fock approximation, i.e., the action of extra electrons over electrons 1 and 2 is taken into account through a mean-field approximation.

_{XB}may be written for the fragment X–B (with H

_{AX}= H

_{XB}when A = B). The electron coming from the external shell of X is labeled 3, and the one coming from B is labeled 4. All the physical quantities derived from the Hamiltonian H

_{AX}(respectively, H

_{XB}) are labeled Q

_{AX}for part A–X (respectively, Q

_{XB}for part X–B).

#### 2.2.2. Assumption 2

**r**) commutes with any symmetry operator $\mathcal{O}$ whose action is described by the properties of a double group $\mathcal{G}$ with the identity operation . As we deal with a two-electron problem on both sides of ligand X, separately, the two low-lying states of the bonds A–X and X–B are a spin-singlet and a spin-triplet. Indeed, if the spin wave function χ(

**s**

_{1},

**s**

_{2}) =|

**s**

_{1},

**s**

_{2}> =|

**s**

_{1}>⊗|

**s**

_{2}> = |${s}_{1}^{\mathrm{z}}$>⊗|${s}_{2}^{\mathrm{z}}$> = |${s}_{1}^{\mathrm{z}}$,${s}_{2}^{\mathrm{z}}$> describes the spin states, we have four possible pairings: |↑↑>,|↑↓>, |↓↑> and |↓↓>. If

**S**=

**s**

_{1}+

**s**

_{2},${S}^{z}={s}_{1}^{z}+{s}_{2}^{z}$ we have two classes of possibilities for writing the spin wave function:

- |0, 0> =$\frac{1}{\sqrt{2}}$(|↑↓> − |↓↑>), S = 0 (singlet state),
- $\begin{array}{l}|1,1>=|\uparrow \uparrow >\\ |1,0>=\frac{1}{\sqrt{2}}(|\uparrow \downarrow >+|\downarrow \uparrow >),\\ |1,-1>=|\downarrow \downarrow >\end{array}\}$ S = 1 (triplet state).

**s**

_{1}and

**s**

_{2}, while the triplet state is even.

#### 2.2.3. Assumption 3

_{a}(

**r**

_{1}), respectively, Φ

_{b}(

**r**

_{2}) describes the eigenstate of the Hamiltonian H

_{1}= T

_{1}= ${p}_{1}^{2}$/2m

_{e}, where m

_{e}is the electron mass, respectively, H

_{2}= T

_{2}= ${p}_{2}^{2}$/2m

_{e}, the secular equation det(H−E ) = 0 (where 1 is the identity matrix and H = T

_{1}+ T

_{2}+ U(

**r**

_{1},

**r**

_{2})) can be solved (cf. Equation (1)). The solutions are spatially symmetric and antisymmetric wave functions, i.e., ${\Phi}_{\mathrm{S}/\mathrm{A}}({r}_{1},{r}_{2})=({\Phi}_{a}({r}_{1}){\Phi}_{b}({r}_{2})\pm {\Phi}_{a}({r}_{2}){\Phi}_{b}({r}_{1}))/\sqrt{2}$, with <Φ

_{S}(

**r**

_{1},

**r**

_{2})|Φ

_{A}(

**r**

_{1},

**r**

_{2})> = 0 by construction.

#### 2.2.4. Assumption 4

**u**

_{1},

**u**

_{2}) = Φ(

**r**

_{1},

**r**

_{2})χ(

**s**

_{1},

**s**

_{2}), where u

_{i}= (

**r**

_{i},

**s**

_{i}). When combining the respective parity properties of functions Φ(

**r**

_{1},

**r**

_{2}) and χ(

**s**

_{1},

**s**

_{2}) with Pauli’s exclusion principle, we must have:

- S = 0 χ(
**s**_{1},**s**_{2}) = |**s**_{1},**s**_{2}> odd, Φ(**r**_{1},**r**_{2}) = Φ_{S}(**r**_{1},**r**_{2}) even, - S = 1 χ(
**s**_{1},**s**_{2}) = |**s**_{1},**s**_{2}> even, Φ(**r**_{1},**r**_{2}) = Φ_{A}(**r**_{1},**r**_{2}) odd.

#### 2.2.5. Assumption 5

_{1/2}is the representation of a spin ½, and if coupling a pair of these spins, we then have to consider the operation Γ

_{1/2}⊗ Γ

_{1/2}= Γ

_{0}⊕ Γ

_{1}, where ⊕ is the direct sum symbol. Γ

_{0}and Γ

_{1}are the corresponding irreducible representatives (irrep) with dim Γ

_{0}= 1

_{,}dim Γ

_{1}= 3 and dim (Γ

_{1/2}⊗ Γ

_{1/2}) = 4.

#### 2.2.6. Assumption 6

**r**) as well as that of energy for the AXB entity. The three corresponding atomic orbitals which are magnetic orbitals [28,29,30] are Φ

_{A}and Φ

_{B}centered on A and B, respectively, and Φ

_{X}centered on the bridge, X. The corresponding states are |A>, |B> and |X>. Φ

_{A}, Φ

_{B}and Φ

_{X}are assumed to be real and are considered as starting (non-disturbed) wave functions i.e., free atomic wave functions that give a spatial description of each of the states |A>, |B> or |X>. In the present case, Φ

_{A}and Φ

_{B}are cationic d-orbitals and Φ

_{X}is an anionic (s or p) orbital.

#### 2.2.7. Assumption 7

#### 2.2.8. Assumption 8

- The states are normalized but not orthogonal (except |A> and |B>):

- The overlap between A and X on the one hand, and X and B on the other one, are defined as follows:

**s**

_{A}, <X|B> =

**s**

_{B},

**s**

_{A}> 0,

**s**

_{B}> 0.

**s**

_{A}=

**s**

_{B}= s > 0.

- Both magnetic sites A and B have a cationic energy level higher than the anionic one, as is generally the case for transition metal compounds; the energy difference between A and X levels (respectively, X and B levels) is 2δ
_{A}E_{A}, with E_{A}> 0 (respectively, 2δ_{B}E_{B}, with E_{B}> 0), so that we have for the fragment A–X linked to X–B:

_{1}+ V

_{1}) |A> = − (1 − δ

_{A})E

_{A}, <B|(T

_{4}+ V

_{4})|B> = − (1 − δ

_{B})E

_{B},

<X|(T

_{2}+ V

_{2})|X> = − (1 + δ

_{A})E

_{A}, E

_{A}> 0, <X|(T

_{3}+ V

_{3})|X> = − (1 + δ

_{B})E

_{B}, E

_{B}> 0¸ A ≠ B,

<A|(T

_{1}+ V

_{1})|A> = <B|(T

_{4}+ V

_{4})|B> = − (1 − δ)E,

<X|(T

_{2}+ V

_{2})|X> = <X|(T

_{3}+ V

_{3})|X> = − (1 + δ)E, E > 0, A = B.

_{A}< 1 for fragment A–X (respectively, δ

_{B}for fragment X−B) but is not necessarily small. δ

_{i}> 0 (with i = A or B) is a very common scenario, and δ

_{i}< 0 corresponds to the particular case of the dihydrogen molecule (with A = B, X being absent in that particular case).

_{AB}, which is defined by an analogy with respect to δ

_{A}and characterizes the link between fragments A–X and X–B. In Figure 3, we also note that, due to the stabilization of AXB during the creation of the bonds A–X and X–B, we do have −(1 + δ

_{A})E

_{A}= −(1 + δ

_{B})E

_{B}. Of course, when A = B, δ

_{A}= δ

_{AB}= δ

_{B}.

- The transfer integrals between |A> and |X> on the one hand, and |X> and |B> on the other hand, are given by:

_{2}+ V

_{2})|X> = − t

_{AX}E

_{A}, <X|(T

_{3}+ V

_{3})|B> = − t

_{XB}E

_{B}, <A|(T

_{i}+ V

_{i})|B> = 0, i = 1,4.

_{AX}= t

_{XB}> 0, but when A ≠ B, t

_{AX}≠ t

_{XB}> 0. The previous equation, stating that there is no transfer between A and B but exclusively between A and X or X and B, is a consequence of the condition imposed by Equation (2). However, as previously noted, the case <A|(T

_{1}+ V

_{1})|B> ≠ 0 may be introduced in a more general model, without difficulty, notably by the bias of an π orbital between A and B.

**s**_{A}, t_{AX}and t_{XB}are small compared to unity, and t_{AX}or t_{XB}is mainly related to the potential interaction between the anion and the cation, so that:

#### 2.3. Expression of the Intermediate “Cationic” States

- The first rule consists of treating the extra electrons in terms of the simple one-electron Hartree–Fock functions;
- The second rule is treating them as excitations of a many-body system; this operation is achieved while keeping a constant value for the total spin involved, S
_{tot}; this leads us to consider an ionic part for centers A and B and an excited one for the ligand X; - As we deal with weak energies involved in the process of excitation, the orbital part ψ(S
_{tot}) may be written as the following hybridization: ψ(S_{tot}) = aψ_{ionic}(S_{tot}) + bψ_{excited}(S_{tot}), where the coefficients a and b must remain small.

_{i}+ V

_{i}(i = 1,2) in the reduced basis {|A>, |X>} for the fragment A–X (respectively, T

_{i}+ V

_{i}(i = 3,4) in the reduced basis {|X>, |B>} for the fragment X−B, separately). The goal of such an operation is to obtain the new cationic (antibonding) normalized eigenstates |$A$> and |$B$>, such as:

_{i}and β

_{i}(with i = A or B) are real numbers. As we deal with a weak chemical bond between A and X (respectively, between X and B), β

_{i}must remain small, and α

_{i}, close to unity. In addition, if using the normalization condition <A|A> = <B|B> = 1, as well as <A|X> = s

_{A}, <X|B> =

**s**

_{B}(

**s**

_{A}> 0,

**s**

_{B}> 0), the new normalization condition <$A$|$A$> = 1 and <$B$|$B$> = 1 leads to the following equation:

_{i}= – β

_{i}

**s**

_{i}+ $\sqrt{1-{\mathsf{\beta}}_{i}^{2}+{({\mathsf{\beta}}_{i}{\mathit{s}}_{i})}^{2}}$ (with i = A or B). If defining the new direct overlap

**S**, we derive owing to Equation (2):

**S**

_{A}and

**S**

_{B}are functionals of the various overlaps corresponding to the bonds A–X and X−B, respectively, we have:

_{A}≠ $|E>$

_{B}) but, if A = B, we are dealing with a degenerate state ($E$= $E$

_{A}= $E$

_{B}). Introducing the definition of |$A$> and |$B$> given by Equation (7), we may write, owing to Equations (4) and (5):

_{i}(with i =A or B), may now be chosen so that $E$

_{i}is minimum, i.e., owing to Equation (14), the equation ∂$E$

_{i}/∂β

_{I}= 0, after a few calculations, yields:

_{A}and S

_{B}are given by Equation (10). Thus, β

_{i}and 1 − α

_{i}(with i = A or B) may show the same sign or opposite signs: if β

_{i}> 0 (respectively, β

_{i}< 0) the state |$A$ > or |$B$> will be represented by a spatially symmetric wave function (respectively, spatially antisymmetric). In addition, as β

_{i}is small, 1 − α

_{i}is also small, and α

_{i}is close to unity, as expected. Then, using the particular value of β

_{i}given by Equation (15), the ground state energy is:

_{AX}= t

_{XB}. The sign ± comes from that of β. We finally define the transfer integral $T$

_{AB}as:

_{AB}= $T$

_{BA}. When A ≠ B, we always have $T$

_{AB}= $T$

_{BA}on the condition that (1 + δ

_{A})E

_{A}= (1 + δ

_{B})E

_{B}(cf. Figure 3). This illustrates the principle of indistinguishability of electrons: the electronic transfer may indifferently occur not only from X to A (case 1) but also from X to B (case 2), with the same physical effect. However, in both cases, we deal with the coupling of 2 electrons on the side of A or on the side of B.

_{A}= α

_{B}= 0, β

_{A}= β

_{B}= 0. From Equations (14) and (18), we derive, as expected, ${E}_{A}=-(1-{\mathsf{\delta}}_{A}){E}_{A}$$,$ ${E}_{\mathrm{B}}={E}_{A\mathrm{B}}=-(1-{\mathsf{\delta}}_{\mathrm{B}}){E}_{\mathrm{B}}$$,T$

_{AB}= 0 (no transfer between A and B).

_{i}, (with i = A or B) given by Equation (15), we have:

_{i}> 0 and – holds for β

_{i}< 0. As E

_{i}> 0, 1 − S

_{i}> 0, 1 + δ

_{i}> 0 whatever the sign of δ

_{i}E

_{A}≈ E

_{B}> 0, and |t|<< 1, $T$

_{AB}(if A ≠ B) or $T$ (if A = B) is negative. Thus, before constructing the collective states, it is clear that S

_{A}, $E$

_{A}and $T$

_{AB}(respectively, S

_{B}, $E$

_{B}and $T$

_{BA}) appear as the basic parameters of the bond A–X (respectively, X−B) and finally characterize the collective states of AXB.

#### 2.4. Construction of the Collective States

_{i}= (

**r**

_{i},

**s**

_{i}) and p

_{1}, p

_{2}, …, p

_{N}characterize the states occupied by these fermions. It is given by the following Slater determinant:

_{i}and p

_{j}

_{,}is as p

_{i}= p

_{j}, the determinant shows two identical lines and vanishes, in agreement with Pauli’s principle, which states that two identic fermions cannot occupy the same state.

^{−1/2}are self-evident normalizing factors and σ = ± recalls the nature of the corresponding ½ spin state (“up” or “down”). At this point, due to the orthogonality condition <$\mathcal{X}$,σ|$\mathcal{Y}$,σ’ > = δ

_{XY}δ

_{σσ}’ (with $\mathcal{X}$ = $A$ or $B$), now, we must have <X,σ|X,σ’ > = δ

_{σσ}’, with X = g or u. Then it is easily shown that the related energies are:

_{i}, s

_{i}(with i = A or B), t, t

_{AX}, t

_{XB}(A = B or A ≠ B) and S are small:

_{Ag}− $E$

_{Au}is independent of the sign of β

_{i}, as expected, and remains very small. In the particular case where there is no superexchange, we recall that S = 0 and $T$ = 0, so that $E$

_{Ag}= $E$

_{Au}.

^{4}determinantal collective states may then be built on each side of the fragment A–X–B. However, due to Pauli’s exclusion principle, coupled to one of indistinguishability, when dealing with 4 fermions coupled in pairs (one per centers A and B, 2 for X with one electron possibly transferred to A, and one possibly transferred to B), the number of states reduces to $\left(\begin{array}{c}4\\ 2\end{array}\right)=6$.

_{S,SZ}> the collective states. We have X = U (ungerade) or X = G (gerade) if referring to the symmetry of the orbital part with respect to the interchange of |$A$> and |$B$>; S and S

^{z}describe the total spin configuration. We define the collective basic state |X,σ; Y,σ’> as the corresponding Slater determinant:

_{11}>,|U

_{10}>,|U

_{1-1}>,|U

_{00}>,|${G}_{0,0}^{g}$> and |${G}_{0,0}^{u}$> on the basis of the following Slater determinants for the fragment A–X−B:

#### 2.5. The Hamiltonian Matrix and Energy Spectrum

_{0,0}>, |U

_{1,1}>, |U

_{1,0}>, |U

_{1,}

_{−1}>}. The non-vanishing terms are those existing between states belonging to the same irreducible representation (irrep) of the orbital $\mathcal{G}$ and spin $\mathcal{R}$ symmetry groups so that the final group is $\mathcal{G}$ ⊗$\mathcal{R}$. As a result, one may expect:

- Diagonal and off-diagonal terms between |${G}_{0,0}^{g}$> and |${G}_{0,0}^{u}$>;
- Only diagonal terms for the states |U
_{S,SZ}> with S = 0 (S^{z}= 0) and S = 1 (S^{z}= 0, ±1); - All the diagonal terms of the states |U
_{1,SZ}> are equal because we deal with the irrep Γ_{1}⊗Γ_{3,u}. Under these conditions, the Hamiltonian matrix is:

_{1}−X−A

_{2}(A = B). Then, we define the following quantities:

_{12}=|

**r**

_{1}−

**r**

_{2}|, the electron labeled 1 is coming from A, the one labeled 2, from X. The physical meaning of the parameters U, C, γ

_{1}and γ

_{2}is simply the following one:

- U is the Coulomb energy for an electron pair occupying the same site;
- C is the Coulomb energy for two electrons occupying neighboring sites;
- γ
_{1}is the Coulomb self-energy of the exchange charge distribution −eΦ_{A}(**r**)Φ_{B}(**r**) and is, thus, referred to as the exchange integral; - γ
_{2}appears as the Coulomb energy between the exchange charge distribution and an electron charge localized on one site. γ_{2}is a transfer integral between two cationic orbitals, resulting from the effective coulombic potential created by the charge of another electron involved in the secular problem; - When there is no superexchange, i.e., no exchange between A and B through X, we have γ
_{1}≠ 0 (the exchange charge distribution is restricted to the bond between A and X, X and B), γ_{2}= 0 as there are no more cationic orbitals and U ≠ 0, C ≠ 0 (the Coulomb energy for two electrons is restricted to first neighboring sites: A and X or X and B).

_{A}≠ U

_{B}. C, γ

_{1}and γ

_{2}conserve the same definition but not the same value as in the case where A = B.

_{Ag}, $E$

_{Au}are given by Equation (22) for A = B or A ≠ B and U, C, γ

_{1}and γ

_{2}by Equation (32). When A ≠ B U is replaced by (U

_{A}+ U

_{B})/2, the contribution of C, γ

_{1}and γ

_{2}is unchanged, though showing a different value than in the case where A = B. In addition, by diagonalizing the upper 2 × 2 matrix in Equation (30), we have the following eigenvalues:

^{p}and E

^{np}for |${G}_{0,0}^{p}$> and |${G}_{0,0}^{np}$>, respectively:

## 3. Physical Interpretation

#### 3.1. Expression of J_{m,m′}

- The states |U
_{S,Sz}> with S = 1 (S^{z}= 0, ±1) that are associated with a “triplet state”, characterized by the eigenvalues ${E}_{0}^{U}$ and ${E}_{1}^{U}$(three-times degenerated); and - The states |${G}_{0,0}^{g}$> and |${G}_{0,0}^{u}$> , with S = 0, are associated with a “singlet state”, characterized by the eigenvalues ${E}_{0,0}^{\pm}$.

_{S}

_{,0}and E

_{T}

_{,0}the low-energy levels of the singlet and triplet spectra, respectively, it is worth recalling that the exchange energy may be defined according to two conventional writings:

- J
_{m}_{,m′}= E_{S}_{,0}– E_{T}_{,0}with the corresponding Hamiltonian exchange H^{ex}= −J_{m}_{,m′}**s**_{1}.**s**_{2}(convention 1); in that case, J < 0 corresponds to an antiferromagnetic arrangement, with E_{T}_{,0}> E_{S}_{,0}, whereas J > 0 corresponds to a ferromagnetic one, with E_{T}_{,0}< E_{S}_{,0}, where m and m′ are the name of d bands located on each side of the ligand X. - J
_{m}_{,m′}= E_{T}_{,0}– E_{S}_{,0}with the corresponding Hamiltonian exchange H^{ex}= J_{m}_{,m′}**s**_{1}.**s**_{2}(convention 2); in that case, J > 0 corresponds to an antiferromagnetic arrangement, with E_{T}_{,0}> E_{S}_{,0}, whereas J < 0 corresponds to a ferromagnetic one, also with E_{T}_{,0}< E_{S}_{,0}.

**s**

_{i}.

**s**

_{j}, here J = E

_{T}

_{,0}– E

_{S}

_{,0}, is submitted according to a couple of conditions:

- J << Δ
- k
_{B}T << Δ

_{B,}the Boltzmann’s constant, and Δ = E

_{S}

_{,1}– E

_{S}

_{,0}; E

_{S}

_{,1}is the first excited-state energy in the singlet spectrum.

_{m,m}

_{′}may be written for the fragment A–X−A:

_{1}and γ

_{2}are, respectively, given by Equations (16), (19) and (32). ${E}_{Ag}-{E}_{Au}\approx 2T$ represents the electronic energy transfer between the magnetic cations and the non-magnetic ligand, in the case of weak overlap (S << 1).

_{m,m}

_{′,0}be the corresponding exchange energy, always given by Equation (41), in which $T$ = 0, U ≠ 0, C ≠ 0, γ

_{1}≠ 0 but γ

_{2}= 0. We retrieve from Equation (41) that J

_{m,m}

_{′,0}= −2γ

_{1}, as expected. We just have an exchange between A and X on the one hand, and X and B on the other hand, without a connection between A and B.

^{ex}= J

**s**

_{1}.

**s**

_{2}), the exchange energy magnitude is J ≈ −(j – uS

^{2}), where u is the Hartree term (direct term) and j the Fock term (exchange term), respectively, defined as:

_{AX}= J

_{XB}≈ j. The magnitude of J

_{AX}(respectively, J

_{XB}) reduces to the Fock (exchange) term, as expected, so that for the two bonds, A–X and X−B, considered separately, we have J

_{0}≈ −2j. If comparing with Equations (32) and (33), we have j ≈ γ

_{1}and u = U so that finally we retrieve:

_{m,m}

_{′,0}= J

_{0}≈ −2γ

_{1}.

_{m,m}

_{′,0}= J

_{0}):

_{m,m}

_{′}for the most commonly encountered physical case. Indeed, from the definitions of γ

_{2}, C, γ

_{1}and U, given by Equation (32), we have the following classification:

_{2}<< C ≈ γ

_{1}<< U

_{1}and r

_{2}(cf. Equations (32) and (33)) appearing in the arguments of the functions, giving the spatial behavior of the involved atomic orbitals.

_{1}with respect to the difference |$E$

_{Ag}−$E$

_{Au}|, which is very small, γ

_{2}being strongly negligible. Thus, two kinds of situations may occur: $\left|{E}_{\mathrm{A}g}-{E}_{\mathrm{A}u}\right|\approx 2\left|T\right|<<U$ or $\left|{E}_{\mathrm{A}g}-{E}_{\mathrm{A}u}\right|\approx 2\left|T\right|>>U$.

_{m}

_{,m′,1}may take values of order, varying between one eV to tenths of eV. We deal with ferromagnetic couplings. Thus, the coulombic interaction favors a ferromagnetic coupling. This is due to a subtle mechanism that is a consequence of the first Hund’s rule. This aspect is detailed in Section 3.2.1.

**Case 2:**$\left|{E}_{\mathrm{A}g}-{E}_{\mathrm{A}u}\right|\approx 2\left|T\right|>>U$ (see Figure 5).

_{m}

_{,m’,2}shows a small value in eV, as suggested by Anderson’s model. We are dealing with antiferromagnetic couplings. This mechanism is detailed in Section 3.2.2. because we now deal with a molecular orbital model. We may deal with Class I compounds (S << 1). However, this is always the case with Class II compounds because, due to the more or less important length of ligand X, the involved electrons show a very small coulombic interaction magnitude. $\left|{E}_{\mathrm{A}g}-{E}_{\mathrm{A}u}\right|\approx 2\left|T+SE\right|$ is greater now, with S < 1.

_{Ag}−$E$

_{Au}| is smaller and we again find case 1: the couplings are now ferromagnetic, but the corresponding value of J is very small.

_{2}+ $T$)

^{2}/(U – C), involving the transfer integral $T$.

#### 3.2. Physical Comments Regarding the Sign of J

#### 3.2.1. Hund’s First Rule

_{Ag}+ $E$

_{Au}= 2$E$, $E$ being given by Equation (16). For an atom, γ

_{2}= 0 and, for a molecule or a polyatomic ion, γ

_{2}≠ 0. These eigenvalues must be compared to ${E}_{0}^{U}$ and ${E}_{1}^{U}$:

_{T}

_{,0}and is stabilized by the factor 2γ

_{1}(few eV for intra-atomic exchanges) with respect to the first excited level, E

_{S}

_{,0}, even in the presence of orbital degeneracy (cf. Figure 4). The ferromagnetic interaction based on the Coulomb exchange integral γ

_{1}is called the Heisenberg exchange.

#### 3.2.2. Molecular Orbital Model

^{z}= 0, ± 1.

#### 3.3. Superexchange Hamiltonian for a 3d^{1} ion. Generalization to an nd^{m} ion (3 ≤ n ≤ 5, 1 ≤ m ≤ 10)

**τ**’s represent the fundamental translations of the lattice. b

_{m,m}

_{’}(

**τ**) is the transfer integral between d bands labeled m and m’; (m,m’) is the corresponding pair of involved bands; U > 0 is the Coulomb repulsion energy. Anderson has called this contribution the “kinetic” part of the exchange J

_{m,m}

_{’}((kinetic). The “potential” part of the exchange, J

_{m,m}

_{’}(potential), is defined at the end of Section 2.1.2, and the total exchange is given by J

_{m,m}

_{’}= J

_{m,m}

_{’}((kinetic) + J

_{m,m}

_{’}(potential).

_{T}and P

_{S}are projectors:

^{t}is a transposed matrix, and is the 4 × 4 identity matrix. Under these conditions, it is easy to show:

_{T}−3P

_{S}= (P

_{T}− P

_{S})(P

_{T}− 3P

_{S}) − 6P

_{S}

^{∗}= <Ψ| because |Ψ> is real. Finally, using the fact that <Ψ|Ψ> = 1, we may finally write:

**q**(t), the action S must be minimal with respect to

**q**(t). The action S may be expressed with the Lagrangian system,$L\left(q(t),\stackrel{\u2022}{q(t)}\right)$, where $\stackrel{\u2022}{q(t)}=dq(t)/dt$ and, after an adequate Legendre transform, with the Hamiltonian system H. As a result, minimizing S means that, at the equilibrium, H must be minimized. In gauge field theories, notably applied in particle physics, this process is called a minimal coupling scheme. Under these conditions, the equilibrium of any dynamic situation is described by the following Hamiltonian:

^{1}characterized by bands m and m’ involved on each side of ligand X:

_{m}

_{,m’}is given by Equation (40).

^{1}to nd

^{5}, we have different half-filled electronic shells, so that the value of the couple (m,m’) varies between 1 and 5. If this value is greater than 5, only the half-filled shells intervene in the calculation of J

_{m}

_{,m’}so that an ion of nd

^{6}(L = 2, S = 2), nd

^{7}(L = 3, S = 3/2), nd

^{8}(L = 3, S = 1) and nd

^{9}(L = 2, S = 1/2) is treated as an ion nd

^{4}, nd

^{3}, nd

^{2}, and nd

^{1}, respectively.

#### 3.4. Introduction of Crystal Field Theory

#### 3.4.1. Expression of J_{m,m′}; Physical Discussion of the Crystal Field Effect

_{k,}due to the fact that there are very weak overlaps between first-nearest neighbors. We do not discuss in this paper the symmetry properties of the cage and the correlative simplifications that may occur.

_{1}, OX

_{2}and OX

_{3,}three orthonormal axes and

**r**= (x

_{1}, x

_{2}, x

_{3}) the Cartesian coordinates of any point M bearing an electron (with the usual correspondence, x

_{1}= x, x

_{2}= y, x

_{3}= z). As a result, the potential describing the crystal field effect may be written as:

**R**

_{k}= (X

_{1},X

_{2},X

_{3}) is the position vector associated with each ion k of the surrounding cage (R

_{k}= C

^{st}). Considering the two cases, q

_{k}e > 0 or q

_{k}e < 0, means that we envisage the possibility not only of an electrically neutral entity constituted by the cage and the inserted cation A or B but also that the set cage + cation may show a positive or a negative electrical charge. Finally, we may write the full electrostatic contributions of the crystal field for the fragment A–X (electron 1 coming from cation A and electron 2 coming from anion X):

^{iso}in the absence of a crystal field are altered. We then have:

_{A}and $T$, $E$

_{Ag}and $E$

_{Au}, respectively, as given by Equations (1), (16), (18) and (22), the coulombic terms ${U}_{A}^{\mathrm{iso}}$, ${C}_{}^{\mathrm{iso}}$, ${\gamma}_{1}^{\mathrm{iso}}$ and ${\gamma}_{2}^{\mathrm{iso}}$, given by Equation (32), and the coulombic contributions of crystal field ${U}_{\mathrm{A}}^{\mathrm{C}\mathrm{F}}$, ${U}_{\mathrm{C}}^{\mathrm{C}\mathrm{F}}$, ${U}_{{\gamma}_{1}}^{\mathrm{C}\mathrm{F}}$ and ${U}_{{\gamma}_{2}}^{\mathrm{C}\mathrm{F}}$ defined in Equation (63) can be labeled by the generic term Y

_{k}and decomposed as a sum of their respective Cartesian components along the axes OX

_{1}, OX

_{2}and OX

_{3,}called ${Y}_{k;{x}_{i}}$(as a scalar is a zero-rank tensor):

_{i}-component ${Y}_{{k}_{{x}_{i}}}$, components ${Y}_{k}$ characterized by different k values are not equal (except when the site in which the cation is inserted shows particular symmetry properties). Due to the definitions of ${U}_{\mathrm{A}}^{\mathrm{C}\mathrm{F}}$, ${U}_{\mathrm{C}}^{\mathrm{C}\mathrm{F}}$, ${U}_{{\gamma}_{1}}^{\mathrm{C}\mathrm{F}}$ and ${U}_{{\gamma}_{2}}^{\mathrm{C}\mathrm{F}}$(cf Equation (63)), we may give the same mathematical argument regarding the respective values of r and R

_{k}appearing in the argument of the atomic orbitals (cf Equation (A12) in Appendix A). We have (cf. Equation (42)):

_{1}, OX

_{2}and OX

_{3}, according to Equation (64).

_{m,m}

_{’}) and no superexchange (J

_{m,m}

_{’,0}) is:

**Case 1:**$\left|{E}_{\mathrm{A}g}^{\mathrm{anis}}-{E}_{\mathrm{A}u}^{\mathrm{anis}}\right|=\left|{E}_{\mathrm{A}g}^{\mathrm{iso}}-{E}_{\mathrm{A}u}^{\mathrm{iso}}\right|\approx 2\left|T\right|<<{U}^{\mathrm{anis}}$.

_{k}e < 0), as ${\gamma}_{1}^{\mathrm{iso}}$> 0 and ${J}_{m,{m}^{\prime},1}^{\mathrm{iso}}=-2{\gamma}_{1}^{\mathrm{iso}}<0$ (cf. Equation (43)), we always have ${J}_{m,{m}^{\prime},1}^{\mathrm{anis}}<0$ according to Equation (71), and all the coulombic interactions favor a ferromagnetic coupling. This effect is enhanced by the crystal field (CF) contribution, independently of its magnitude. This is also due to a subtle mechanism that is a consequence of the first Hund’s rule, in spite of the presence of the crystal field. This aspect has been detailed in Section 3.2.1. This allows us now to justify the fact that Class I and Class II compounds enter this case.

_{k}e > 0), we have two possibilities, when examining Equation (71):

- ${\gamma}_{1}^{\mathrm{iso}}>>-2{U}_{{\gamma}_{1}}^{\mathrm{C}\mathrm{F}}>0$; we deal with a weak CF contribution (in the case of 3d
^{m}ions); ${J}_{m,{m}^{\prime},1}^{\mathrm{anis}}\approx -2{\gamma}_{1}^{\mathrm{iso}}<0$; the surrounding cage is mainly characterized by an important geometrical size: we may deal with Class I compounds. This is also the case when using organic ligands whose long length may be adapted to the magnetic system that one wishes to build up [15,16,17,18]; this is a good way to obtain isotropic (Heisenberg) spin-spin couplings for Class II compounds; we always have ferromagnetic couplings, including in the particular case of ${\gamma}_{1}^{\mathrm{iso}}\approx -2{U}_{{\gamma}_{1}}^{\mathrm{C}\mathrm{F}}>0$ so that, finally, the ferromagnetic coupling is strongly enhanced and ${J}_{m,{m}^{\prime},1}^{\mathrm{anis}}\approx -4{\gamma}_{1}^{\mathrm{iso}}\approx -4{U}_{{\gamma}_{1}}^{\mathrm{C}\mathrm{F}}<0$ (in the case of ions 4d^{m}and 5d^{m}); - ${\gamma}_{1}^{\mathrm{iso}}<<-2{U}_{{\gamma}_{1}}^{\mathrm{C}\mathrm{F}}>0$; we deal with a strong CF contribution (case of 5d
^{m}ions); now we have ${J}_{m,{m}^{\prime},1}^{\mathrm{anis}}\approx -2{U}_{{\gamma}_{1}}^{\mathrm{C}\mathrm{F}}>0$; antiferromagnetic couplings are favored and this only concerns Class I compounds.

_{3}= Oz is favored (with respect to OX

_{1}= Ox and OX

_{2}= Oy) or the axes OX

_{1}= Ox and OX

_{2}= Oy are equally favored by the crystal field (with respect to OX

_{3}= Oz). Then, we have:

**Case 2:**$\left|{E}_{Ag}^{\mathrm{anis}}-{E}_{Au}^{\mathrm{anis}}\right|=\left|{E}_{Ag}^{\mathrm{iso}}-{E}_{Au}^{\mathrm{iso}}\right|\approx 2\left|T\right|>>{U}^{\mathrm{anis}}$

_{Ag}−$E$

_{Au}|, which is insensitive to the crystal field by definition. This is in the case of 3d ions.

#### 3.4.2. Expression of the Hamiltonian

_{T}and P

_{S}are projectors (defined by Equation (49)) and are acting in the triplet and singlet subspaces, respectively. ${\left({P}_{T}-3{P}_{S}\right)}_{xy}$ is the projector acting in the subspace corresponding to the x-y plane and ${\left({P}_{T}-3{P}_{S}\right)}_{zz}$ acts along the z-axis.

## 4. Conclusions

^{1}ion. For the first time, this result has allowed us to predict the sign of J and its magnitude: when coulombic interactions are dominant, our model follows Hund’s rule and we explain why couplings are automatically ferromagnetic (J < 0); when coulombic interactions are no longer dominant, our model is equivalent to the molecular orbital one and couplings are always antiferromagnetic (J > 0), except in one particular case where couplings are ferromagnetic but with a small absolute value of J.

**s**

_{1}.

**s**

_{2}. The model has been easily generalized in the case of the nd

^{m}ion (3 ≤ n ≤ 5, 1 ≤ m ≤ 10, orbital degeneracy).

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Conflicts of Interest

## Appendix A

_{1}, OX

_{2}and OX

_{3}three orthonormal axes and (x

_{1},x

_{2},x

_{3}) the coordinates of any point M bearing a mass m

_{e}and a charge −e. The kinetic energy T, defined by Equation (1), may be written as:

**r**j is labeled x

_{j}

_{;i}for the sake of simplicity. The potential V, also defined in Equation (1), represents the interaction of one electron with respect to the rest of the cation. If dealing with a spherical system, we have:

_{i}, i = 1, 3. As a result, after the adequate integration of the first of Equation (A2), V is invariant by the permutation of x

_{1}, x

_{2}and x

_{3,}and may be artificially written:

_{12}= |

**r**

_{1}−

**r**

_{2}|, K = e

^{2}for U

_{A}, U

_{B}, γ

_{1}and γ

_{2}given by Equation (33); d

**r**

_{i}represents the elementary volume in spherical coordinates i.e.,

**r**

_{i}

^{2}sinθ

_{i}dθ

_{i}dφ

_{i}. In all cases, K > 0.

**r**

_{1}and

**r**

_{2}are the position vectors representing electron 1 from cation A, respectively, electron 2 from anion X, for instance. In a spherical coordinate system, we deal with the triplet (r

_{j},θ

_{j},φ

_{j}), defined as follows:

_{12}may be expressed under the symmetric form with respect to the exchange of indices 1 and 2 [39]:

_{l}

_{,m}(θ,φ) represents the well-known spherical harmonics [39].

_{c}may be decomposed into 3 Cartesian components, ${U}_{{\mathrm{c}}_{{x}_{i}}}$, so that:

_{c}may be decomposed into a system of Cartesian coordinates along each axis OX

_{i}, i = 1,3.

**R**

_{k}= (X

_{1},X

_{2},X

_{3}) is the position vector associated with each ion k (R

_{k}= C

^{st}) and

**r**= (x

_{1},x

_{2},x

_{3}) is that of electron belonging to cation A (or B) or anion X. Under these conditions, the corresponding crystal field contribution may be written as:

**R**

_{k}−

**r**|

^{−1}has been examined by Hutchings [40] in the assumption r < R

_{k,}near an origin of cubic point symmetry corresponding to the three most-encountered situations (see Figure 2 in [40]):

- The charges composing the cage are located at the corners of an octahedron (sixfold coordination); in that case,
**R**_{k}= (a,0,0), (−a,0,0), (0,a,0), (0,−a,0), (0,0,a) and (0,0,−a); R_{k}= a is the distance of each corner from the origin; - The charges are located at the corners of a cube (eightfold coordination);
**R**_{k}= (a,a,a), (−a,a,a), (a,−a,a), (−a,−a, a), (a,−a,−a), (a,a,−a), (−a, a,−a) and (−a,−a,−a);${R}_{k}=a\sqrt{3}$; - The charges are located at the corners of a tetrahedron (with two tetrahedra per cube, fourfold coordination);
**R**_{k}= (a,a,a), (−a,−a,a), (a,−a,−a), (−a,a,−a) for tetrahedron 1 and**R**_{k}= (a,−a,a), (−a,a,a), (−a,−a, −a), (a,a,−a) for tetrahedron 2;${R}_{k}=a\sqrt{3}$.

**R**

_{k}−

**r**|

^{−1}in Cartesian coordinates, with the assumption that r/R

_{k}< 1 [40]. Under these conditions, this author obtains a perturbation series analogous to the famous dipolar expansion. He shows that ${V}^{\mathrm{C}\mathrm{F}}(r)$ may be written with Cartesian coordinates under the generic form:

_{1}(respectively, x

_{2}, x

_{3}).

_{1}, x

_{2}, and x

_{3}but, according to site symmetries, we may have not only ${V}_{{x}_{1}}^{\mathrm{C}\mathrm{F}}$=${V}_{{x}_{2}}^{\mathrm{C}\mathrm{F}}$=${V}_{{x}_{3}}^{\mathrm{C}\mathrm{F}}$ but also ${V}_{{x}_{1}}^{\mathrm{C}\mathrm{F}}$≠${V}_{{x}_{2}}^{\mathrm{C}\mathrm{F}}$≠${V}_{{x}_{3}}^{\mathrm{C}\mathrm{F}}$.${V}_{mix}({x}_{1},{x}_{2},{x}_{3})$ is a polynomial characterized by terms composed of products separately involving two variables x

_{1}and x

_{2}, x

_{2}and x

_{3}, x

_{3}and x

_{1,}showing even different powers, but it is impossible to separate variables x

_{1}, x

_{2}, and x

_{3}(if odd powers intervene in higher-order terms, the final result of integration vanishes). As a result, Equation (A13) may be rewritten as:

_{1}, x

_{2}and x

_{3,}${U}_{c}^{\mathrm{C}\mathrm{F}}$ appears under the form:

_{c}given by Equation (A12), in which |

**R**

_{k}−

**r**|

^{−1}is expanded in terms of spherical harmonics. The matrix element U

_{c}and correlated general rules of calculation may be found from this expansion, using vector coupling coefficients (see [41], p. 37). Hutchings recalls the rules for determining nonzero matrix elements, which follow from the Wigner–Eckart theorem (see [41], p. 75).

_{c}may be also expressed owing to Cartesian components. This method allows us to stop the expansion vs. spherical harmonics Y

_{l}

_{,m}(θ,φ) for small values of l and m, if using symmetries derived from that of the host site, whereas the direct calculation of the integral giving ${U}_{\mathrm{c}}^{\mathrm{C}\mathrm{F}}$ is longer because it is exclusively ruled by the convergence of the series, giving ${V}^{\mathrm{C}\mathrm{F}}({x}_{1},{x}_{2},{x}_{3})$ (cf. Equation (A9)).

## References

- Anderson, P.W. Antiferromagnetism. Theory of superexchange interaction. Phys. Rev.
**1950**, 79, 350. [Google Scholar] [CrossRef] - Anderson, P.W. New approach to the theory of superexchange interactions. Phys. Rev.
**1959**, 115, 2. [Google Scholar] [CrossRef] - Anderson, P.W. Theory of magnetic exchange interactions: Exchange in insulators and semiconductors. Solid State Phys.
**1963**, 14, 99. [Google Scholar] - Anderson, P.W. Magnetism; Rado, G.T., Suhl, H., Eds.; Wiley: New York, NY, USA, 1962; Volume I, p. 25. [Google Scholar]
- Kanamori, J. Theory of the magnetic properties of ferrous and cobaltous oxides, I. Prog. Theor. Phys.
**1957**, 17, 177. [Google Scholar] [CrossRef] [Green Version] - Kanamori, J. Theory of the magnetic properties of ferrous and cobaltous oxides, II. Prog. Theor. Phys.
**1957**, 17, 197. [Google Scholar] [CrossRef] [Green Version] - Van Vleck, J.H. Note on the use of the Dirac vector model in magnetic materials. Rev. Mat. Fis. Teor.
**1962**, 14, 189. [Google Scholar] - Levy, P.M. Rare-earth-iron exchange interaction in the garnets. I. Hamiltonian for anisotropic exchange interaction. Phys. Rev.
**1964**, 135, A155. [Google Scholar] [CrossRef] - Levy, P.M. Rare-earth-iron exchange interaction in the garnets. II. Exchange Potential for Ytterbium. Phys. Rev.
**1966**, 147, 311. [Google Scholar] [CrossRef] - Curély, J. Magnetic orbitals and mechanisms of exchange. II. Superexchange. Mon. Für Chem.
**2005**, 136, 1013. [Google Scholar] [CrossRef] - Goodenough, J.B. Direct cation-cation interactions in several oxides. Phys. Rev.
**1960**, 117, 1442. [Google Scholar] [CrossRef] - Goodenough, J.B. Theory of the role of covalence in the perovskite-type manganites [La,M(II)]MnO
_{3}. Phys. Rev.**1955**, 100, 564. [Google Scholar] [CrossRef] [Green Version] - Goodenough, J.B. An interpretation of the magnetic properties of the perovskite-type mixed crystals La
_{1−x}Sr_{x}CoO_{3−λ}. J. Phys. Chem. Solids**1958**, 6, 287. [Google Scholar] [CrossRef] - Kanamori, J. Superexchange interaction and symmetry properties of electron orbitals. J. Phys. Chem. Solids
**1959**, 10, 87. [Google Scholar] [CrossRef] - Escuer, A.; Vicente, R.; Goher, M.A.S.; Mautner, F.A. Synthesis and structural characterization of [Mn(ethyl isonicotinate)
_{2}(N_{3})_{2}]_{n}, a two-Dimensional alternating ferromagnetic-antiferromagnetic compound. Magnetostructural correlations for the end-to-end pseudohalide-manganese system. Inorg. Chem.**1996**, 35, 6386. [Google Scholar] [CrossRef] - Escuer, A.; Vicente, R.; Goher, M.A.S.; Mautner, F.A. A new two-dimensional manganese(II)-azide polymer. Synthesis, structure and magnetic properties of [{Mn(minc)
_{2}(N_{3})_{2}}]_{n}(minc = methyl isonicotinate). J. Chem. Soc. Dalton Trans.**1997**, 22, 4431. [Google Scholar] [CrossRef] - Goher, M.A.S.; Morsy, A.M.A.-Y.; Mautner, F.A.; Vicente, R.; Escuer, A. Superexchange interactions through quasi-linear end-to-end azido bridges: Structural and magnetic characterisation of a new two-dimensional manganese-azido system [Mn(DENA)
_{2}(N_{3})_{2}]_{n}(DENA = diethylnicotinamide). Eur. J. Inorg. Chem.**2000**, 8, 1819. [Google Scholar] [CrossRef] - Escuer, A.; Esteban, J.; Perlepes, S.P.; Stamatatos, T.C. The bridging azido ligand as a central “player” in high-nuclearity 3d-metal cluster chemistry. Coord. Chem. Rev.
**2014**, 275, 87, and references therein. [Google Scholar] [CrossRef] - Shull, C.G.; Strauser, W.A.; Wollan, E.O. Neutron diffraction by paramagnetic and antiferromagnetic substances. Phys. Rev.
**1951**, 83, 333. [Google Scholar] [CrossRef] - Tinkham, M. Paramagnetic resonance in dilute iron group fluorides. I. Fluorine hyperfine structure. Proc. R. Soc.
**1956**, A236, 535. [Google Scholar] - Shulman, R.G.; Jaccarino, V. Effects of superexchange on the nuclear magnetic resonance of MnF
_{2}. Phys. Rev.**1956**, 103, 1126. [Google Scholar] [CrossRef] - Jaccarino, V.; Shulman, R.G. Observation of nuclear magnetic resonance in antiferromagnetic Mn(F19)2. Phys. Rev.
**1957**, 107, 1196. [Google Scholar] [CrossRef] - Shulman, R.G.; Jaccarino, V. Nuclear magnetic resonance in paramagnetic MnF
_{2}. Phys. Rev.**1957**, 108, 1219. [Google Scholar] [CrossRef] - Kondo, J. Band theory of superexchange interaction. Prog. Theor. Phys.
**1957**, 18, 541. [Google Scholar] [CrossRef] [Green Version] - Blount, E.I. Formalisms of band theory. Solid State Phys.
**1962**, 13, 305. [Google Scholar] - Jahn, H.A.; Teller, E. Stability of polyatomic molecules in degenerate electronic states-I—Orbital degeneracy. Proc. Roy. Soc.
**1937**, A161, 220. [Google Scholar] - Bates, C.A. Jahn-Teller effects in paramagnetic crystals. Phys. Rep.
**1978**, 35, 187. [Google Scholar] [CrossRef] - Kahn, O. Magneto-Structural Correlations in Exchange Coupled Systems; Willett, R.D., Gatleschi, D., Kahn, O., Eds.; NATO ASI Series, Series C: Mathematical and Physical Sciences; Reidel Publishing Company: Dordrecht, The Netherlands, 1985; Volume 140, p. 37. [Google Scholar]
- Kahn, O. Molecular Magnetism; VCH Publishers: New York, NY, USA, 1993. [Google Scholar]
- Girerd, J.J.; Journaux, Y.; Kahn, O. Natural or orthogonalized magnetic orbitals: Two alternative ways to describe the exchange interaction. Chem. Phys. Lett.
**1981**, 82, 534. [Google Scholar] [CrossRef] - Curély, J. Magnetic orbitals and mechanisms of exchange. I. Direct exchange. Special Issue. Mon. Für Chem.
**2005**, 136, 987. [Google Scholar] - Van Vleck, J.H. Theory of the variations in paramagnetic anisotropy among different salts of the iron group. Phys. Rev.
**1932**, 41, 208. [Google Scholar] [CrossRef] - Moriya, T. Magnetism; Rado, G.T., Suhl, H., Eds.; Wiley: New York, NY, USA, 1962; Volume I, p. 85. [Google Scholar]
- Moriya, T. Anisotropic superexchange interaction and weak ferromagnetism. Phys. Rev.
**1960**, 120, 91. [Google Scholar] [CrossRef] - Dzialoshinskii, I.E. Thermodynamic theory of “weak” ferromagnetism in antiferromagnetic substance. Sov. Phys. JETP
**1957**, 6, 1259. [Google Scholar] - Dzialoshinskii, I.E. A thermodynamic theory of “weak” ferromagnetism of antiferromagnetics. J. Phys. Chem. Solids
**1958**, 4, 241. [Google Scholar] [CrossRef] - Katsnelson, M.I.; Kvashnin, Y.O.; Mazurenko, V.V.; Lichtenstein, A.I. Correlated band theory of spin and orbital contributions to Dzialoshinskii-Moriya interactions. Phys. Rev.
**2010**, 82, 100403. [Google Scholar] [CrossRef] [Green Version] - Qi, X.; Guionneau, P.; Lafon, E.; Pérot, S.; Kaufmann, B.; Mathonière, C. New photomagnetic ionic salts based on [Mo
^{IV}(CN)_{8}]^{4−}and [W^{IV}(CN)_{8}]^{4−}anions. Magnetochemistry**2021**, 7, 97. [Google Scholar] [CrossRef] - Varshalovich, D.A.; Moskalev, A.N.; Kherkonskii, V.K. Quantum Theory of Angular Momentum; World Scientific: Singapore; Teaneck, NJ, USA; Hong Kong, China, 1988; p. 166. [Google Scholar]
- Hutchings, M.T. Point-charge calculations of energy levels of magnetic ions in crystalline electric fields. Solid State Phys.
**1964**, 16, 227. [Google Scholar] - Edmonds, A.R. Angular Momentum in Quantum Mechanics; Princeton University Press, Ed.; Princeton University Press: Princeton, NJ, USA, 1957. [Google Scholar]

**Figure 1.**Values of spin S and orbital L momenta of an isolated transition ion, characterized by electronic shells 3d

^{m}, 4d

^{m}or 5d

^{m}(1 ≤ m ≤ 10).

**Figure 2.**“Ground” and “excited” configurations in the original superexchange process for the sequence A–X–A; the electronic configuration of the valence shell has been added for each ion (in our case, A = Mn, X = O, for instance); due to the weak overlap, coefficients a and b are small. In the present case, the electronic transfer is from X to A but it can also occur from X to B, indifferently (here, B = A).

**Figure 3.**Radial behavior of energy for the centrosymmetric entity AXB in the general case of different magnetic sites (A ≠ B) on both sides of ligand X.

**Figure 4.**Energy level scheme for the AXB centrosymmetric system (with here A = B for the sake of simplicity) in the case of dominant coulombic interactions; the difference |$E$

_{Ag}−$E$

_{Au}| has been artificially zoomed for clarity. J is given by Equations (39) and (43).

**Figure 5.**Energy level scheme for the AXB centrosymmetric system (with, here, A = B for the sake of simplicity) when Coulomb interactions are negligible (molecular orbital model). J is given by Equations (39) and (44).

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Curély, J.
The Microscopic Mechanisms Involved in Superexchange. *Magnetochemistry* **2022**, *8*, 6.
https://doi.org/10.3390/magnetochemistry8010006

**AMA Style**

Curély J.
The Microscopic Mechanisms Involved in Superexchange. *Magnetochemistry*. 2022; 8(1):6.
https://doi.org/10.3390/magnetochemistry8010006

**Chicago/Turabian Style**

Curély, Jacques.
2022. "The Microscopic Mechanisms Involved in Superexchange" *Magnetochemistry* 8, no. 1: 6.
https://doi.org/10.3390/magnetochemistry8010006