Vibronic Model for Intercommunication of Localized Spins via Itinerant

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C: Physical Processes in Nanomaterials and Nanostructures

Vibronic Model for Intercommunication of Localized Spins via Itinerant Electron Andrew V. Palii, Sergey M. Aldoshin, Boris Tsukerblat, Juan Modesto Clemente-Juan, and Eugenio Coronado J. Phys. Chem. C, Just Accepted Manuscript • DOI: 10.1021/acs.jpcc.8b12380 • Publication Date (Web): 31 Jan 2019 Downloaded from http://pubs.acs.org on February 2, 2019

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The Journal of Physical Chemistry

Vibronic Model for Intercommunication of Localized Spins via Itinerant Electron Andrew Palii,*1,2 Sergey Aldoshin,1 Boris Tsukerblat,*3,4 Juan Modesto Clemente-Juan,5 Eugenio Coronado5 1 Laboratory

of Magnetic Nanomaterials,, Institute of Problems of Chemical Physics, 142432 Chernogolovka, Moscow Region, Russia

2 Laboratory

of Physics of Semiconductor Compounds, Institute of Applied Physics, MD-2028 Chisinau, Moldova

3Department

of Chemistry, Ben-Gurion University of the Negev, 84105 Beer-Sheva, Israel

4Deprtment 5Instituto

of Chemical Sciences, Ariel University, 40700 Ariel, Israel

de Ciencia Molecular, Universidad de Valencia, 46980 Paterna, Spain

*Corresponding

authors: [email protected] (A.P.); [email protected] (B.T.)

Abstract In this article we propose a vibronic pseudo Jahn-Teller model for partially delocalized mixed valence molecules aimed to the description of the magnetic coupling between the localized spins mediated by the delocalized electron. The simplest partially delocalized system which retains the main studied features is assumed to consist of an one-electron mixed-valence dimer which is connected to the two terminal magnetic ions. The model involves the following key interactions: electron transfer in the spin-delocalized subsystem of a mixed valence molecule which is mimicked by a dimeric unit, coupling of the itinerant electrons with the molecular vibrations and isotropic magnetic exchange between the localized spins and delocalized electron. The proposed model which can be referred as the vibronic “toy” model, is intentionally restricted to the named basic interactions and, therefore, it is aimed to describe only the key features of a wide class MV clusters exhibiting partial electronic delocalization without entering the details of their electronic and vibrational structures. The pseudo JahnTeller vibronic coupling which is considered in the framework of the Piepho, Krausz and Schatz model adapted to the case of partially delocalized mixed valence molecules, is shown to give rise to the specific patterns of the adiabatic potentials and spin-vibronic levels. It is revealed (qualitatively and quantitively) how the vibronic coupling affects the connection of the localized spins via the itinerant electron. In particular, the vibronic coupling acts as a localization factor and significantly influences the conditions for realization of physically different limits of the double exchange and indirect exchange. The vibronic coupling in 1 ACS Paragon Plus Environment

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partially delocalized mixed valence systems is also shown to produce a strong impact on the selection rules for the optical transitions and gives rise to the specific shapes of the intervalence absorption profiles. 1. Introduction Mixed valence (MV) molecular systems

1-8

comprise metal ions in different oxidation

states. Along with the localized electrons which form the spin cores such molecules also include itinerant electrons whose mobility can be controlled by an external electric field giving rise to fascinating possibilities of usage such systems in molecular spintronics.9-13 Here we will focus on important class of magnetic MV clusters in which localized spins, such as, metal ions in inorganic complexes or nitroxide radicals in organic systems are magnetically coupled through the spin-delocalized subsystem contacting the localized spins. Some examples of such partially delocalized inorganic MV systems are provided by the reduced polyoxometalates (POMs) comprising mobile electrons (spin-delocalized subsystem) interacting with the localized spins forming spin-localized subsystem.12-16 Such molecular systems

can

be

exemplified

by

the

POMs

[PMoVI11MoVO40(VO)2]

and

[PMoVI11MoVO40{Ni(phen)2(H2O)}2] in which the direct magnetic coupling between the two hosted remote VIV ions (localized spins S=1/2) or two NiII ions (localized spins S=1) is forbidden because of very large distances between these metal centers. At the same time the indirect coupling takes effect due to the presence of the extra electron delocalized over the central Keggin anion core,

12-13

which gives rise to a long-range interaction between the

localized spins mediated by the mobile electron. As another example of inorganic partially delocalized MV systems one can mention the octanuclear iron cluster composed of inner tetranuclear valence-delocalized iron core coupled to the four outer localized spins of iron ions. 17-22

One could mention also wide classes of other instances where a spin-polarized rather than

closed-shell entity connects two localized spins.23-26 These can be radical ligand-containing single-molecule magnets represented by the mononuclear transition metal-radical complexes, radical-bridged transition metal single-molecule magnets,23

radicals in metal–organic

frameworks.24 The influence of a third spin on charge- and spin-transfer dynamics in electron donor-bridge-acceptor molecules with bridging nitronyl nitroxide radicals is described in ref. 25.

Along with mentioned inorganic systems, the class of MV molecules with partial electron

delocalization is also exemplified by the multispin organic systems, which are of current interest in molecular spintronics.

27-36

Finally, some aggregate composed of quantum dots

(which can be referred to as “physical molecules” ) accommodating different specific numbers 2 ACS Paragon Plus Environment

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of unpaired electrons can be also regarded as physical analogs of partially delocalized MV clusters. Recently we have reported a simplified “toy” model for the description of the magnetic coupling between two localized spins in partially delocalized linear MV systems comprising four metal sites. 37 In this model system the outer (terminal) sites are occupied by the localized spins, while the central dimer comprises extra electron delocalized over two diamagnetic cores. Such spin-delocalized dimer plays a role of an effective bridge mediating the magnetic coupling between the terminal localized spins. The proposed model includes the electron transfer between the two metal centers of the central dimer (described by the hopping parameter t) and the exchange coupling between the mobile electron and the outer spins (parameter J). The combination of these two interactions was shown to result in spin-polarization mechanism that is physically quite similar to the double exchange (DE) coupling proposed by Zener, Anderson and Hasegawa 38,39. According to this mechanism the spin of the excess electron is coupled via strong intra-center ferromagnetic Hund type exchange interaction with the spins of paramagnetic ions (spin cores), thus aligning them parallel to the spin of the mobile electron. Since the intra-ionic exchange produces the coupling of the two spins (spin of the extra electron and that of the spin-core) belonging to the same metal ion, the spin-cores involved in the conventional DE can be also termed “internal spin cores”. As distinguished from this conventional DE concept, the mechanism proposed in Ref. 37 involves the inter-center exchange coupling of the mobile electron with the alien (“external”) spin cores represented by the two outer centers of the tetramer. To distinguish this spin-polarization mechanism from the conventional DE mechanism we suggested to term it “external core double exchange” (ECDE). In spite of physical similarity of the two mentioned spin-polarization mechanisms, there are at least two important differences between them mentioned in Ref. 37. First, as distinguished from the Hund-type intra-center exchange between the extra electron and the internal spin core that is always ferromagnetic, the inter-center exchange between the extra electron and the alien spin core can be either ferro- or antiferromagnetic depending on the character of the bridging ligands, participating orbitals and topology of the system. Second, the role of the non-Hund configurations is much more important in the ECDE problem as compared with the case of conventional DE. This is because the intra-center exchange giving rise to the DE in MV clusters strongly exceeds the magnitude of the transfer parameter and so the non-Hund configurations can be often neglected. In contrast, in the ECDE problem the electron transfer can be comparable or even stronger than the intercenter exchange interaction leading thus to a rather strong mixing of Hund and non-Hund configurations. 3 ACS Paragon Plus Environment


A significant physical restriction of the toy model developed in Ref

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37

is that this model

involves only the electronic interactions (such as electron transfer and exchange coupling), and ignores the interaction of the mobile electron with the molecular vibrations, i. e. the vibronic coupling. Meanwhile the vibronic coupling is known to reprsent an inherent ingredient of the description of the properties of MV systems

40-45

because it tends to create a barrier for

tunneling of the extra electron, which reduces its mobility giving rise to the so-called vibronic self-trapping effect. This means that although the electronic toy model is quite useful for better understanding of the properties of partially delocalized MV systems, its applicability to the analysis of the real MV molecules is restricted to the systems exhibiting weak vibronic coupling. To overcome these limitations of the electronic model, here we develop the vibronic toy model of partially delocalized MV systems with ECDE. The article is organized as follows. At the beginning we describe the vibronic model that represents an extension of the well-known Piepho, Krausz and Schatz (PKS) model 40-45 to the case of tetrameric partially delocalized MV systems, consisting of internal spin-delocalized dimer and two outer localized spins. We define the full Hamiltonian consisting of the electronic terms and the terms involving the molecular PKS vibrations. The analysis of the vibronic effects is performed both in the framework of the adiabatic semiclassical approximation and with the aid of more exact quantum-mechanical approach based on the numerical solution of the pseudo-Jahn-Teller (JT) dynamic vibronic problem. First, we analyze the adiabatic potentials for the two physically different limits, namely, for the limit of strong exchange coupling termed also the DE limit in Ref.37 and for limit of strong transfer referred to as indirect exchange limit. The consideration of the adiabatic semiclassical picture is completed by the analysis of the trapping effect caused by the exchange interaction. Then, we proceed to the consideration of the dynamic vibronic problem, and discuss the spin-vibronic energy levels as functions of the vibronic coupling parameter. Finally, based on the numerical solution of the vibronic problem accompanied by the schemes of the adiabatic potentials and established selection rules for the intervalence transitions, we analyze the main peculiarities of the intervalence light absorption spectra for the partially delocalized MV molecules. 2. Hamiltonian of partially-delocalized mixed valence molecule In this Section we introduce the Hamiltonian of the system which includes electronic and vibronic interactions. Since the former are described in Ref.37 where the “toy” model have been proposed, here we give only a brief decription of the electronic model underlying the vibronic approach. The main idea of the toy model is to imitate a 4 ACS Paragon Plus Environment

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complex multiroute spin-delocalized subsystem of partially delocalized MV molecule by a dimer in which the extra electron is delocalized over two spinless sites B and C, as schematically shown in Figure 1a. The localized spins SA=SD=S0 occupy the two terminal positions A and D adjacent to the two sites B and C of the MV dimer (the array may not necessarily be linear).

The Hamiltonian corresponding to proposed vibronic model

includes the following basic interactions:

Hˆ  Hˆ tr  Hˆ ex  Hˆ vib .

(1)

The Hamiltonian in Eq. (1) contains the following three basic interactions constituting the model: 1. The first term Hˆ tr represents the one-electron transfer over two orbitals 𝜑𝐵 and 𝜑𝐶 located on the sites B and C of the MV dimer B-C: Hˆ tr  t

  cˆ  cˆ   cˆ  cˆ   ,  1 2

 B

C

 C

(2)

B

where cî and cî  are the creation and annihilation operators of the electron on the site i=A or

(a)

(b) Figure 1. The model spin sites in partially delocalized system: B-C-unit-MV dimer separating two localized spins A and D (a); image of the out-of-phase PKS vibration 𝑞 ― . Expanded and compressed (as compare to their average sizes in Figure 2a) sites are denoted by the enlarged and decreased balls. In the phase 𝑞 ― > 0 when electron leaves site B (which was expanded) and jumps to site C (which is still compressed), in 𝑞 ― < 0 the displacements are of the opposite sign (b). 5 ACS Paragon Plus Environment


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B and 𝑡 ≡ 𝑡𝐵𝐶 is the transfer integral which is assumed to effectively incorporate all multiroute transfer pathways in real complex systems. Figure 1a illustrates the transfer processes in the dimeric B-C subunit of the system. 2. The term Hˆ ex describes exchange coupling between the localized spins SA and SD and the spin of the mobile electron:





Hˆ ex  2 J nˆB Sˆ A sˆB  nˆC Sˆ D sˆC .

(3)

In Eq. (3) the following notations are introduced: 𝑺𝐴 and 𝑺𝐷 are the vector spin operators for the localized spins, while 𝒔𝐵 and 𝒔𝐶 are the spin operators for the mobile electron, sB  1 2 and sC  1 2 are the values of the spins of the extra electron, and the symbols B and C are temporarily retained to mark the site of localization. Since SA and SD are the spins of the identical terminal metal ions, 𝑆𝐴 = 𝑆𝐷 ≡ 𝑆0. The exchange contribution to the Hamiltonian, Eq. (3), contains factors nˆ B and nˆC which are the operators of the site populations, nˆ B  nˆC  1 (Figure 1a). They are introduced to indicate the site of localization of the itinerant electron and to switch on the exchange with the neighboring spin, A or D when the electron is instantly localized at the adjacent site B or C. In fact, when the extra electron is instantly localized on the sites B or C it is coupled to the terminal spins SA or SD through the Heisenberg-Dirac-VanVleck (HDVV) exchange interaction terms so far indicated. HDVV coupling gives rise to the two spin states for each localization,

S AB  S *  S0  1 2 for the A-B pair and

SCD  S *  S0  1 2 for the C-D pair. The two states with maximal spin S *  S0  1 2 appear when the itinerant electron instantly residing at the sites B and C is coupled ferromagnetically to the terminal ions. These states can be referred as Hund type states. Alternatively, the case of antiferromagnetic coupling, S *  S0  1 2 , corresponds to the non-Hund states for the C-D pair. 3. The third term in Eq. (1) describes the vibronic interaction. The full vibronic problem for the MV compounds represents a multimode task complicated by the presence of the mobile electron. The vibronic model proposed and discussed in the basic papers 40-45 assumes that the redox cites accommodating the extra electron are weakly connected by the relatively long bridges through which the extra electron is allowed to jump between the sites. Under this condition (which is, for example, well met in Creutz and Taube complexes)

the itinerant

electron, is known to produce a significant deformation of the local surrounding of the site of 6 ACS Paragon Plus Environment

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instant localization in MV molecules as shown schematically in Figure 1b. This inherent property of MV compounds necessarily requires to take into consideration the electronvibrational (vibronic) coupling. We will employ the PKS model, initially formulated for a MV dimer,

40-43

which takes into account the coupling between the extra electron and the fully

symmetric (“breathing”) vibrations 𝑞𝐴 and 𝑞𝐶 of the ligand surroundings of donor and acceptor sites. Then “in-phase” mode, 𝑞 + = (1 2)(𝑞𝐵 + 𝑞𝐶), and “out-of-phase” mode, 𝑞 ― = (1 2)

(𝑞𝐵 ― 𝑞𝐶) , built as symmetric and antisymmetric linear combinations of local breathing vibrations, are regarded as the normal modes of the system. Only the antisymmetric vibration 𝑞 ― is interrelated with the electron transfer processes. It is important to note that the vibronic coupling is a necessary ingredient of the model. In fact, the electronic levels of the “rigid” system correspond to the stationary states with a definite parity which describes fully delocalized system. On the contrary, the vibronic coupling in MV compounds leads (under certain conditions) to the “broken symmetry” states in which the itinerant electron is partially or fully trapped. It is to be noted that the PKS model reduces the multimode problem to the single mode one and, therefore, the conditions implied by the PKS model exclude more complicated effects such as structural changes of the molecule in course of vibrations, elongation or shortening of the bonds. The last effects which could be important appear in a generalized PKS model as the result of modulation of the transfer integrals by the metal-metal (in metal complexes) movements 45 or dependence of the transfer parameters upon the angles in the non-linear bridges (regarding the extension of PKS model towards the inclusion of the role of the ligands, see also Ref. 46). Here we apply the basic PKS model to the delocalized unit in the system under consideration (Figure 1a) neglecting the effects so far mentioned. The image of the “out-ofphase” vibration of spin-delocalized dimeric fragment in the adopted model is shown in Figure 1b. It is to be underlined that PKS model assumes that only the ligands participate in the breathing motion, while the positions of the metal sites in MV unit remain fixed in course of PKS vibrations. This means that in the model so far discussed we consider the influence of the vibrations on the transfer processes, while the exchange coupling is assumed to remain unaffected by the “breathing” displacements of the ligands. Refinement of the PKS model with due account for displacements of the metal sites is given in 44,45, but here we will use the basic version in order to avoid excessive complications. Of course, the local vibrations of the sites are able to modify the exchange interactions via the ligands but this effects seems to be small

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enough to be taken into account within the approximate approach adopted here.

The

Hamiltonian Hˆ vib can be presented as: h  2  2  ˆ ˆ ˆ Hˆ vib   q  2   H ev  H osc  H ev , q  2 

(4)

where the first term ( Hˆ osc ) is the Hamiltonian of the harmonic oscillator, 𝑞 ― ≡ 𝑞 is the dimensionless coordinate of the active “out-of-phase” vibration, 𝜔 is the frequency, 𝐻𝑒𝑣 is the vibronic interaction which is defined by the 2 × 2-matrix in the basis composed of the oneelectron orbitals 𝜑𝐵 and 𝜑𝐶 of MV unit:

1 0  Hˆ ev   q   ,  0 1

(5)

where 𝜐 is the vibronic coupling parameter. Since the sign of 𝜐 does not influence the physical results, for the sake of definiteness we assume that 𝜐 > 0. PKS model is essentially simplified and it is easier to list what this model does not take into account rather than what it considers. But it is worth to note and what is the most important issue, is that the PKS model is able to describe the main physical factor that determines the properties of MV compounds, namely, the barrier between localized configurations and degree of localization which is manifested in all physical properties of such type of compounds. That is why the PKS model became conventional and, in particular, Robin and Day classification of MV compound according to the degree of localization is essentially based on this model. Due to its simplicity the PKS model is invaluable tool in comparative consideration of the series of MV compounds with different structures and bridging ligands. In this regard it can be considered as a “toy” model for the vibronic problem and therefore, the PKS approach just well suited to the basic model we propose here. 3. Spin polarization mechanism and vibronic matrices of partially-delocalized mixed valence molecule The electronic Hamiltonian, Hˆ tr  Hˆ ex describes the spin polarization mechanism within which the itinerant spin se keeps its orientation while travelling forth and back and, therefore, forces two localized spins SA and SD (i. e. two external spin-cores) to be aligned parallel. This situation is schematically shown in Figure 2 in terms of classical spins. Such polarization mechanism represents the ECDE mechanism proposed in Ref. 37.


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Figure 2. Pictorial representation of the electron transfer and effect of polarization of the localized spins by the moving electron in terms of classical spins. In the case under consideration the three spins 𝑆𝐴 , 𝑆𝐶 and 𝑠𝑒 (spin of the itinerant electron) can be coupled in the two alternative ways which correspond to the two possible sites of the instant localization of the itinerant electron:

Sˆ A + sê  Sˆ AB , Sˆ AB  Sˆ D  Sˆ ,

Sˆ D + sê  Sˆ CD , Sˆ CD  Sˆ A  Sˆ ,

and

(6)

where 𝑆𝐴𝐵 and 𝑆𝐶𝐷 are the intermediate spins in the three-spin coupling schemes and S is the 1

1

full spin of the system ( 𝑆𝐴𝐵 = 𝑆0 ± 2 , 𝑆𝐶𝐷 = 𝑆0 ± 2 ). Subsequently, the basis set is given by the following spin-coupled representation:

S A , se S AB  S D , S , M S  S AB  S0 

1 2

S , M S

,

S D , se SCD  S A , S , M S  SCD  S 0  12  S , M S .

(7)

Using the spin recoupling equations for a three-spin system

39

one can obtain the matrix

elements of the transfer Hamiltonian which can be found in Ref.

37.

The matrix of the full

Hamiltonian is blocked accordingly to the values S of the full spins of the system and, therefore, one arrives at the 2 × 2 matrices. The analytical expressions for the electronic levels (eigenvalues of the Hamiltonian Hˆ tr  Hˆ ex ) are given in Ref. 37. Using the adopted spin coupling schemes one can prove that the matrix of the vibronic coupling is diagonal with the following matrix elements:

 S AB  S0  12  S , M S Hˆ ev  S AB  S0  12  S , M S    SCD  S0  12  S , M S Hˆ ev  SCD  S0  12  S , M S

 q .

(8)

It is worth to note that the matrix of the vibronic coupling in the spin coupled basis is the same as in the basis of localized orbitals of the spin-delocalized B-C unit (Eq. (5)). 4. General expressions for the adiabatic energies At the first step of our consideration we will take into account the vibronic interaction in the framework of the adiabatic approximation which neglects the vibrational kinetic energy in 9 ACS Paragon Plus Environment


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the Hamiltonian Hˆ vib (Eq. (4)) but gives descriptive physical results. It is worth to note that the adiabatic approach is quite efficient in modelling of the magnetic properties (interrelated with a few low-lying levels) but it fails in the description of the intervalence vibronically assisted optical bands arising from the transitions to the highly excited vibronic levels. That is why in Section 8 the quantum-mechanical solution of the vibronic problem will be presented which allows us to model the profiles of the light absorption bands (Section 9). In the framework of PKS model the adiabatic potentials U(q), which can be associated with the energies of the system (vs coordinates of ions q) within the adiabatic approximation, can be found analytically. Like the electronic levels, the branches of the adiabatic potential are enumerated by the total spins of the system. The final results of calculations are the following:

(

)

1 1 1 𝑈 ± 𝑞, 𝑆 ∗= 𝑆0 + ,𝑆 = 𝑆𝑚𝑎𝑥 = 2𝑆0 + = ℏ𝜔 𝑞2 ― 𝐽𝑆0 ± 𝜐2𝑞2 + 𝑡2 , 2 2 2 (9) 𝑈 ―± (𝑞,𝑆 < 𝑆𝑚𝑎𝑥)

[

( ) ] .

1 1 = ℏ𝜔 𝑞2 + 𝐽 ± 𝐽2(2𝑆0 + 1)2 + 4𝜐2𝑞2 + 4𝑡2 ― 4𝐽 9 𝜐2𝑞2 + 𝑡2 𝑆 + 2 2

+ The superscripts “ ― “ and “ + “ in 𝑈 ― ± and 𝑈 ±

2

mean that the corresponding adiabatic

levels originate from the non-Hund’s and Hund’s configurations, while the subscripts " ± ” denote the upper and lower branches for each pair of the levels with a certain spin S. Provided that 𝜐 = 0 expressions in Eq. (9) give the set of electronic energy levels which are studied in detail in Ref. 37. The results can be illustrated by consideration of a particular case of 𝑆𝐴 = 𝑆𝐷 = 𝑆0 = 1 which preserves all main features of the basic model. In this particular case the general expressions, Eq. (9), for the potential curves can be reduced to the following ones:

(

3

)

5

𝑈 ± 𝑞,𝑆 ∗ = 2,𝑆 = 2 =

(

) 1[

5

ℏ𝜔 2 2𝑞

― 𝐽 ± 𝑡2 + 𝜐2𝑞2 , 2

]

2

2 2 2 2 2 2 2 1 𝑈― ± 𝑞,𝑆 < 2 = 2 ℏ𝜔𝑞 + 𝐽 ± 9𝐽 + 4𝑡 + 4𝜐 𝑞 ― 4𝐽 9𝜐 𝑞 + 𝑡 (𝑆 + 2) ,

(10)

(

) 1[

5

2

]

2

2 2 2 2 2 2 2 1 𝑈+ ± 𝑞,𝑆 < 2 = 2 ℏ𝜔𝑞 + 𝐽 ± 9𝐽 + 4𝑡 + 4𝜐 𝑞 + 4𝐽 9𝜐 𝑞 + 𝑡 (𝑆 + 2) .


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In Sections 5 and 6 we will give a detailed analysis of the adiabatic potentials for the limiting cases of strong exchange coupling (|J|>>|t|) and strong electron transfer (|J||t|

and (ii) strong transfer, |J| ℏ𝜔 𝑡 (2𝑆 + 1)/6 , the adiabatic potential possesses two minima in which the mobile electron is mostly localized on the sites B and C, while in the opposite case of weak coupling/strong transfer the system preserves symmetric configuration and fully delocalized, 11 ACS Paragon Plus Environment


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𝑞𝑚𝑖𝑛(𝑆) = 0. The positions 𝑞𝑚𝑖𝑛(𝑆) as well as the conditions for localization are spindependent (which is a common consequence of the double exchange), so that the conditions of instability for 𝑆 = 1 2, 3 2 and 5 2 are 𝜐2 > ℏ𝜔 𝑡 /3, 𝜐2 > 2ℏ𝜔 𝑡 /3 and 𝜐2 > ℏ𝜔 𝑡 correspondingly. This instability can be referred to as spin-dependent pseudo JT effect which has been recognized in MV systems exhibiting DE (see 1, 2 and refs. therein). Since the vibronic coupling and electron transfer are competitive, the states with a larger spin are more stable with respect to nuclear reorganization to the self-consistent states or, in other words, less localized. These conclusions (especially the quantitative criteria) are limited by the assumption of a very large gap between the two groups (Hund’s and non-Hund’s) of levels, but they prove to be qualitatively (and quantitively) modified under more general conditions implied by the ECDE which mixes these two groups of the levels. A more generic results of the adiabatic toy model are presented in Figure 3. Providing weak coupling, υ = 0.6 ℏ𝜔 (Figures 3a), the system is stable in the symmetric configuration for all spin states and consequently delocalized. Since in the case of a weak vibronic coupling the minima correspond to the initial symmetric configuration (i.e., 𝑞 = 0), the energy levels of the system (associated with the minima of the adiabatic potentials) are unaffected by the vibronic coupling. This pattern (shown in Figure 4a) consists of the two exchange multiplets separated by the gap 3|J|, each split by the electron transfer into the pairs of levels of the opposite parity:





E S   S 0  1 2, S  J S 0  t S  1 2  2 S 0  1

(12)

with S  1/ 2, ... S max  1, S max , and E  S   S0  1 2, S    J  S0  1  t  S  1 2   2 S0  1

(13)

with S  1 2,...S max  1 . Such spin-dependence of the energy levels is peculiar for the DE

35

justifying thus the term

ECDE proposed in Ref. 37 for the considered spin-polarization mechanism. For the same reason the strong exchange limit leading to the pattern of the energy levels described by Eqs. (12), (13) has been also called “double exchange limit”. In the case of the JT instability the energy levels of the system can also be associated (although, in more rough approximation) with the minima of the adiabatic potentials, which give also a visual representation of the character and nature of localization of the mobile electron. This imaginative representation of the energy levels can be used for the qualitative


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consideration in the limits of weak and strong coupling. In the last case this can be justified due to the fact that the system mostly localized in the deep wells of the adiabatic potentials. Further on we will use such pattern of the minima (Figure 4) along with the adiabatic potentials (Figure 3). In the case of intermediate (closer to weak) coupling υ = 1.1ℏ𝜔 (Figure 3b) the lower sheet of the adiabatic potentials possesses two shallow minima only for 𝑆 = 1 2 , while in the remaining spin states (𝑆 > 1 2) the system is fully delocalized. This result remarkably demonstrates some of physical consequences of spin dependence of the degree of localization in the partially delocalized MV system comprising spin-delocalized unit. With the further increase of the vibronic coupling from intermediate υ = 2.2ℏ𝜔 to strong υ = 4.0ℏ𝜔 (Figures 3c to 3f) the lower branches of the adiabatic curves acquire distinct minima of the increasing depth. The two peculiarities of this transformation are to be mentioned: (1) the gap between the low lying 𝑆 = 1 2 and 3 2 levels in the minima point (in which the system is mostly localized) decreases with the increase of υ , so that finally in the limit of strong vibronic coupling these two levels become degenerate (Figures 3 and 4); (2) providing moderate vibronic coupling (υ = 2.2ℏ𝜔 and 2.8ℏ𝜔) the first excited minima correspond to the delocalized states with 𝑆 = 1 2 and 3 2 , while the next excited states 𝑆 = 1 2 , 3 2, 1/2 are localized ((Figure 3c, d and 4d); (3) further increase of the vibronic coupling results in an usual situation of the coexistence of the localized and delocalized minima (Figures 3e). In the limit of strong coupling, υ = 4.0ℏ𝜔 , the system has two energetically equivalent lowest minima comprises 𝑆 = 1 2, 3 2 states, while the first excited minima comprise 𝑆 = 1 2, 3 2,5 2 levels. The positions 𝑞𝑚𝑖𝑛(𝑆) of the two low lying minima become spin- independent, i.e. they correspond to the same deformation of the system, 𝑞𝑚𝑖𝑛(1 2) = 𝑞𝑚𝑖𝑛(3 2) =± 𝜐 ℏ𝜔 , while the energies of the minima are separated by the gap 3|J| . It is notable that the ferromagnetic state comes from the excited Hund’s configuration due to strong vibronic mixing of the two groups of levels. This composition of the spin levels reproduces just the pattern of the electronic levels (Figure 3a) at 𝑡 = 0 which can be referred to as effect of transfer quenching due to full vibronic trapping of the itinerant electron. The system in this case can be imagined as a superposition of the two disconnected isomers A-B*-C-D and A-B-C*-D in which the itinerant electron is localized on the sites B (C) and coupled to the metal



Page 14 of 37

Figure 3. Adiabatic energy levels calculated in the strong exchange limit with the parameters ℏ𝜔 = 200𝑐𝑚 ―1, 𝑡 = 0.2 ℏ𝜔, 𝐽 = ―0.5ℏ𝜔. Coloring of spin values for the adiabatic curves with S=3/2, S=5/2 and S=1/2 is indicated in Figure. ions A (D), while the spins D (A) are magnetically independent. Therefore, the vibronic coupling can be attributed to the factors interfering the magnetic coupling mediated by the itinerant electron due to loss of its mobility. This phenomenon has a direct analogy with the concept of polaron in dielectric crystals. In fact, a mobile electron in crystals shifts the neighboring atoms from their equilibrium positions and moves together with the polarization produced by its field which, in turn, increases the effective mass of the electron and reduces its mobility. The “molecular polaron” considered here has the same physical root and behaves in


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Figure 4. Energy pattern for the minima of the potential curves in the case of strong exchange. Coloring is the same as in Figure 3. Notations: (d)-delocalized minima, (l)-localized minima (double degenerate states). Patterns a, d and f are evaluated for the same sets of parameters as the adiabatic potentials in Figures 3a, 3d and 3f. Energy pattern in Figure 4f corresponds to full localization of the system. a similar manner, while an essential feature of “molecular polaron” in MV systems is the existence (along with the ionic polarization) of the vibronically-dependent spin polarization effect. 6. Adiabatic vibronic problem in the case of strong transfer Now we will analyze the adiabatic potentials in the strong transfer limit defined by the inequality |J|

Vibronic Model for Intercommunication of Localized Spins via Itinerant

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