# Can the Double Exchange Cause Antiferromagnetic Spin Alignment?

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

_{X}Ca

_{1-x})(Mn

^{III}

^{I}Mn

^{IV}

_{1-x})O

_{3}containing MV fragment Mn

^{III}-O

^{2−}-Mn

^{IV}. The double exchange can be referred to as a spin-dependent electron transfer over the magnetic metal sites whose spins are polarized by the mobile electron giving rise to the ferromagnetic spin alignment. Migration of these ideas from solid state physics to chemistry was stimulated by the study of molecular systems of biological significance, such as two-iron (Fe

^{2+}-Fe

^{3+}) ferredoxin, protein with [Fe

_{3}S

_{4}] core [5] and also of other complex polynuclear MV systems like reduced polyoxometalates with Keggin and Wells–Dawson structures [6,7,8]. Molecular applications of the concept of double exchange gave impact to the generalization of the theory as applied to the systems with arbitrary number of mobile electrons and to large multicenter systems [9,10]. Success of the theory was proved by the treatment of polynuclear MV clusters such as hexanuclear octahedral clusters [Fe

_{6}(μ

_{3}-X)

_{8}(PEt

_{3})

_{6}]

^{+}(X = S, Se and Et = C

_{2}H

_{5}) [11] and giant reduced polyoxovanadates [V

_{18}O

_{48}]

^{n−}(n = 4 ÷ 18) [12,13]. Finally, formulation of the symmetry assisted approach to the solution of multidimensional vibronic problem in nanoscopic MV systems completed the theoretical development of this stage of the field [14,15].

## 2. Basic Model for a Mixed-Valence Tetrameric Square-Planar System

_{4h}point group. There are six possible distributions of the two excess electrons over the four sites as shown in Figure 1. In two of these distributions (electronic configurations) the extra electrons are localized on the antipodal sites forming diagonals of the square, while in the remaining four configurations the electrons occupy neighboring positions forming its edges. The two diagonal electronic configurations (denoted as D

_{1}and D

_{2}in Figure 1) minimize the Coulomb energy of the electronic repulsion due to the fact that in these configurations the two electrons occupy the most distant positions from each other and thus form the ground Coulomb manifold, while the remaining four configurations (D

_{3}…D

_{6}) with shorter interelectronic distance give rise to the excited Coulomb levels separated from the ground manifold by the energy gap U. Note that the two ground diagonal configurations are transformed to each other under the action of the operation of rotation around C

_{4}axis within the D

_{4h}point group, as well as the four excited configurations. At the same time, the excited configurations cannot be obtained from the ground ones by the D

_{4h}group operations thus showing that these two kinds of configurations are physically different.

_{0}= n/2. When the excess electron is trapped in some metal site (i.e., this site is occupied by the ${d}^{n+1}$-ion) its spin is coupled ferromagnetically with the core spin S

_{0}to give the spin S

_{0}+ 1/2 as schematized in Figure 2a,b. For the sake of simplicity only the transfer processes between the neighboring sites located along the sides of the square tetramer are assumed to be nonvanishing, consequently $t$ is the transfer parameter. Figure 2a,b shows the transfer processes which produce mixing of the two kinds of charge configurations, for example, mixing of the ground neighboring ${D}_{3}$ (${d}_{1}^{n+1}-{d}_{2}^{n+1}-{d}_{3}^{n}-{d}_{4}^{n}$) configuration with the antipodal excited ${D}_{1}$ (${d}_{1}^{n+1}-{d}_{2}^{n}-{d}_{3}^{n+1}-{d}_{4}^{n}$) one. In Figure 2c the orbital scheme illustrating the transfer of the excess electron from the site i to the site k is shown for the simplest case when n = 1. Electrons of spin cores occupy orbitals denoted as ${\phi}_{i}$ and ${\phi}_{k}$, while the excess electron may move over the upper orbitals denoted as ${\psi}_{i}$ and ${\psi}_{k}$ resulting in the polarization of the spin cores in accordance with the conventional double exchange mechanism. In general case each metal site forming the MV tetramer should contain n + 1 orbitals, with n of these orbitals (core orbitals) being single occupied and the highest (n + 1)-th orbital being either empty or single occupied depending on the position of the excess electron. Note that in the present consideration of the double exchange the excited non-Hund states of each ion are assumed to be separated from the ground Hund states by the energy gaps strongly exceeding both the value of the electron transfer parameters and the Coulomb energy U. Under such assumption (that seems to be reasonable in many cases) one can truncate the double exchange problem defining it within the space involving only the ground states of each ions as is it is usually accepted in the modelling the properties of MV clusters exhibiting double exchange.

_{ii}involved in such Hamiltonian represent the on-site interelectronic Coulomb energies, while in the Hamiltonian, Equation (1), the electronic configurations with two excess electrons occupying the same metal site are excluded, and the parameters ${U}_{i\hspace{0.17em}k}$ describe the Coulomb repulsion between the excess electrons occupying different metal sites i and k. According to Figure 1 the Hamiltonian, Equation (1), contains two different Coulomb energies ${U}_{d}\equiv {U}_{13}={U}_{24}$ (diagonal configurations D

_{1}and D

_{2}) and ${U}_{n}\equiv {U}_{12}={U}_{23}={U}_{34}={U}_{14}$ (nearest neighboring distributions D

_{3}…D

_{6}) separated by the energy gap $U\equiv {U}_{n}-{U}_{d}$ as discussed above. As to the states with two excess electrons per site, they are strongly excited and their mixing with the low-lying electronic configurations D

_{1}…D

_{6}for which each site may contain no more than one excess electron gives rise to the HDVV exchange interaction between the metal ions. Below this HDVV exchange will be taken into account in the framework of the extended model (see next Section 3).

## 3. Extended Model: Exchange Interaction

**|**. In this notations the set of the two intermediate spin values arising upon coupling of the four local spins are indicated as $(\tilde{S}\left({D}_{\lambda}\right)$ (for example, this set can be $\left({S}_{13}\left({D}_{\lambda}\right),{S}_{24}\left({D}_{\lambda}\right)\right)$, S is the quantum number of the total spin, and M is the quantum number of the total spin projection. Note that the numbers of the electrons populating different sites and hence the local spins are defined by the electronic distribution. This is ensured by the symbol ${D}_{\lambda}$ which indicates that the set of spin functions belong to a certain distribution ${D}_{\lambda}$ of the mobile electrons ($\lambda =1,\mathrm{2..6},$ Figure 1). Consequently, the intermediate spins in the four-spin coupling scheme are also defined by a certain distribution ${D}_{\lambda}.$

**|**defined for a certain electronic configuration ${D}_{\lambda}$. This means that the matrix of ${\widehat{H}}_{ex}$ has block-diagonal structure, $\langle {D}_{\lambda}{\tilde{S}}^{\prime}\left({D}_{\lambda}\right),SM\left|{\widehat{H}}_{ex}\left({D}_{\lambda}\right)\right|{D}_{\mu}\tilde{S}\left({D}_{\mu}\right),SM\rangle \sim $ ${\delta}_{\lambda \mu}$, where the Kronecker symbol ${\delta}_{\lambda \mu}$ ensures action of the exchange Hamiltonian within the set of spin states belonging to a definite distribution and excludes off-diagonal matrix elements. On the contrary, the double exchange Hamiltonian ${\widehat{H}}_{DE}$ links states belonging to different distributions ${D}_{\lambda}$ so that $\langle {D}_{\lambda}{\tilde{S}}^{\prime}\left({D}_{\lambda}\right),SM\left|{\widehat{H}}_{DE}\right|{D}_{\mu}\tilde{S}\left({D}_{\mu}\right),SM\rangle \sim 1-{\delta}_{\lambda \mu}$. The notation of spin-operators contains symbol ${D}_{\lambda}$ of configuration in addition to the running symbol $i$ numerating the sites that defines the value of ${s}_{i}$. For example, for the distribution ${D}_{1}$ (Figure 2) ${s}_{1}={s}_{3}={s}_{0}+1/2$, while the two remaining sites have spins ${s}_{0}$. Each distribution ${D}_{\lambda}$ generates a specific network of the exchange interactions whose parameters are reduced to the three independent quantities, $J\left({d}^{n+1}-{d}^{n}\right)\equiv J$, $J\left({d}^{n+1}-{d}^{n+1}\right)\equiv {J}_{1}$ and $J\left({d}^{n}-{d}^{n}\right)\equiv {J}_{2}$ as illustrated in Figure 2a,b for particular cases of distributions ${D}_{3}$ and ${D}_{1}$.

## 4. Combined Effect of the Double Exchange and Coulomb Repulsion

## 5. Double Exchange in Regime of Strong Coulomb Repulsion

_{1}passes into neighboring position D

_{6}via one-electron transfer $1\to 2$ and then the jump $3\to 4$ transforms D

_{6}into the final antipodal configuration D

_{2}. One can see that the first order transfer does not operate within the space of only distant configurations, while the second order process does. That is why the effective bi-electronic transfer parameters that appear in the second order perturbation calculations is expressed as $\tau ={t}^{2}/U$. The second order double exchange separates the energy levels according to the full spin of the system giving rise to the two states with S = 3, six states with S = 2, eight states with S = 1 and four states with S = 0. The energies of these states are listed in the Table 1.

**D**

_{4h}group (which is the excited level) while the S = 2 $\left({S}_{13}=1,{S}_{24}=1\right)$ can be designated as ${}_{}{}^{5}B{}_{1g}$ term. Then by coupling the states of the localized and delocalized units one can conclude that the term of the ${d}^{2}-{d}^{2}-{d}^{1}-{d}^{1}$ tetramer with the maximal spin $S=3$ is ${}_{}{}^{7}E$. One can see from the Table 1 that the degeneracy of two $S=3$ states is related to the orbital degeneracy which means that this is an “exact” degeneracy originating from the point symmetry of the system. A comprehensive discussion of the degeneracies in spin systems and their physical consequences can be found in review articles [31,32]. In particular, one can observe the so-called “accidental degeneracy” interrelated with the unitary symmetry that in general are high than the point one. Regarding the action of the double exchange, one can conclude that two electrons in ${}_{}{}^{3}E$ term produce ferromagnetic double exchange in the system (as schematically shown in Figure 4b).

## 6. Beyond Basic Model: Concomitant Effect of the HDVV Exchange

_{1}and D

_{2}can be written as:

_{1}and ${S}_{1}={S}_{3}={s}_{0}$, ${S}_{2}={S}_{4}={s}_{0}+1/2$ for D

_{2}. The energy levels are expressed in terms of the full spin $S$ and the two intermediate spins ${S}_{13}$, ${S}_{24}$ which are peculiar for each distribution:

## 7. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**Numeration of the sites of the square-planar tetrameric cluster with two mobile electrons, two antipodal distributions denoted as D

_{1}and D

_{2}and four excited neighboring distributions D

_{3}… D

_{6.}The sites occupied by the extra electrons are shown by black balls, the spin cores are indicated as white balls.

**Figure 2.**Schemes of the magnetic sites, exchange and transfer parameters for the tetrameric MV cluster with two mobile electrons shown for the neighboring Coulomb configuration D

_{3}(

**a**) and for distant configuration D

_{1}(

**b**), and also the orbital scheme of the one electron transfer between the two sites i and k shown for the simplest case of one-electron spin cores. ϕ are the orbitals occupied by the localized electrons (spin-core orbitals), and ψ are the upper orbitals available for the transfer of the excess electron (

**c**).

**Figure 3.**Combined effect of double exchange and Coulomb interaction on the energy spectrum of square-planar ${d}^{2}-{d}^{2}-{d}^{1}-{d}^{1}$– tetramer. The low-lying part of the energy spectrum with labelling of the energy levels is shown as insert. The energy levels are labelled as S (f), where S is the total spin of the tetramer and f is multiplicity of the repeated levels with the same S. The energy of the ground state is regarded as a reference energy.

**Figure 4.**Illustration for the spin polarization effects in the tetrameric square-planar systems with two mobile electrons: antiferromagnetic spin alignment of delocalized spins (

**a**); ferromagnetic spin alignment of delocalized spins (

**b**).

**Figure 5.**Visual representation of the sequential 1→2, 3→4 double electron transfer D

_{1}→D

_{2}which acts within the basis spin-functions belonging to the two antipodal localization of the electronic pair in MV tetramer of ${d}_{1}^{n+1}-{d}_{2}^{n+1}-{d}_{3}^{n}-{d}_{4}^{n}$ type. Spin polarization effect is also schematically shown.

**Figure 6.**Correlation diagram ${E}_{ex}/\left|J\right|vs\tau /\left|J\right|$ for the MV square-planar ${d}^{2}-{d}^{2}-{d}^{1}-{d}^{1}$ -tetramer calculated within the strong U-limit. Coloring of spin states is shown in the insert.

**Table 1.**Spin levels $E\left(S\right)/\tau $ belonging to the antipodal charge configuration in the strong U limit. The numbers of the levels having the same spin and energy are indicated in parentheses.

S | $E\left(S\right)/\tau $ |

S = 3 | $-4\left(2\right)$ |

S = 2 | $-5/4$(2); $-3$(2); $-11/2$(1); $-3/2\left(1\right)$ |

S = 1 | $-3/2\left(2\right);-1/2\left(1\right);-9/2\left(1\right);\sqrt{6}\left(2\right);-\sqrt{6}\left(2\right)$ |

S = 0 | $-4\left(1\right);0\left(2\right);-6\left(1\right)$ |

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**MDPI and ACS Style**

Palii, A.; Clemente-Juan, J.M.; Aldoshin, S.; Korchagin, D.; Golosov, E.; Zilberg, S.; Tsukerblat, B.
Can the Double Exchange Cause Antiferromagnetic Spin Alignment? *Magnetochemistry* **2020**, *6*, 36.
https://doi.org/10.3390/magnetochemistry6030036

**AMA Style**

Palii A, Clemente-Juan JM, Aldoshin S, Korchagin D, Golosov E, Zilberg S, Tsukerblat B.
Can the Double Exchange Cause Antiferromagnetic Spin Alignment? *Magnetochemistry*. 2020; 6(3):36.
https://doi.org/10.3390/magnetochemistry6030036

**Chicago/Turabian Style**

Palii, Andrew, Juan M. Clemente-Juan, Sergey Aldoshin, Denis Korchagin, Evgenii Golosov, Shmuel Zilberg, and Boris Tsukerblat.
2020. "Can the Double Exchange Cause Antiferromagnetic Spin Alignment?" *Magnetochemistry* 6, no. 3: 36.
https://doi.org/10.3390/magnetochemistry6030036