Localisation vs. delocalisation in the dimeric mixed-valence clusters in the generalised vibronic model. Magnetic manifestations

Chemical Physics 240 Ž1999. 149–161

Localisation vs. delocalisation in the dimeric mixed-valence clusters in the generalised vibronic model. Magnetic manifestations a J.J. Borras-Almenar , E. Coronado a,1, S.M. Ostrovsky b, A.V. Palii ´ B.S. Tsukerblat a,),2

a,2

,

a

b

Departamento Quimica Inorganica, UniÕersidad de Valencia, Dr. Moliner 50, E-46100 Burjassot, Spain Institute of Applied Physics, Academy of Sciences of MoldoÕa, Academy str. 5, MD-2028, KishineÕ, MoldoÕa Received 8 July 1998

Abstract The problem of localisation–delocalisation in the dimeric mixed-valence clusters is considered in the framework of the generalised vibronic model. The model takes into account both the local vibrations on the metal sites ŽPiepho–Krausz–Schatz model. and the multicenter Žmolecular. vibrations changing the intermetallic distances Žas suggested by Piepho.. In the framework of the semiclassical adiabatic approach the potential surfaces are analysed and different kinds of localised and delocalised states are found. On the basis of the calculated degrees of the localisation the conventional Robin and Day classification of mixed-valence compounds is reconsidered in view of the generalised vibronic model. The magnetic properties of the many-electron mixed-valence dimers are considered as well. The multicenter vibrations are shown to produce a ferromagnetic effect. q 1999 Elsevier Science B.V. All rights reserved.

1. Introduction The vibronic interaction is inherent to the fascinating area of mixed valency that is relevant to material science, catalysis and metallobiochemistry as well as to the vast field of molecular and solid state magnetism. The background for the consideration of the vibronic effects in mixed-valence ŽMV. systems was pioneered by Piepho, Krausz and Schatz and became well known as the PKS model w1–3x. The PKS model deals with the full-symmetric vibrations Ž‘breathing’ modes. belonging to the constituent moieties of the MV cluster. In the dimeric systems the out-of-phase mode built from the local breathing vibrations proves to be relevant to the electron transfer processes. The strength of the vibronic coupling determines the degree of the localisation in the

)

Corresponding author. E-mail: [email protected] E-mail: [email protected] 2 On leave from the Quantum Chemistry Department, Institute of Chemistry, Academy of Sciences of Moldova, MD-2028 Kishinev, Moldova. 1

0301-0104r99r$ - see front matter q 1999 Elsevier Science B.V. All rights reserved. PII: S 0 3 0 1 - 0 1 0 4 Ž 9 8 . 0 0 3 7 5 - 9

J.J. Borras-Almenar et al.r Chemical Physics 240 (1999) 149–161 ´

150

symmetric MV dimers. Depending on the interrelation between the electron transfer parameter and vibronic coupling three classes of MV compounds can be distinguished: valence trapped ŽClass I., partially delocalised ŽClass II. and completely delocalised ŽClass III.. This classification scheme proposed by Robin and Day in 1967 w4x is of fundamental importance for the chemical insight on the MV compounds, especially for the discussion of spectroscopic manifestations of mixed valency. In the many-electron dimeric MV clusters possessing spin cores the moving electron is shown to produce strong ferromagnetic effect through the double exchange mechanism w5x Žsee also review papers w6–8x.. The magnetic properties of MV dimers are determined mainly by the interplay between the double exchange, magnetic exchange and vibronic interaction. The magnetic manifestations of the PKS-type interaction were reported and reviewed in several papers w6–10x. PKS interaction proves to produce a localisation effect so that in the limit of strong PKS coupling the system exhibits the Heisenberg-type magnetic behaviour inherent to fully localised system. During ten years the PKS model remained the only theoretical tool for the consideration of the vibronic effects in MV compounds. Later on Piepho w11,12x suggested a new vibronic model that also takes into account the modulation of the intermetallic distances in course of the molecular vibration. As distinguished from the PKS interaction this kind of vibronic coupling can promote the electron delocalisation. The combined effect of these two types of interactions was considered in the description of the band-shapes of intervalence optical absorption. In this paper we focus on the problem of localisation–delocalisation in MV dimers and refine the Robin and Day classification in the framework of the generalised vibronic model that takes into account both types of the vibronic interactions. The electronic densities of the extra electron on the metal sites are found as a function of the key parameters of the system, namely, vibronic coupling constants and double exchange parameter. The magnetic manifestations of the molecular vibrations introduced by Piepho ŽP-vibrations. are also considered. The multicenter vibrations is shown to produce a ferromagnetic effect in the MV dimers.

2. The model Let us consider a symmetric MV dimer AB in which one ‘extra’ electron is delocalised over two spin cores SA0 s S B0 ' S0 . The double exchange w5x produces the ferromagnetic effect due to the polarisation of the spin cores. The energy levels of the bimetallic system arising from the double exchange and magnetic exchange interaction Žisotropic Heisenberg-type exchange. can be presented as: E " Ž S . s yJS Ž S q 1 . "

t 2 S0 q 1

Ž S q 1r2. ,

Ž 1.

where J is the magnetic exchange parameter and t is the transfer integral, S is the total spin of the dimer. The vibronic coupling with the PKS out-of-phase mode mixes the states with the opposite parity cqs 2y1 r2 Ž wA q w B . Ž wA and w B are the localised states. and cys 2y1 r2 Ž wA y w B . corresponding to two levels of Eq. Ž1. with the same spin S leading thus to a pseudo-Jahn–Teller vibronic problem. The matrix of the PKS coupling is of the form:

ž

cq

cy

0 Õq

Õq , 0

/

Ž 2.

where q s 2y1 r2 Ž qA y q B . is the dimensionless ‘out-of-phase’ vibration, qA and q B stand for the full symmetric coordinates associated with the A and B moieties, Õ is the PKS vibronic coupling constant.

J.J. Borras-Almenar et al.r Chemical Physics 240 (1999) 149–161 ´

151

The P-vibrations change the metal–metal distances resulting thus in the modulation of the transfer integral. To a first approximation one gets: t Ž R. s t Ž R s R0 . q

ž

Et Ž R . ER

/

Q,

Ž 3.

Rs R 0

where R s R AB is the metal–metal distance, R 0 is the metal–metal distance for the system consisting of two spin cores, i.e. for the system without extra electron, Q s R y R 0 is the vibrational coordinate in the P-model, l s yŽEt Ž R .rER .Rs R 0 is the vibronic coupling parameter Žit is to be noticed that t in Eq. Ž1. is just t Ž R s R 0 ... The matrix of the adiabatic potential corresponding to a total spin S of the MV dimer Ž S s 2 S0 q 1r2, 2 S0 y 1r2, . . . , 1r2. can be represented as following Ž " s 1.:

cq v U Ž S . s yJS Ž S q 1 . q

2

q2 q

V 2

Q2

ž

1 0

cq

cy

t S . y lŽ S . Q Õq

Õq , y Ž t Ž S . y lŽ S . Q .

cy 0 1

q

/ žŽ

/

Ž 4. where t Ž S . s Ž S q 1r2. trŽ2 S0 q 1. is the effective Žmultielectron. spin-dependent double exchange parameter, lŽ S . s Ž S q 1r2. lrŽ2 S0 q 1. is the spin-dependent P-coupling parameter possessing the same spin dependence as the double exchange, v and V are the frequencies of the PKS and P-vibrations correspondingly. As distinguished from the PKS coupling the P-coupling is spin-dependent and commutes with the double-exchange term.

3. Adiabatic surfaces Piepho w11,12x has considered the intervalence absorption band on the basis of the numerical solution of the dynamic vibronic problem. In order to derive the analytical expressions for the electronic densities and to give in this way a general insight on the localisation–delocalisation phenomenon we will use here a more simple adiabatic approximation in which the motion of the system in the qQ-space is confined to the lower sheet of the adiabatic surface. Eq. Ž4. results in the following expression for the adiabatic potential relating to a given full spin state S: 1 1r2 2 s U" , Ž q, Q . s yJS Ž S q 1 . q Ž v q 2 q V Q 2 . " Ž t Ž S . y lŽ S . Q . q Õ 2 q 2 Ž 5. 2 where the signs ‘y’ and ‘q’ relate to the lower and upper sheets. The corresponding adiabatic wave-functions are the following: 1r2 s F" Ž q, Q . s

1 2

ž

t Ž S . y lŽ S . Q

1"

2 Ž t Ž S . y lŽ S . Q . q Õ 2 q 2

1r2

/

cq 1r2

"

Õq

1

< Õq < 2

ž

1.

t Ž S . y lŽ S . Q 2 Ž t Ž S . y lŽ S . Q . q Õ 2 q 2

1r2

/

cy .

Ž 6.

Since the physical results are independent of the sign of the parameters for the sake of definiteness we assume that Õ, l, t ) 0. Depending on the relative values of the key parameters several qualitatively different types of the adiabatic surfaces should be distinguished. These different kinds of adiabatic surfaces are shown in Fig. 1.

152

J.J. Borras-Almenar et al.r Chemical Physics 240 (1999) 149–161 ´

J.J. Borras-Almenar et al.r Chemical Physics 240 (1999) 149–161 ´

153

Ž1. PKS coupling exceeds P-coupling (we will refer this case to as the case of strong PKS coupling) Õ2

v

)

l2 Ž S . V

It should be noticed that the vibrational frequencies v and V are also involved in this condition and in the following conditions of such type. Within the case of strong PKS coupling depending on the value of the transfer parameter there are two physically different situations: Ž1a. ComparatiÕely weak transfer: tŽ S. -

Õ2 y

v

l2 Ž S . V

and Ž1b. ComparatiÕely strong transfer: tŽ S. G

Õ2 y

l2 Ž S .

v V SŽ In the case 1a the lower sheet Uy q, Q . of the adiabatic surface possesses two equivalent minima in the 0 0 qQ-plane at the points  "q Ž S ., Q Ž S .4 , where 0

q Ž S. s Q0 Ž S . s

Õ2

v2

y

1r2

t 2 Ž S . Õ 2V 2

Ž Õ 2V y l2 Ž S . v .

2

.

Ž 7.

lŽ S . v t Ž S . l2 Ž S . v y Õ 2V

The positions of the minima are symmetric in the q-subspace and shifted in the positive direction along Q-axis. This situation is shown in Fig. 1a and b and illustrated also by the contour plots in Fig. 2a and b. The energies of these two equivalent minima are the following: U s "q 0 Ž S . , Q 0 Ž S . s yJS Ž S q 1 . y

Õ2

t2 Ž S.

y 2v

2

ž

Õ2 y

v

l2 Ž S . V

.

Ž 8.

/

Two minima are separated by one or two saddle points located on the Q-axes ŽFigs. 1a and 2a.. The coordinates of these points are found to be  0, Qy Ž S .4 and  0, Qq Ž S .4 with Q "s "lŽ S .rV . These two saddles have different energies 0 U s Ž 0, Q . . s yJS Ž S q 1. . t Ž S . y

l2 Ž S . 2V

,

Ž 9.

with the first saddle being deeper. Under the condition t Ž S . ) l2 Ž S .rV the upper saddle point disappears Žcompare Fig. 1a and b..

SŽ Fig. 1. The lower sheet Uy q, Q . of the adiabatic potential for MV dimer Ž v s V .: Ža. t Ž S . s 0.4 v , Õ s 2 v , lŽ S . s v ; Žb. t Ž S . s 1.5 v , Õ s 2 v , lŽ S . s v ; Žc. t Ž S . s 4v , Õ s 2 v , lŽ S . s v ; Žd. t Ž S . s 0.4v , Õ s v , lŽ S . s 2 v ; Že. t Ž S . s 1.5 v , Õ s v , lŽ S . s 2 v ; and Žf. t Ž S . s 4v , Õ s v , lŽ S . s 2 v .

154

J.J. Borras-Almenar et al.r Chemical Physics 240 (1999) 149–161 ´

SŽ Fig. 2. The contour plots of the lower sheet Uy q, Q . for MV dimer Ž v s V .. The parameters are the same as those in Fig. 1a–f.

J.J. Borras-Almenar et al.r Chemical Physics 240 (1999) 149–161 ´

155

The electronic densities on the ions A and B in the minima points given by Eq. Ž7. can be obtained from the adiabatic wave-function Fy Ž q, Q . of the lower sheet ŽEq. Ž6... These electronic densities are found to be:

rA Ž "q 0 , Q 0 . s r B Ž .q 0 , Q 0 . s

1 2

1 . Ž1 y g 2 Ž S . .

1r2

,

Ž 10 .

where

g Ž S. s

tŽ S. Õ

2

y

v

l2 Ž S .

.

Ž 11 .

V

Eq. Ž10. shows that in the  yq 0 Ž S ., Q 0 Ž S .4 minimum the ‘extra’ electron is mostly localised on the ion A meanwhile in the  qq 0 Ž S ., Q 0 Ž S .4 minimum it is localised on B. One can see that the degree of the localisation in the case 1a is governed by the only parameter g Ž S . w g Ž S . - 1 under the condition of the case 1ax. Providing g Ž S . s 0 the system is fully localised, meanwhile for g Ž S . s 1 the system is fully delocalised. Eq. Ž10. demonstrates the physical role of the key interactions: increase of t Ž S . andror lŽ S . tends to delocalise the system, meanwhile increase of PKS interaction increases the degree of localisation. With the increase of g Ž S . the two minima  qq 0 Ž S ., Q 0 Ž S .4 move toward the deeper saddle point  0, Qy Ž S .4 , while the barrier between the minima decreases and the system becomes more and more delocalised. At the limit g Ž S . s 1 these two minima are transformed into one minimum so that instead of saddle point we have now the only minimum in  0, Qy Ž S .4 position. Contour plots in Fig. 2 clearly illustrate these transformations. In this way we arrive at the type 1b of the adiabatic surfaces as shown in Fig. 1c. The further increase of g Ž S . leads to the stabilization of this only minimum in which the extra electron is fully delocalised. It should be noticed that Eq. Ž9. is inapplicable to the case 1b Žwhen g Ž S . ) 1.. Under the condition g Ž S . ) 1 the electronic densities rA and r B are obtained by the substitution of the coordinates  0, Qy Ž S .4 of the only minimum into adiabatic wave-function Fy ŽEq. Ž6... This substitution gives FyS Ž0, Qy . s cy so that in this minimum

rA Ž 0, Qy . s r B Ž 0, Qy . s

1 2

.

Ž 12 .

One can see that within the condition of the case 1b the system is fully delocalised irrespectively of the transfer parameter and both vibronic parameters. Ž2. P-coupling exceeds the PKS coupling Õ2

v

-

l2 Ž S . V

Similar to the case 1 within the case of strong P-coupling the two situations should be considered, namely: Ž2a. t Ž S . - l2 Ž S .rV y Õ 2rv Ž weak transfer . and Ž2b. tŽS. G l2 ŽS.rV y v 2rv Ž strong transfer .. SŽ Providing weak transfer the adiabatic surface Uy q, Q . possesses two minima with different energies given Ž .  Ž .4 by Eq. 9 just in the same positions 0, Q " S where in the case 1a we have observed the saddle points ŽFig. 1d and e.. At the same time now  "q 0 Ž S ., Q 0 Ž S .4 are the coordinates of two energetically equivalent saddle points given by Eq. Ž8.. The adiabatic wave-functions in the minima points are cy Ždeepest minimum. and cq

J.J. Borras-Almenar et al.r Chemical Physics 240 (1999) 149–161 ´

156

Žshallow minimum. so that under the condition of the relatively strong P-coupling and weak transfer Žcase 2a. the system proves to be always fully delocalised. The localised states correspond now to the saddle points and so they are unstable. Increase of g Ž S . leads to the transformation of the adiabatic surface in such a way that the saddle points move toward the shallow minimum  0, Qq Ž S .4 and provided that t Ž S . s l2 Ž S .rV y Õ 2rV this minimum disappears. This is shown by the contour plots ŽFig. 2d–f.. It should be stressed that independently of the key parameters defining the position of the minima and the saddle points Žas well as the heights of the barriers. the system remains fully delocalised in the minima of the adiabatic potential. Finally, in the case 2b the adiabatic surface possesses the only minimum in the position  0, Qy Ž S .4 ŽFigs. 1f and 2f.. Ž3. In the particular case Õ 2rv s l2 Ž S .rV providing t / 0 we face the situation similar to the case 2b with one minimum in the position  0, Qy Ž S .4 . The results obtained shows that only one kind of minima may exist: this can be either localised states Ždelocalised states are unstable. or delocalised states Žwith unstable localised states.. The coexistence of localised and delocalised minima proves to be impossible. This is similar to the well-known situation in the classical Jahn–Teller T2 m Ž e q t 2 . problem where either tetragonal or trigonal minima can exist but never both of them simultaneously w13x. It should be stressed that the inequalities mentioned above are closely related to the spin state. The degree of localisation decreases with the increase of the full spin of the system, the high-spin states of the magnetic cluster are more delocalised.

4. Discussion of the Robin and Day classification in the generalised vibronic model The Robin and Day classification scheme proposed in 1967 w4x distinguishes the MV compounds accordingly to the degree of the localisation of the ‘extra’ electron on the metal sites. The original form of this classification was based on the parameter W that is the difference in energies of the ‘extra’ electron localised on two metal sites in an asymmetric MV dimers. Later on Piepho, Krausz and Schatz w1,3x realized to full extent the crucial role of the vibronic interaction in the description of the MV compounds in view of the problem of localisation–delocalisation and especially for the spectroscopic manifestation of mixed valency. In the symmetric MV dimers Žequivalent sites, W s 0. the vibronic coupling proves to be the only physical interaction that is able to localise the ‘extra’ electron Žlike W interaction in an asymmetric system.. According to the Robin and Day classification refined in the framework of the PKS model for the symmetric MV dimers the following criteria was established w2x: Class I: t Ž S . - Õ 2rv

Ž 13 .

SŽ . Ž1. Ž S is the ground state spin.. The lower potential curve Uy q possesses two deep minima. Ž . This is the case of strongly localised valence trapped systems exhibiting the broad intervalence optical bands in the high-energy region. These bands are of low or negligible intensities.

Class II: t Ž S . - Õ 2rv

Ž 14 .

SŽ . Ž2. This is still the case of double-well adiabatic potential Uy q , but the barrier is low so that the systems belonging to Class II are partially delocalised.

Class III : t Ž S . ) Õ 2rv

Ž 15 .

J.J. Borras-Almenar et al.r Chemical Physics 240 (1999) 149–161 ´

157

Ž3. ŽStrong transfer andror weak vibronic PKS coupling.. The low branch of the adiabatic potential possesses the only minimum corresponding to fully delocalised state. The intervalence bands are expected to be narrow and intensive. So the PKS model predicts that the intervalence absorptions bands become broader and less intensive with the increase of the degree of localisation. Let us consider now the Robin and Day classification scheme from the point of view of the generalised vibronic model involving both PKS and P-vibrations. We start with the case 1 of PKS coupling exceeding P-coupling Ž Õ 2rv ) l2 Ž S .rV .. In this case depending on the magnitude of the electron transfer parameter MV compounds can belong to Class I, II or III. So providing t Ž S . < Õ 2rv y l2 Ž S .rV we have strongly localised system belonging to Class I, for t Ž S . - Õ 2rv y l2 Ž S .rV the system can be assigned to Class II and for t Ž S . G Õ 2rv y l2 ŽS.rV we arrive at the fully delocalised system ŽClass III.. These conditions are formally similar to those in the PKS model Žcriteria Ž13. – Ž15... However, one can observe the essential difference between these criteria. In fact, in the generalised model instead of pure PKS vibronic contribution Õ 2rv we are dealing with the combined term Õ 2rv y l2 Ž S .rV . One can see that the vibronic localisation effect is effectively reduced due to the fact that the P-coupling promotes the delocalisation. In fact, the tunneling of the system between two minima ŽFig. 1a and b. is expected to occur through the saddle point rather than along the q-axis where the barrier is higher. This can be clearly seen from the contour plots in Fig. 2a and b. As the result for strong enough P-coupling the MV system can belong to the Class II or III even providing weak electron transfer Žor strong PKS coupling.. On the other hand, strong localisation ŽClass I. is achieved only for weak transfer andror weak P-coupling. Let us pass now to the case when P-interaction exceeds PKS interaction Ž Õ 2rv - l2 Ž S .rV .. As it was shown above within this case the system is fully delocalised independently of the relative values of t Ž S . and Õ 2rv . This means that the system in which the dominating vibronic interaction is of P-type always belongs to Class III even providing very small t Ž S .. This result is in striking contradiction with the prediction of the PKS model in which the degree of the delocalisation in the symmetric MV dimers is determined by the interplay between the electron transfer and PKS coupling. The intervalence absorption in the generalised dynamic vibronic model was considered by Piepho w11,12x. Here we note only that the correlation between the degree of localisation and parameters of intervalence bands Žwidth, position and intensity. established in the PKS model is to be reconsidered. Particularly, in the contrast to the conclusion based on the PKS model the fully delocalised system ŽClass III. can exhibit strong broad Žbut not narrow. intervalence band if the P-coupling is dominant. This situation is illustrated by the vertical section of the adiabatic surfaces ŽFig. 3. along the Q-axis. One can see that the broad intervalence band may arise from the transition between fully delocalised states.

Fig. 3. The vertical section UŽ q s 0, Q ., scheme of the intervalence absorption.

J.J. Borras-Almenar et al.r Chemical Physics 240 (1999) 149–161 ´

158

5. Magnetic properties The magnetic manifestation of the PKS coupling is well studied w7,9x. This kind of vibronic coupling localises the system and thus effectively reduces the ferromagnetic effect arising from the double exchange Žsee fig. 4 in Ref. w9x.. In the strong PKS coupling limit the system becomes fully localised so in the deep minima of the adiabatic potential Lande’s ´ rule for the spin levels is restored with the effective exchange parameter JX sJq

t 2v Õ2

,

Ž 16 .

involving second-order ferromagnetic contribution arising from the reduced double exchange. This contribution is small for strongly localised systems. In order to study the magnetic properties of the MV systems in the framework of the generalised vibronic model we use the semiclassical approach developed in w14x. The spin levels in this approach may be identified with the adiabatic surfaces U S Ž q, Q . so that the usually accepted Van Vleck equation becomes: `

Ý Ý S Ž S q 1. Ž 2 S q 1. H 2 meff

ŽT .

s g 2m2B

s

y`

"

s "

Hexp yU

,

`

s "

Ý Ý Ž 2 S q 1. H s

Hexp yU y`

"

Ž q, Q . rkT d q dQ Ž 17 .

Ž q, Q . rkT d q dQ

where meff ŽT . is the temperature-dependent magnetic moment. As it was shown in w14x the semiclassical approach describes the temperature dependence meff ŽT . of the vibronic systems with very high accuracy. Fig. 4a illustrates the effect of P-vibronic coupling for the simplest d1 –d 2 MV dimer Ž S0 s 1r2, S s 1r2, 3r2. in the case of relatively small double exchange Žcompare with the antiferromagnetic isotropic exchange.. The competition between the isotropic exchange and the double exchange gives the antiferromagnetic Ž Sgr s 1r2. ground state ŽFig. 4, l s 0.. As one can see from Fig. 4 a P-coupling increases the magnetic moments so that the double exchange proves to be effectively enhanced. Providing l s 2 v the ground state remains antiferromagnetic Ž m ŽT s 0. s 1.73 B.M.. meanwhile for l s 3 m ŽT s 0. s 3.87 B.M. so that in this case the ground state is ferromagnetic as one can observe providing strong double exchange. The effect of the P-coupling can be understood in the limiting case of negligible PKS interaction. In the case of strong P-coupling the energy levels of the system can be roughly identified with the energies UminŽ S . of the minima of the adiabatic potentials. Substituting the positions of the minima Q "s "lŽ S .rV in Eq. Ž5. one finds " Umin Ž S . s yJeff S Ž S q 1 . .

t 2 S0 q 1

ž

Sq

1 2

/

,

Ž 18 .

where

l2 Jeff s J q

2V

.

Ž 19 .

Energy scheme given by Eq. Ž18. is formally similar to that given by Eq. Ž1., the last being obtained providing that the vibronic coupling is switched off. One should stress that the vibronic P-coupling results in a strong ferromagnetic contribution l2r2 V in the exchange parameter. Fig. 4b illustrates the combined effect of the PKS and P-types of vibronic coupling in the case of relatively strong double exchange. In absence of the vibronic coupling Ž Õ s 0, l s 0. the system is ferromagnetic due to strong double exchange. Inclusion of the PKS interaction Ž Õ s 2 v . reduces the magnetic moment so that m ŽT s 0. s 1.73 B.M. Ž Sgr s 1r2.. At the same time the P-coupling increases m ŽT . and providing l G 2 the

J.J. Borras-Almenar et al.r Chemical Physics 240 (1999) 149–161 ´

159

Fig. 4. Magnetic moments of d 2 –d1 MV dimer Ž v s V .: Ža. t s 0.5 v , J sy v , Õ s 0; and Žb. t s v , J s 0.1 v , Õ s 2 v .

ground state possesses Sgr s 3r2 that is peculiar for the case of strong double exchange Žorrand weak PKS interaction.. This conclusion is also in the line of the fact that the delocalisation of the extra electron favours the ferromagnetic effect due to polarisation of spin cores. In this sense the double exchange and the vibronic P-coupling are equivalent.

6. Concluding remarks Our results obtained in the framework of the semiclassical adiabatic approach demonstrate how the localisation–delocalisation phenomenon and magnetic properties of symmetric MV dimers are influenced by the combined action of vibronic interactions with local and multicenter vibrations. Providing relatively strong PKS coupling Ž Õ 2rv ) l2 Ž S .rV . and small electron transfer Ž t Ž S . - Õ 2rv y l2 Ž S .rV . the system is localised in equivalent minima. The systems obeying these conditions can be assigned to the Class I or II in Robin and Day

160

J.J. Borras-Almenar et al.r Chemical Physics 240 (1999) 149–161 ´

classification depending on the strength of the electron transfer. Providing t Ž S . G Õ 2rv y l2 Ž S .rV the system possesses single-well adiabatic surface and exhibits full delocalisation ŽClass III.. Finally, for the relatively strong coupling with the molecular vibrations Õ 2rv - l2 Ž S .rV the adiabatic potential possesses in general two nonequivalent minima corresponding to completely valence detrapped states. It should be stressed that in this case the system belongs to the Class III irrespectively of the electron transfer and vibronic PKS coupling parameters. Therefore the generalised vibronic model allows to give a new insight on the Robin and Day classification scheme and on the interrelation between characteristics of the intervalence optical bands and degree of the localisation. With respect to the magnetic properties of many-electron dimers the essential difference between the roles of local and molecular vibrations should be emphasised. In fact, PKS vibrations reduce the double exchange destroying thus the ferromagnetic spin alignment. On the contrary, the molecular vibrations effectively enhance the double exchange giving rise to a strong ferromagnetic contribution. Some remarks concerning the applicability of the refined vibronic model to the real MV systems should be made. In general the multicenter vibronic interaction plays an important role in the systems for which the molecular orbital basis is more relevant then the valence bond type basis of the PKS model. The first application of the generalized model was made by Piepho in her pioneering paper w12x dealing with the Creutz–Taube ion Žm-pyrazine. bisŽpentaammine-ruthenium.Žq5.. The importance of the multicenter vibrations is supported by the conclusion that the Creutz–Taube ion is a delocalised MV system with the relatively broad intervalence band arising from b 2g )Žd, p . ™ b 3uŽd, p ). excitation. The analysis of the band shape supports also the conclusion that this band is formed primarily from coupling to a molecular mode rather than to a localised one Žalthough the interaction with the localised vibration is also significant.. A clear indication on an importance of the multicenter vibronic coupling is given in w15x where the electronic and geometric structure of a trinuclear MV Cu 2 ŽII.CuŽIII. cluster was studied. In fact, the authors arrived at the conclusion that the real structure of the cluster is C 2v rather than the idealized one D 3h due to the distortion of the symmetric network of the metal–metal distances. It should be emphasized that the vibronic model dealing with the multicenter and local vibrations can be considered as more or less good approximation reflecting the reality occurring in MV clusters. The division in two types of vibrations seems to be relevant when the whole system consists of well-isolated subunits that can be exemplified by the Creutz–Taube complex. A case of a MV cluster consisting of strongly coupled moieties is represented by the wNEt 4 xwCu 2 Cl 4 x compound prepared by Willett w16x and studied theoretically in w17x. In fact, the structure of wNEt 4 xwCu 2 Cl 4 x involves CuŽI.CuŽII. MV clusters and can be described as a chain of CuCl 4 tetrahedra sharing edges. Apparently in this case the multicenter and local vibrations are mixed and the concept of two types of active vibrations taken separately can be applied for qualitative analysis only. As far as the magnetic properties of MV compounds are concerned it should be noted that the magnetic susceptibility data alone cannot permit the unique determination of the key parameters involved, namely, t, J, Õ and l. The determination of these parameters, particularly the estimation of the role of two kinds of active vibrations requires the detailed experiments including Žalong with the magnetochemistry. also EPR and Mossbauer spectroscopy. In this view the quantum-mechanical calculations of the key parameters including the ¨ vibronic ones seems to be of great help w18x.

Acknowledgements This work was supported by the Spanish DGICYT ŽPB94-0998. and the European Union ŽNetwork on Molecular Magnetism, ERB 4061 PL970197.. AP thanks Generalitat Valenciana for a visiting grant. BST thanks University of Valencia for a visiting professor grant. We thank the CIUV for the use of its computer facilities. The authors thank A. Bencini for providing his paper w18x prior to publication.

J.J. Borras-Almenar et al.r Chemical Physics 240 (1999) 149–161 ´

161

References w1x w2x w3x w4x w5x w6x w7x w8x w9x w10x w11x w12x w13x w14x w15x w16x w17x w18x

S.B. Piepho, E.R. Krausz, P.N. Schatz, J. Am. Chem. Soc. 100 Ž1978. 2996. P.N. Schatz, in: D.B. Brown ŽEd.., Mixed-Valence Compounds, NATO Adv. Stud. Inst. Ser. Reidel, Dordrecht, 1980, p.115. K.Y. Wong, P.N. Schatz, Prog. Inorg. Chem. 28 Ž1981. 369. M.B. Robin, P. Day, Adv. Inorg. Radiochem. 10 Ž1967. 247. P.W. Anderson, H. Hasegawa, Phys. Rev. 100 Ž1955. 675. B.S. Tsukerblat, M.I. Belinskii, V.E. Fainzilberg, in: M.E. Vol’pin ŽEd.., Sov. Sci. Rev. B Chem. 9, Overseas Sciience, New York, 1987, p.337. J.J. Borras-Almenar, E. Coronado, B.S. Tsukerblat, R. Georges, in: E. Coronado, P. Delhaes, D. Gatteschi, J.S. Miller ŽEds.., Molecular Magnetism: From Molecular Assemblies to the Devices, NATO Adv. Stud. Inst. Ser. 321, Kluwer, Dordrecht, 1996, p.105. G. Blondin, J.J. Girerd, Chem. Rev. 90 Ž1989. 1359. B.S. Tsukerblat, H.M. Kishinevsky, A.V. Palii, V.Ya. Gamurari, A.S. Berengolts, Mol. Phys. 76 Ž1992. 1103. G. Blondin, S. Borshch, J.J. Girerd, Comments Inorg. Chem. 12 Ž1992. 315. S.B. Piepho, J. Am. Chem. Soc. 110 Ž1988. 6319. S.B. Piepho, J. Am. Chem. Soc. 112 Ž1990. 4197. I.B. Bersuker, The Jahn–Teller Effect and Vibronic Interactions in Modern Chemistry, Plenum, New York, 1984. J.J. Borras-Almenar, E. Coronado, H.M. Kishinevsky, B.S. Tsukerblat, Chem. Phys. Lett. 217 Ž1994. 525. D.E. Root, M.J. Henson, T. Machonkin, P. Mukherjee, T.D.P. Stack, E.I. Solomon, J. Am. Chem. Soc. 120 Ž1998. 4982. R.D. Willett, Inorg. Chem. 26 Ž1987. 3423. C.J. Calzado, J.F. Sunz, J. Am. Chem. Soc. 120 Ž1998. 1051. V. Barone, A. Bencini, I. Ciofini, C.A. Daul, F. Tott, J. Am. Chem. Soc. Žin press..