Vibronic model of hyperfine interaction in dimeric mixed-valence clusters

Chemical Physics 255 (2000) 51±57

www.elsevier.nl/locate/chemphys

Vibronic model of hyper®ne interaction in dimeric mixed-valence clusters A.V. Palii a, M.I. Belinsky b,*, B.S. Tsukerblat a b

a Institute of Chemistry, Academy of Sciences of Moldova, Academy Street 3, 2028 Kishinev, Moldova School of Chemistry, Tel Aviv University, Sakler Faculty of Exact Sciences, Ramat-Aviv, 69978 Tel Aviv, Israel

Received 13 December 1999

Abstract The problem of ``vibronic'' hyper®ne coupling parameters in spin-coupled states of the dimeric mixed-valence clusters is considered in the framework of the adiabatic approach. The employed generalised vibronic model involves the interaction of the ``extra'' electron with the local (``breathing'' modes) and molecular (intermetallic) vibrations. The ®rst type of vibrations produces a localisation e€ect while the second one promotes delocalisation. The explicit expressions for the hyper®ne parameters in terms of the double exchange and two vibronic parameters are given, and the conditions are found under which the e€ect of partial delocalisation can manifest itself in the hyper®ne structure of EPR for di€erent spin states of the system. Ó 2000 Elsevier Science B.V. All rights reserved.

1. Introduction Polynuclear mixed-valence (MV) metal clusters are the active centres of many biological systems and their models [1±15]. They are also relevant to the vast ®eld of molecular and solid state magnetism [16±18]. Hyper®ne interactions observed in the M ossbauer, EPR, ENDOR and NMR spectroscopy experiments provide direct information about the degree of localisation of the moving electron [19±23]. The hyper®ne parameters of the EPR spectra appear as the result of the interplay between the key interactions in MV clusters, namely, double exchange, magnetic exchange and vibronic coupling. * Corresponding author. Tel.: +972-3-640-7942; fax: +972-3640-9293. E-mail addresses: [email protected] (A.V. Palii), [email protected] (M.I. Belinsky).

The majority of MV dimers in biological systems exhibit the trapped valency [3,4,15,21±23] belonging thus to class I in Robin±Day classi®cation [24]. The examples of the most investigated clusters of this kind are the biological and model systems Fe(II)Fe(III) [1±4,15,21±23,25±29] and Mn(III)Mn(IV) [3±13,30±38]. The double exchange in the trapped states is reduced, so, the isotropic exchange (usually antiferromagnetic) forms the ground state with a de®nite total spin Sgr ˆ 1=2 for iron and manganese clusters. Spin-coupling model gives the interrelation between the intrinsic (local) hyper®ne constants ai and e€ective hyper®ne constants Ai for a given spin state of the cluster [9±13,16,19±23,39] (for the Sgr ˆ 1=2 state of [Fe(III)Fe(II)] cluster: A1 ˆ …7=3†a (F(III)), A2 ˆ ÿ…4=3†a (Fe(II)) and for the Sgr ˆ 1=2 state of [Mn(III)Mn(IV)] cluster: A1 ˆ 2a (Mn(III)), A2 ˆ ÿa (Mn(IV))). These e€ective molecular A values are used for the

0301-0104/00/$ - see front matter Ó 2000 Elsevier Science B.V. All rights reserved. PII: S 0 3 0 1 - 0 1 0 4 ( 0 0 ) 0 0 0 6 0 - 4

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A.V. Palii et al. / Chemical Physics 255 (2000) 51±57

analysis of the hyper®ne structure of EPR and ENDOR and hyper®ne constants observed in M ossbauer spectroscopy of iron clusters [19,21, 22]. The opposite case of strong electron transfer and fully delocalised ``extra'' electron (class III compounds in Robin and DayÕs scheme [24]) was discovered in mixed-valence iron dimers in [Fe2 (OH)3 (tmtacn)2 ]2‡ [40±42] and in Mn(III) Mn(IV) clusters of [Mn2 (l ÿ O)(OAc)2 ]3‡ and [Mn2 (l ÿ O)2 (OAc)2 ]2‡ [43]. Electron delocalisation leads to the averaged hyper®ne constants in the high-spin states of the dimer produced by strong double exchange. The most interesting case of partially delocalised system (class II) was found in the half-met-L derivatives of haemocyanin [44] containing Cu(II)Cu(I) MV dimers. A signi®cant degree of delocalisation for this compound was proposed on the basis of the analysis of the intervalence transfer bands and hyper®ne structure of EPR [44]. The in¯uence of the electron migration on the hyper®ne constants was considered in spincoupling model [32] and in the model of spin-density calculations [51]. However, the action of vibronic coupling on the hyper®ne parameters was not considered. Piepho, Krausz and Schatz (PKS) [45,46] realised to a full extent the crucial role of the vibronic coupling in the problem of localisation in the MV compounds within the vibronic model dealing with the ``breathing'' local modes. Later on, Piepho [47,48] suggested a new vibronic model that also takes into account the molecular (intermetallic) vibrations that promote the electron delocalisation. In Ref. [49], this generalised vibronic model was considered with the emphasis on the magnetic properties of MV dimers and Robin±Day scheme. In the present paper, we aim to consider the hyper®ne coupling parameters in the generalised vibronic model. As distinguished from the Hush formalism [50] dealing with the phenomenological parameter a describing the degree of the localisation, we deduce the explicit expression for the hyper®ne constants for the spin-coupled states in terms of the vibronic parameters in the generalised vibronic model.

2. Hyper®ne parameters for an MV dimer in spin-coupled states Let us consider the hyper®ne interaction in a dimeric MV system dn ±dn‡1 containing one extra electron moving over two spin cores. Since we aim to consider the system in which the extra electron is partially (or completely) delocalised, it is reasonable to write the hyper®ne interaction Hamiltonian in its original form: X X a…ri ÿ RA †siA IA ‡ a…ri ÿ RB †siB IB ; …1† Hˆ i

i

where IA and IB are the spin operators of the nuclei A and B located in the positions RA and RB , siA and siB are the electronic spin operators (i enumerates the electrons of the open shells), a…ri ÿ RA…B† † are the functions of the electronic coordinates, their explicit form depends on the mechanism of the hyper®ne interaction. Let us denote the electronic wave functions of the localised A B and AB states as jSA MA ; SB MB i and jSA MA ; SB MB i, respectively. These many-electron wave functions are expressed in terms of Slater determinants being at the same time the eigenfunctions of S2A , S2B , SAZ and SBZ . Averaging operator (1) over the electronic coordinates and passing from electronic spin operators siA and siB to the ionic spin operators SA and SB (Wigner± Eckart theorem), one can obtain the e€ective spin Hamiltonian of hyper®ne interaction that is operative for A B localisation of the extra electron: HA B ˆ hSA MA ; SB MB jHjSA MA ; SB MB ie ˆ a IA SA ‡ aIB SB :

…2†

Hamiltonian HAB is obtained from Eq. (2) by the change A $ B. The same change is to be done in Eqs. (3) and (5) in order to obtain HAB . The label e indicates that the integration is carried out over the electronic coordinates. In Eq. (2), a and a are the hyper®ne constants for dn‡1 and dn ions, respectively. Using the Wigner±Eckart theorem, one can pass to the spin Hamiltonians for a dimer de®ned in spin-coupled representation jSA ; SB ; SMi and jSA ; SB ; SMi Thus, we get (see, e.g. Ref. [39]):

A.V. Palii et al. / Chemical Physics 255 (2000) 51±57

HA B ˆ hSA SB ; SMjHjSA SB ; SMie  S…S ‡ 1† ‡ SA …SA ‡ 1† ÿ SB …SB ‡ 1†  a IA ˆ 2S…S ‡ 1†

HA B

 S…S ‡ 1† ‡ SB …SB ‡ 1† ÿ SA …SA ‡ 1† aIB S: ‡ 2S…S ‡ 1† …3†

We de®ne a spin core as the ion without an extra electron (dn ion) in the case of less than half-®lled d shell …n 6 4† and the ion with the trapped electron (dn‡1 ion) in the case of more than half-®lled d shell …n > 4†. Denoting spin of the core as S0 one ®nds S A ˆ SB ˆ S0 ;

SA ˆ SB ˆ S0 ‡ 12

…4†

in the case of less than half-®lled d shell and SA ˆ SB ˆ S0 ‡ 12;

SA ˆ SB ˆ S0

…5†

in the case of a more than half-®lled d shell (only high-spin states will be taken into consideration). The energy spectrum of dimeric mixed-valence clusters is formed by the Heisenberg magnetic exchange interaction Hex ˆ ÿ2J SA SB …Hex ˆ ÿ2J SA SB † in the SA SB …SA SB † localisation scheme and by the Anderson±Hasegawa [52] double exchange interaction [15,16,40,41] which describes the spin-dependent resonance splitting of the cluster levels. Under de®nitions (4) and (5) of the spin cores the double exchange parameter tS retains the common form for both cases: tS ˆ hSA SB ; SMjHtr jSA SB ; SMi f

ˆ …ÿ1† t

S ‡ 12 ; 2S0 ‡ 1

…6†

where t stands for the transfer integral, f ˆ 1 for n 6 4 and f ˆ 2S0 ‡ 1 for n > 4. In the localised systems and in the case of a relatively weak double exchange …jtj=…1 ‡ 1=2S0 † < JS…S ‡ 1††, the antiferromagnetic exchange results in the S ˆ 1=2 ground state of MV dimer, which is characteristic for all biological iron±sulphur dimeric clusters and their synthetic model systems. Strong double exchange leads to ferromagnetic ground state with maximal total spin S [40,41,53]. Using de®nitions (4) and (5), one can rewrite the hyper®ne Hamiltonian equation (3) as follows:

   S0 ‡ 34 1 1 ˆ a I A 2 S…S ‡ 1†    S0 ‡ 34 1 1 ‡ aIB S; S…S ‡ 1† 2

53

…7†

where the upper and lower signs relate to the cases of n 6 4 and n > 4, respectively. 3. Vibronic model Here we will employ the generalised vibronic model that takes into account two types of vibrations. The ®rst type involves full-symmetric local vibrations (``breathing'' modes) belonging to the constituent moieties of the MV dimer (PKS model). The vibrations of the second type [47,48] modulate intermetallic distances (molecular or multicentre vibrations). As distinguished from PKS vibrations giving rise to localisation, molecular vibrations (P vibrations) promote the electron delocalisation [49]. The matrix of the adiabatic potential for an MV dimer, involving double exchange, magnetic exchange and vibronic coupling with two named modes can be presented in the following form:   x 2 X 2 US …q; Q† ˆ ÿ JS…S ‡ 1† ‡ q ‡ Q I 2 2 ‡ vqrZ ‡ ‰tS ÿ kS QŠrX :

…8†

Here rZ and rX are the Pauli matrices, I is the 2  2 unit matrix, J is the magnetic exchange paf rameter, kS ˆ …ÿ1† k…S ‡ 1=2†=…2S0 ‡ 1† is the spin-dependent parameter of the vibronic coupling with the molecular vibration Q, v is the parameter of the PKS interaction with the p dimensionless ``out-of-phase'' vibration q ˆ …1= 2†…qA ÿ qB † (qA and qB stand for the full symmetric local coordinates associated to sites A and B). Finally, x and X are the frequencies of the PKS and P vibrations correspondingly. Later on, for the sake of de®niteness, we will assume that t, v, k > 0. The adiabatic potentials of the MV dimer within the generalised vibronic model have been examined in detail in Ref. [49]. Providing that the PKS coupling exceeds P-coupling (case 1), i.e. v2 =x > k2S =X, two physically di€erent situations are to be

54

A.V. Palii et al. / Chemical Physics 255 (2000) 51±57

distinguished: weak transfer (case 1a, tS < v2 =x ÿ k2S =X) and strong transfer (case 1b, tS > v2 =x ÿ k2S =X). In case 1a, the lower sheet of the adiabatic potential UÿS …q; Q† possesses two equivalent minima at the points fq0 ; Q0 g: q v kS q0 ˆ 1 ÿ c2S ; …9† Q 0 ˆ ÿ cS ; X x

Considering, for example, the left minimum we ®nd

where

H…ÿq0 ; Q0 † ˆ qS HA B ‡ …1 ÿ qS †HAB ;

ÿ
ÿ1 cS ˆ tS v2 =x ÿ k2S =X :

Eq. (1) over the electronic coordinates using the wave function (11): H…q0 ; Q0 † ˆ hUSM …q0 ; Q0 †jHjUSM …q0 ; Q0 †ie : …12†

…10†

where

These two minima are separated by one or two saddles (see Fig. 1 in Ref. [49]) located at the points f0; Qÿ g (lower saddle) and f0; Q‡ g (upper) with Q ˆ kS =X. Under the condition tS > k2S =X, the upper saddle disappears. In case 1b …cS > 1†, UÿS …q; Q† has the only minimum on the Q axis. Providing P-coupling exceeds PKS-coupling (case 2, v2 =x < k2S =X), the adiabatic potential has two minima with di€erent energies in the positions f0; Q g (case 2a, tS < k2S =X ÿ v2 x) or the only minimum (case 2b, tS > k2S =X ÿ v2 =x). In case 2a, two minima are separated by two saddles located at the points fq0 ; Q0 g.

1 2

4. Hyper®ne coupling parameters in the generalised vibronic model We will treat the problem of hyper®ne coupling constants in the adiabatic approximation considering the minima points as the stable states of the system. Providing strong PKS coupling and relatively weak transfer (case 1a), the adiabatic wave functions of the lower sheet in the minima points are found to be p 1 ÿ cS USM …q0 ; Q0 † ˆ 12 p  1 ‡ cS jSA ; SB ; SMi …11† p 1 ÿ cS ‡ 12 p  1 ‡ cS jSA ; SB ; SMi: In order to derive the e€ective Hamiltonian of hyper®ne interaction, one should average H in

 q qS ˆ 1 ‡ 1 ÿ c2S :

…13†

…14†

The values qS and 1 ÿ qS are the electronic densities on sites A and B at the minima point fÿq0 ; Q0 g. Since cS 6 1; qS P 1=2, and hence, in this minimum, the extra electron is mainly localised on site A and to a lesser extent on site B. The distribution of the electronic densities in the minimum f‡q0 ; Q0 g is opposite. Providing cS ˆ 0, the system is fully localised …qS ˆ 1†. With the increase of cS , the two minima fq0 ; Q0 g move toward the deeper saddle point f0; Qÿ g and at the limit cS ˆ 1, there is only one minimum in f0; Qÿ g position. In this minimum, qS ˆ 1=2, i.e. the system is fully delocalised. Using Eqs. (7) and (13), one can represent H…ÿq0 ; Q0 † in the following ®nal form: H…ÿq0 ; Q0 † ˆ ‰AS …ÿq0 ; Q0 †IA ‡BS …ÿq0 ; Q0 †IB ŠS;

…15†

where AS and BS are the ``vibronic'' hyper®ne constants. They are found to be the following: S0 ‡ 34 1 AS …ÿq0 ; Q0 † ˆ ‰qS …a ÿ a† ‡ aŠ  2S…S ‡ 1† 2  ‰qS …a ‡ a† ÿ aŠ; S0 ‡ 34 1 BS …ÿq0 ; Q0 † ˆ ‰qS …a ÿ a † ‡ a Š  2S…S ‡ 1† 2  ‰qS …a ‡ a † ÿ a Š; …16† where the upper and lower signs relate to the cases of n 6 4 and n > 4, respectively. One can show that the hyper®ne constants in the minimum

A.V. Palii et al. / Chemical Physics 255 (2000) 51±57

f‡q0 ; Q0 g are related to those given by Eqs. (16) as follows: AS …‡q0 ; Q0 † ˆ BS …ÿq0 ; Q0 †; BS …‡q0 ; Q0 † ˆ AS …ÿq0 ; Q0 †:

…17†

In the limit case of cS ˆ 0, qS ˆ 1, Eqs. (16) take the following approximate form:   S0 ‡ 34 1 a ; AS …ÿq0 ; Q0 †  1  2 2S…S ‡ 1† …18†   S0 ‡ 34 1 a: BS …ÿq0 ; Q0 †  1  2 2S…S ‡ 1† Eq. (18) gives the same hyper®ne parameters as we have already found for the localised spin-coupled representation (Eq. (3)), so we are dealing with the vibronic trapping of the extra electron. Application of Eqs. (16) to the ground state …S ˆ 1=2† of Fe(II)Fe(III) pair (S(Fe(II)) ˆ S0 ˆ 2, S(Fe(III)) ˆ 5/2) with antiferromagnetic exchange and relatively weak double exchange gives the following expressions: A1=2 …ÿq0 ; Q0 † ˆ ÿ43qS a ‡ 73a…1 ÿ qS †; B1=2 …ÿq0 ; Q0 † ˆ ÿ43…1 ÿ qS †a ‡ 73aqS ;

…19†

55

AS …0; Qÿ † ˆ BS …0; Qÿ † S0 ‡ 34 1 …a ÿ a†: ˆ …a ‡ a†  4 4S…S ‡ 1†

…22†

Considering as an example the case of S ˆ 1=2 state of Fe(II)Fe(III) dimer and applying Eq. (22) we get A1=2 …0; Qÿ † ˆ B1=2 …0; Qÿ † ˆ ÿ23a ‡ 76a:

…23†

We consider here the case when double exchange does not destroy the antiferromagnetic ground state. 5. Remarks concerning g-factors Let us consider the Zeeman interaction for a dimer in the localised (say A B) state: H ˆ b…g SA ‡ gSB †H:

…24†

Model (24) corresponds to the cluster with isotropic local g-factors and to the model with parallel orientation of the axis of local g-tensors. Within this localisation scheme A B, one can introduce molecular g-factor for a given S-state as follows [39]: S…S ‡ 1† ‡ SA …SA ‡ 1† ÿ SB …SB ‡ 1†  g 2S…S ‡ 1† S…S ‡ 1† ‡ SB …SB ‡ 1† ÿ SA …SA ‡ 1† ‡ g: 2S…S ‡ 1†

where a ˆ a (Fe(II)), a ˆ a (Fe(III)). In the limit case of qS ˆ 1 (A B localisation), we arrive at the well-known hyper®ne Hamiltonian for S ˆ 1=2 of Fe(II)Fe(III) dimer:
ÿ …20† HA B ˆ ÿ 43a IA ‡ 73aIB S:

gS …A B† ˆ

The case cS < 1 and v2 =x > k2S =X (case 1a) is the only one when the system is partially localised …q0 6ˆ 0†. Providing strong transfer (case 1b) and/ or strong P-coupling (case 2), PKS coordinate q is unshifted in the minima, and the system is fully delocalised:

Molecular g-factor gS (AB ) in localization scheme AB is obtained from Eq. (25) by means of change A$B. For homonuclear MV dimer, SA ˆ SB and SA ˆ SB , so

USM …0; Q † ˆ p12…jSA ; SB ; SMi  jSA ; SB ; SMi†; …21† where the sign ÿ…‡† relates to the lower (upper) minimum in case 2a. In the lower fully delocalised minimum f0; Qÿ g, the averaged hyper®ne constants for n 6 4 (upper sign) and n > 4 (lower sign) are found to be

…25†

gS …A B† ˆ gS …AB †:

…26†

Since g-values are independent of the site of localisation, the expectation value of Zeeman interaction will be the same in both localised and delocalised minima, and hence, one can introduce the e€ective molecular g-factor for MV dimer by S0 ‡ 34 1 …g ÿ g†; gS ˆ …g ‡ g†  2S…S ‡ 1† 2

…27†

with ‡…ÿ† sign corresponding to n 6 4 …n > 4†.

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A.V. Palii et al. / Chemical Physics 255 (2000) 51±57

The important point is that, as distinguished from the molecular hyper®ne parameters, the molecular g-factor in model (24) is independent of the degree of localisation. This is due to the fact that Zeeman interaction (24) does not feel the distribution of the electronic densities over metal sites, and thus, the molecular gS is determined only by spin S and isotropic local gS . 6. Concluding remarks The results obtained in the framework of the semiclassic adiabatic approach demonstrate how the hyper®ne coupling parameters are in¯uenced by the combined action of vibronic interactions with local and multicentre vibrations. We have shown that the vibronic localisation e€ect is expected to occur only providing relatively strong PKS coupling …v2 =x > k2S =X† and relatively weak double exchange …tS < v2 =x ÿ k2S =X†, i.e. under the conditions corresponding to Class II in the re®ned Robin and Day classi®cation [49]. Since these conditions contain spin-dependent parameters tS and kS , the isotropic magnetic exchange also affects the degree of localisation. Eq. (13) shows that the ``vibronic'' hyper®ne parameters are presented as a mixture of localised ones. It should be pointed out that the only combined parameter qS of the electronic density, which includes the vibronic effects, can be extracted from the hyper®ne structure of EPR. In order to ®nd separately the double exchange parameter t and the vibronic parameters k and v one has to go further and to analyse the intervalence band transitions.

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