LocalizationâDelocalization in Bridged Mixed-Valence Metal Clusters

Subscriber access provided by - Access paid by the | UCSB Libraries

Article

Localization-delocalization in Bridged Mixedvalence Metal Clusters: Vibronic PKS Model Revisited Andrew Vladimir Palii, Boris S. Tsukerblat, Juan Modesto Clemente-Juan, and Sergey M. Aldoshin J. Phys. Chem. A, Just Accepted Manuscript • DOI: 10.1021/acs.jpca.5b05186 • Publication Date (Web): 25 Aug 2015 Downloaded from http://pubs.acs.org on September 5, 2015

Just Accepted “Just Accepted” manuscripts have been peer-reviewed and accepted for publication. They are posted online prior to technical editing, formatting for publication and author proofing. The American Chemical Society provides “Just Accepted” as a free service to the research community to expedite the dissemination of scientific material as soon as possible after acceptance. “Just Accepted” manuscripts appear in full in PDF format accompanied by an HTML abstract. “Just Accepted” manuscripts have been fully peer reviewed, but should not be considered the official version of record. They are accessible to all readers and citable by the Digital Object Identifier (DOI®). “Just Accepted” is an optional service offered to authors. Therefore, the “Just Accepted” Web site may not include all articles that will be published in the journal. After a manuscript is technically edited and formatted, it will be removed from the “Just Accepted” Web site and published as an ASAP article. Note that technical editing may introduce minor changes to the manuscript text and/or graphics which could affect content, and all legal disclaimers and ethical guidelines that apply to the journal pertain. ACS cannot be held responsible for errors or consequences arising from the use of information contained in these “Just Accepted” manuscripts.

The Journal of Physical Chemistry A is published by the American Chemical Society. 1155 Sixteenth Street N.W., Washington, DC 20036 Published by American Chemical Society. Copyright © American Chemical Society. However, no copyright claim is made to original U.S. Government works, or works produced by employees of any Commonwealth realm Crown government in the course of their duties.

Page 1 of 30

The Journal of Physical Chemistry

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Localization-Delocalization in Bridged Mixed-Valence Metal Clusters: Vibronic Pks Model Revisited A. Palii,1, 4,* B. Tsukerblat,2,* J.M. Clemente-Juan,3 S.M. Aldoshin 4 1

Institute of Applied Physics, Academy of Sciences of Moldova, Kishinev, Moldova 2 Department of Chemistry, Ben-Gurion University of the Negev, P.O. Box 653, Beer-Sheva 84105, Israel 3 Instituto de Ciencia Molecular (ICMol), Universidad de Valencia, C/Catedrático José Beltrán, 2, 46980 Paterna, Spain 4 Institute of Problems of Chemical Physics, Russian Academy of Sciences, 1, Acad. Semenov av., Chernogolovka 142432, Moscow Region, Russia * Authors to whom correspondence should be addressed: [email protected] (A.P.), [email protected] (B.T.)

Abstract Here we describe a new vibronic model of mixed valence (MV) dimer inspirited by the conventional Piepho, Krausz and Schatz (PKS) approach. We attempted to partially lift the main restriction of the PKS model dealing with the vibronically independent moieties of a MV molecule. The refined version of PKS model in which the bridging ligands are included deals with the three main interactions: electron transfer (integral t0) related to the high-symmetric ligand configuration, on-site vibronic coupling (parameter  ) arising from the modulation of the crystal field on the metal sites by the breathing displacements of their nearest ligand surroundings, and inter-center vibronic coupling (parameter  ) describing the dependence of the electron transfer on ligand positions in course of their breathing movement. We apply the modified model to the analysis of the adiabatic potentials and electronic density distributions in the minima of their lower sheets for the cases of one-electron MV dimer with long and short bridges and for the two-electron MV dimer exhibiting valence disproportionation effect. The inclusion of the inter-center interaction in addition to the conventional PKS coupling is shown to produce a strong effect on the degree of localization in MV dimers, and, in particular, on the assignments to the Robin and Day classes, and on the conditions of stabilization of valence disproportionated states in the bielectron transfer systems. 1. Introduction Contemporary developments of the emerging fields of nanoelectronics and molecular spintronics breathed new life into the evolution of the long standing problem of mixed1 ACS Paragon Plus Environment


Page 2 of 30

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

valency marked by

the landmark discovery of

MV Creutz-Taube ion.1

The unique

possibilities offered by the molecular systems with itinerant electrons can be employed through manipulation of their conducting, magnetic, electric and optical properties. As examples one can mention the experimental discovery of the electron transport in a cobalt complex2, highspin molecular magnets3, spin transport through a binuclear iron dimers4 complex. The electric field control over the optical absorption in magnetic MV molecules5 and electrically and magnetically switchable molecular multiferroics6 have been recently proposed. MV molecules (in particular, MV polyoxometalates) are able to generate multifunctional molecular materials in which two or several useful properties coexist (see review article 7).

Redox-induced

electron transfer materials are poised for use as detectors, sensors, etc. in future-generation electronic devices.8 Finally, MV compounds play an important role in the problem of the vibronic trapping

9

of charge polarized states in molecular quantum cellular automata, an

emerging paradigm for nanoelectronics proposed by Lent et al.10-13 in which the binary information is encoded in charge configuration of the molecular cell. These contemporary trends gave impact to the development of the new models and concepts of mixed-valency. Many efforts have been applied toward better understanding of the main mechanisms governing electron delocalization and electronic intervalence absorption bands

in

MV

complexes.14-31 Even at the earlier stage of these studies it has been realized that along with the electronic coupling between the metal sites (electron transfer) the vibronic coupling plays a crucial role as a source of electron localization and, as a particular consequence, in the features of the intervalence (electron transfer) bands and Mössbauer spectra. The full ab-initio evaluation of the electronic states of MV compounds (see exhaustive review article by Malrieu et al. 14) coupled to the full set of the exact vibrational degrees of freedom of a polyatomic system represents a complicated task. Although the ab-initio approach provide the most informative tool for the “bottom-up” study based on the first principles, such type of calculations cannot be always directly used as working tools for chemists and experimentalists. That is why along the development and applications of the abinitio approaches, many efforts have been applied to the elaboration of possibly simple phenomenological models which would capture the main features of mixed valency (like barrier between localized configurations, electronic densities on the sites, etc.) and which are able at the same time to provide a pictorial representation of the electron transfer and vibronic effects in terms of semiempirical parameters. The majority of the works dealing with the vibronic coupling in MV systems were based on the semi-classical approach explored by Sutin 2 ACS Paragon Plus Environment

Page 3 of 30


1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

et al.

15-18

and Hush

19-21

(see also reviews

22,23

). The first quantum-mechanical vibronic model

aimed at the description of the intervalence transfer absorption bands in MV complexes was proposed by Piepho, Krausz and Schatz and is known as PKS-model.

24

Within PKS model

the donor and acceptor possess a single orbital each and a coupling between the extra electron and the fully symmetric (“breathing”) vibrations of the ligand surroundings of donor and acceptor sites are taken into account. Then the “in-plase” mode Q+ and “out-of-phase” mode Qbuilt as symmetric and antisymmetric linear combinations of local breathing vibrations QA and QB (where A and B are the two metal sites of the dimeric MV cluster) are regarded as the normal modes of the system. The PKS model was extended to include multiple single-site modes,25,26 solvent effects, 27

and also multicenter symmetric vibrations which change the inter-site distances thus

modulating the electron transfer parameters and resulting in the two-state two-mode model proposed by Piepho. 28,29 As was first pointed out by Hush, 30 (see also the discussion in31 and refs. therein) the main shortcoming of the PKS model is that it ignores the bridging ligand which are of crucial importance in mediating the electron transfer between the metal centers in the most typical situation when no direct overlap between the donor and acceptor orbitals may exist. In order to overcome the limitations of the PKS model Ondrechen et al

31-47

elaborated

the three-state vibronic model which explicitly includes the electronic and vibrational participation of the bridging ligand by taking into account an electronic state  L localized on the bridging ligand (in addition to the states  A and  B ) and also the symmetric bridge vibrations QL (in addition to the local breathing modes QA and QB involved in the PKS model). Although the PKS model as well as the three-state Ondrechen model were and continue to be the efficient tools in the theoretical analysis of the observed intervalence absorption spectra and also serve for justification of the phenomenological Robin and Day classification of MV compounds, 48 the important features of local breathing vibrations have not been taken into account in the both mentioned models. Indeed, one of the key assumption in both models is that the inter-center electronic coupling matrix element  is independent of the local coordinates QA and QB, and therefore the “breathing modes only modulate the crystal fields acting on the sites A and B.

However, the bridging ligands belonging to the nearest

environments of the sites A and B do participate in the local QA and QB vibrations and hence in the molecular Q vibrations as well. The shifts of the bridging ligands from their equilibrium positions in course of PKS vibrations should also modify the inter-center electronic coupling 3 ACS Paragon Plus Environment


Page 4 of 30

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

matrix element that is expected to be in general dependent on Q and different from its equilibrium value  0 . Such a possibility was first mentioned in Refs

49

and

50

where it was

indicated that the vibronic coupling and the electron transfer parameters are not independent and the electron transfer should depend on the displacements of the bridging ligand. The modulation of the electron transfer can be described by introducing an additional vibronic coupling parameter (denoted below by  ) that can be naturally incorporated in both Ondrechen’s and in PKS models. In this article we will consider this effect restricting ourselves with a simple modified PKS model treated within the semiclassical adiabatic approximation. We apply the parametric modified vibronic model incorporating the intercenter effect of the PKS vibrations to the analysis of localization-delocalization in the two representatives of MV clusters, namely, in one-electron MV dimer with short and long bridges in which the valences of the constituent ions differ by one and in two-electron MV dimer in which the valences differs by two. This interesting case of valence disproportionation has been observed in Pt(II)Pt(IV), Pd(II)Pd(IV) compounds and crystals (Bi(III)Bi(V), Sb(III)Sb(V), etc.). We will show that the inclusion of the inter-center vibronic interaction affecting the electron transfer, in addition to the PKS coupling, strongly influences the degree of localization (in particular, Robin and Day assignment) and the stabilization of the valence disproportionated states. 2. Modified PKS model for one-electron mixed valence dimer: the case of a short bridge To briefly discuss the PKS model let us note that this model deals with the electronic basis composed of two orbitally non-degenerate localized orbitals  A and  B which are coupled by the inter-site electron transfer. The last is described by the parameter

   A hˆ  B , with hˆ being the one-electron Hamiltonian of the system. The extra electron interacts with the breathing vibrations QA and QB on the sites A and B which are assumed to be vibronically independent and therefore represent (within this approximation) the normal modes

of



2

Q  1

the

 Q

A

system.

Then

the

antisymmetric

(out-of-phase)



 QB  and the symmetric (in-phase) mode, Q  1

2

molecular

 Q

A

mode

 QB  can be

build up, with the symmetric vibration being decoupled from the electronic motion.

24

The

coupling between the electronic and nuclear motions on each of the two equivalent sites is determined by the only vibronic coupling parameter VA  VB  V and therefore the PKS model 4 ACS Paragon Plus Environment

Page 5 of 30


1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

can be regarded as the two-state one-mode vibronic model with the three parameters (  , V and vibrational frequency  ). A significant feature of the PKS mode is that it does not affect the positions of the metal sites which remain to be fixed in course of this vibration. This definition of the PKS coordinate implies serious natural limitations to the applicability of the PKS model and other models dealing with the on-site vibronic coupling as a main source of the electron localization. Indeed, the Q mode is, in general, well defined only provided that the two single-site modes QA and QB are vibronically independent. Intuitively, the latter can be justified, at least, qualitatively, for MV systems in which the sites A and B are coupled through a long bridge and hence there are no common ligands belonging to the first coordination spheres of the metal sites. On the contrary, if the nearest ligand surroundings of the metal sites share bridging ligands we face a complicated problem in which all atoms of the molecule participate in the active vibrations and therefore the PKS vibrations cannot be referred to as normal ones even approximately. It goes without saying that in this case the applicability of the PKS model for the quantitative discussion is doubtful. Nevertheless, in an actual case of a special topology when the two first coordination spheres of the sites A and B share one common bridging ligand and the bridging A-L-B angle is 180o one can reach an essential progress in the generalization of the PKS model. cases of corner shared bridged systems were reported in Refs

51-54

Many

, although the edge shared

and face-shared bridged structures are more typical for the MV dimers (see review article

51

and Refs. therein). To demonstrate a possibility to generalize the PKS model for the linear AL-B dimer let us consider a simple model system composed of the two square-planar subunits sharing a bridging ligand as shown in Fig. 1. In line with the PKS concept let us introduce the antisymmetric (out-of-phase) coordinate Q which is represented by the simultaneous expanding of one site and compression of another one while the positions of the metal ions are fixed (Fig. 1). This displacement can be approximately referred to as a normal coordinate of the system providing that the metal ions are much heavier than the ligands and therefore do not move in course of the vibration. It is worthwhile to underline the difference between the vibration Q so far defined in the modified PKS model and mode Q . As distinguished from the genuine PKS modes composed of the independent local vibrations, no in-phase vibration of the type of Q keeping

metal ions in fixed positions can be defined for the MV dimer

sharing the corner. Indeed, the Q vibration in this case change the metal-metal distance since 5 ACS Paragon Plus Environment


Page 6 of 30

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

it produces simultaneous expansion or reduction of all bond lengths (Fig. 1b). This vibration should be ruled out from the consideration if one wishes to remain in the framework of the PKS model in which the metals are assumed to be fixed.

28,29

It is to be noted that the Q

Fig. 1. Conventional vs. modified PKS model as applied to a bi-square MV cornershared dimer: (a) out-of-phase vibration Q in the modified PKS model (the distance between the atoms remain unchanged in course of vibration); (b) inphase vibration Q changing the metal-metal distance. mode does appear in the framework of the Piepho model which allows the movement of the metal ions). A remarkable feature of the Q vibration in the corner shared dimer is that it simultaneously produces the local effect modulating the crystal field around each metal ion (like genuine PKS mode) and a significant intercenter effect which manifests itself in the modulation of the electronic transfer parameter (like metal-metal vibration introduced by Piepho

28,29

). In the model so far described we follow the assumption that the energies

corresponding to the ligand orbitals are much higher than the metal energies and, correspondingly, the former only virtually participate in the electron transfer. In other words, we remain within the initial assumption that there are only two metal orbitals in the system and the pseudo Jahn-Teller coupling involving ligand orbitals is suppressed by a large gap (see 6 ACS Paragon Plus Environment

Page 7 of 30


1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

review on Jahn-Teller and pseudo Jahn-Teller effects in Refs. 55-58). Under this condition the disproportionation of the metal-ligand bonds cannot occur and therefore the transfer matrix element is maximal when the bridging ligand is located at the inversion center of the dimer and it is decreased when the ligand is shifted from the center. The vibronic coupling appears as a result of the expansion of the transfer integral  in a series in the coordinate Q for which the reference configuration corresponds to the symmetric molecule ( Q  0 ). It is convenient to introduce the dimensionless units which are useful for the pictorial representations of the main results. The transfer parameter having the dimension of the energy can be measured in the units of the vibrational energy  (  is the frequency of PKS vibration) , i.e. the dimensionless transfer parameter will be defined as

t     . One can pass to the dimensionless vibrational coordinate q  Q M   , where M  is the effective mass associated with the PKS mode. Then up to first non-vanishing term one can obtain: 1   2t  t  q   t  q  0    2  q2  t0   q2 , 2  q  q 0 

(1)

where  is the dimensionless (in  units) vibronic coupling parameter defined in Eq. (1) and

t0  t q  0 is the dimensionless electron transfer integral corresponding to the

undistorted ligand environment. The linear vibronic coupling vanishes ( t q q 0  0 ) due  to the reason so far mentioned.

The transfer parameter t q  reaches the maximum value

t0 at q  0 and therefore the interaction with the out-of-phase mode in the system under consideration proves to be quadratic with respect to q coordinate. For the same reason the sign of the vibronic parameter is related to the sign of the transfer integral:   0   0 if

t0  0 t0  0 . In the basis of the localized orbitals ( A ,  B ) the vibronic coupling is off-diagonal (i.e. proportional to the  x matrix) as well as the electron transfer term. The total Hamiltonian of the MV dimer in the matrix form within the  A ,  B basis can be presented as:

1   2  Hˆ     q2  2    q  z  t0   q2  x  ,  q  2  





(2) 7

ACS Paragon Plus Environment


Page 8 of 30

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

where the first term represents the free harmonic vibration (in dimensionless units) and the remaining terms are the electronic coupling (proportional to t0) and the vibronic coupling. The diagonal vibronic coupling (proportional to the matrix  z ) arises from the interaction of the extra electron with the full-symmetric site vibrations qA and qB which form the q vibration (like in PKS model).

In Eq. (2)   V

M 3 is the dimensional vibronic coupling

parameter describing the local effect of the PKS mode. Neglecting the kinetic energy of the nuclear motion we arrive at the adiabatic potentials of the form:



1 U  q     q2   2 q2  t 0   q2 2



2

 .

(3)

We will focus of the lower branch of the adiabatic potential U  q  and corresponding adiabatic wave-function:

   q   cd

1 0 A dB

 q   A  cd

0 1 A dB

 q   B ,

(4)

The eigenvectors are found as: 2 d 1A  d B0

c

cd20  d1 A

B

t

 q  

0

  q2 

2

2  t0   q    q   2 q2   t0   q2    q   1  cd21 d 0  q  . 2 2 

A

2

, (5)

B

2 The analysis of the adiabatic potentials allows us to conclude that for  t0  2   1 and

  1 2 the lower branch of the adiabatic potential U  q  possesses two equivalent minima at the positions:  qmin  

 2  2 t0   2  4 t0  . 2 2 2 2 1  4 2

(6)

2 On the contrary, at  t0  2   1 and   1 2 the lower branch has the only minimum at 0 qmin  0 . It is seen from Eq. (5) that in this minimum the electronic density is equally

distributed between two centers (fully delocalized minimum), while in the minima located at  the extra electron is presumably localized at one of the two sites. qmin

8 ACS Paragon Plus Environment

Page 9 of 30


1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

These results allow us to reformulate the conventional conditions for the localization degree in the Robin and Day classification scheme as applied to the corner shared systems (although qualitatively they are applicable to more complicated systems). Conventionally, within the PKS model the system is considered as delocalized provided that the vibronic 2 coupling is weak enough and/or the transfer is strong which can be expressed as  t0  1 . 2 The condition  t0  2   1 ,   1 2 for the existence of the only minimum in the lower

sheet of the adiabatic potential represents a modified criterion for the assignment of a MV dimer to fully delocalized class III of MV systems according to the Robin and Day 2 classification scheme. Similarly, the condition  t0  2   1,   1 2 under which the

lower sheet possesses two minima, serves as an indication that the system belongs to classes I or II (localized or partially delocalized MV systems). We thus conclude that the inclusion of the influence of the PKS vibrations on the electron transfer mediated by the bridging ligands results in the modification of the criteria underlying the Robin and Day classification of MV compounds. Finally, one can observe instability with respect to PKS distortion providing   1 2 . The instability criterion is independent of both vibronic coupling constant  and transfer



integral t0 . Using the molecular orbitals    1



2  A   B  as the basis and setting   0

and t0  0 one can represent the Hamiltonian, Eq. (2) as follows:

 1  2  1  2 2    2 Hˆ      q  2    q2 Z       q    2    

 ~2 2   q   ~2    q    0

    1  2  ~ 2  2   q  2  2  q~   0

(7)

In deriving this Hamiltonian it was taken into account that the quadratic vibronic coupling ~ and to the rescaled vibrational coordinates q~ leads to the new effective frequencies    corresponding to the two molecular orbitals   and   :

    1  2 , q  1  2 

14

q

(8)



Page 10 of 30

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

~ ~  The instability criterion   1 2 means that for   0   0 the effective frequency    becomes imaginary. The combined effect of PKS vibration is illustrated in Fig. 2. The adiabatic potential U  q  has two equivalent minima in the case of   0 which corresponds 2 to conventional PKS model in accordance with the standard criteria  t0  1. The inclusion

of the intercenter vibronic coupling

significantly changes the character of the adiabatic

potential and the electronic density distribution in its minima. 2 Thus, for moderate values of  satisfying condition 1   t0 /  2 (cases of   2.5,

and   3 provided that t0  5 ) the increase of  tends to approaching of two minima





  located at qmin (Fig. 2a) with simultaneous decrease of the barrier  b  U  qmin   U  0

 

separating these minima. This is accompanied by the decrease of the difference between the probabilities to find the extra electron on the sites A and B in the minima, for example in the   right minimum  A qmin   cd21 d 0 qmin  is decreased with the increase of  A

B

(Fig. 2c) while

    1   A qmin  is increased.  B qmin

Finally, at some critical value of  (that is bigger for bigger values of  ) the two minima merge and the double-well adiabatic potential is transformed into the single-well one, 0 with the occupation probabilities (electronic densities) in the only minimum located at qmin 0

0 0 being equally distributed between the two sites (  A qmin    B qmin   0.5 ). These features

of the adiabatic potentials arise from the competition between the intracenter

vibronic

coupling that tends to distort the molecule in order to obtain a gain in the crystal field stabilization energy and the intercenter vibronic coupling that precludes the dimer from being distorted because in high-symmetric case q  0 the electron transfer stabilization energy reaches its maximal value. An additional illustration of the evolution the adiabatic potential

U  q  from the double-well to the single-well shaped is provided by Fig. 3 showing U  q  calculated at   3, t0  5 and different values of  . It is also seen that for values of 


Page 11 of 30


1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

5  qmin

b

4

  3.5

3

1.0

 3

0.5

  2 .5

1

  3 .5

1.5

 3

2

2.0

  2 .5 0.0

0 0.0

0.2

0.1



0.3

0.4

0.5

0.0

0.1

0.2





0.3

0.4

0.5

(b)

(a)

(c)  of the right minimum Fig. 2.  - dependences (calculated at t0  5 ) of the position qmin of U  q  (a); barrier  b separating two minima (b); occupation probability   A  qmin   cd2

1 0 A dB

 q  on the site A in the right minimum (c).  min

exceeding the borderline  =1/2 for stability/instability (cases of   0.54 and 0.8 in Fig. 3) instead of the infinite parabolic potential well we find that U  q  increases with the increase of q in the proximity of the central minimum, passes through the maximum and then infinitely decreases. This indicates that it is energetically preferable for the MV molecule to be decomposed. 2 Providing strong on-site type interaction when  t0 /  2 (case of   3.5 in Fig. 2  ,  b and occupation probabilities are more complicated. and Fig. 4) the  -dependences of qmin

Thus, it follows from Fig. 2 and Fig. 4 that at relatively weak intercenter interaction the distances between the minima and the barrier are slightly decreasing with the increase of  . This is accompanied by the change of the electronic density distribution in the minima towards stronger localization, i.e. the asymmetry of the electronic density distribution between the two sites is increased. Such type of behavior can be assigned to the fact that in case 11 ACS Paragon Plus Environment


Page 12 of 30

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

 2 t0 /  2 the  - type interaction dominates, and the system has to be distorted along PKS coordinate, despite the loss in electron transfer stabilization energy. This loss in transfer splitting energy is bigger for stronger intercenter coupling (  ) and so the system is getting more localized even in spite of the fact that the energy barrier is diminishing. At large  (close to the stability/instability borderline) the minima start to move apart and the barrier starts to increase. In spite of that the extra electron in the minima becomes more delocalized

Fig. 3. Lower sheet U  q  of the adiabatic potential calculated at   3, t0  5 and different values of  (curves marked).

Fig. 4. Lower sheet U  q  of the adiabatic potential calculated at   3.5, t0  5 and different values of  (curves marked).


Page 13 of 30


1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

because in this case the classically allowed region becomes wider (compare cases of   0.45, and 0.49 in Fig. 4). Finally at   1 2 (case of   0.51 in Fig. 4) we arrive at the situation when U  q  exhibits instability. 3. Modified PKS model for one-electron mixed valence dimer: the case of a long bridge

In this Section we consider a dimer in which the metal ions are connected by a long bridge and therefore it is assumed that there are no common bridging ligands belonging to the first coordination spheres of the ions A and B. Since the in-phase vibration q (as well as q-) can modulate the electron transfer occurring via the bridge this vibration contributes also to the overall vibronic problem along with the out-of-phase vibration. The two active PKS coordinates are illustrated in Fig. 5. It is assumed that both in-phase coordinate q and out-ofphase coordinate q are counted from the high-symmetric (non-distorted) ligand configuration

q0 corresponding to the maximal value t0 of the electron transfer parameter. The total Hamiltonian this case has the form: 1  2 Hˆ      q2  2 q  2   ,  

  2 2    q  z   t0    q    q   x  .  

(9)

It can be shown that the two vibrations involved in the model have the same frequencies which coincide with the frequency of the local breathing mode,      .

In Eq. (9) the

interaction with the q mode ( parameter  ) is eliminated by the shift of the reference point because it is proportional to the unit matrix. The dependence of the transfer integral from the vibrational coordinate q and q- which can be referred to as the non-Condon effect can be estimated by the ab initio calculation. A similar dependence of the transfer parameters on the metal-metal distance was studied by Piepho in Refs

28,29

in which it was shown that the

relative contributions of the breathing mode and the metal-metal displacements depends on the degree of covalence. The increase of covalence was shown to decrease the contribution of the breathing modes in favor of the contribution of the metal-metal vibrations. It is to be noted that the non-Condon effect for the Creutz-Taube ion was shown to be small.59 From Eq. (9) one can find the following expression for the adiabatic surfaces:



Page 14 of 30

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

U   q , q  

2  2 q  q2     2 q2   t0    q2    q2  .  2

(10)

One can show that the minima of the lower adiabatic surface U  q , q  are located at q  0 and the same coordinates q as in the case when only q mode is operative. This is illustrated in Fig. 6 for a particular case   =   when in-site type coupling with the q mode is relatively strong and for this reason all minima are located along q axis. We obtain the same regularities for the positions of the minima, height of the barrier, electronic densities in the minima and stability criteria with respect to out-of-phase distortion as those shown in Figs. 24. The section U  q , q  0 at    0 it represents a parabola with the minimum at q  0 .

Fig. 5. Out-of-phase and in-phase PKS-vibrations in MV dimer with a long bridge Providing    1 2 the curve U  q , q  0 demonstrates a monotonic growth with the increase of q , and the same is evidently true for any sections of the adiabatic potential

U  q , q  . This means that in this case the system is stable with respect to the in-phase distortion because it is confined in a “canyon” with infinitely high walls. On the contrary 14 ACS Paragon Plus Environment

Page 15 of 30


1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

when    1 2 the growth of the function U  q , q  0 with q changes to the decrease when q passes through the breaking point.

(a)

(b)

(c)

(d)

Fig. 6. Lower adiabatic surface U  q , q  calculated at   3.5, t0  5 and the following equal values of   and   :       0 (a);       0.3 (b);       0.45 (c);

      0.53 (d). This is also valid for any vertical section of U  q , q  but in this case q passes through the maximum (see Fig. 6). Thus at    1 2 the system is instable with respect to q . Note that in case      (Fig. 6) the system is simultaneously stable (or unstable) with respect of both distortions (along q and q ). The situation can be different when      (Fig. 7). Thus



Page 16 of 30

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

for

   1 2 ,    1 2 (Fig. 7a) the vertical section U  q  0, q  represents a typical

double- well adiabatic potential (e.g., stability with respect to q ), but the system is unstable

(b)

(a)

Fig. 7. Lower adiabatic surface U  q , q  calculated at   3.5, t0  5 and the following non-equal values of   and   :    0.53,    0.3 (a);    0.3,    0.53 (b).

due to instability along q (“canyon” with finite walls). On the contrary, providing    1 2 ,

   1 2 (Fig. 7b) the system is instable due to q -instability (no minima along q ; continuous decrease with the increase of q ). Since we employ a semiempiric parametric model for a dimeric MV system it is worthwhile to summarize the parameters involved in the model. In the case of a short bridge the incorporation of the bridging ligand leads to a two-site one-mode model with three parameters (namely, electron transfer , vibronic coupling parameter and the frequency of the active vibration). In the case of long bridge the two-site two-mode model involves four parameters, namely, electron transfer , two vibronic parameters and the vibrational frequency which is common for the two active vibrations. The number of the dimensionless parameters so far used for presentation of the results is 2 and 3 correspondingly. 4. Modified PKS model for a two-electron valence disproportionated dimer In this section we consider MV compounds of the d2-d0 type with a two-electron difference in oxidation state. The majority of such MV systems of such type can be considered 16 ACS Paragon Plus Environment

Page 17 of 30


1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

as structural units of either pseudo-1D chains (Pt(II)Pt(IV), Pd(II)Pd(IV) etc.) or 3Dcompounds (Bi(III)Bi(V), Sb(III)Sb(V), etc.). Among these systems the most studied (both theoretically and experimentally

60-62

) are the Wolfram´s red salt (WRS) compounds which

represent a chloride bridged chain of Pt: [PtIIIL4] [PtIVL4Cl2] Cl4ˑ4H2O consisting of square planar and tetragonally distorted octahedral molecules. Different spectroscopic measurements 63-67

as well as conductivity measurements

68

have shown that WRS are Robin and Day class II

MV systems. Assuming weak coupling between Pt(II)-Pt(IV) “unit cells” a single “unit cell” approach was used in

69-71

for the theoretical description of the absorption, resonance Raman

and luminescence spectra of WRS. This description was based on the PKS model generalized to the case of two-center two-electron problem. As to the isolated MV d2-d0 clusters, such systems are relatively scarce. As examples one can mention Ru2+-Ru0 and Rh2+-Rh0 dimers showing distinct intervalence transfer bands

72,73

arising from the d2-d0 d1-d1 transition as

well as dinitrogen-bridged Fe2+-Fe0 complex. 74 In the approach developed in

69-71

the symmetric stretching mode Cl- PtIV-Cl in

Pt(II)Pt(IV) dimers played the role of breathing local modes, which were used to compose the out-of-phase molecular vibration q as illustrated in Fig. 8. A

B

Pt(II)

Cl

Cl

Cl

Pt(IV)

Pt(III)

Cl

Pt(IV)

q  0

Cl

Pt(II)

Cl

Cl

Pt(III)

Cl

Cl

q  0 q  0

Fig. 8. Out-of-phase vibration in Pt(II)-Pt(IV)-dimer

Using as a basis the states   d A2  d B0  ,   d 1A  d B1  and   d A0  d B2  we can present the matrix of the Hamiltonian of d 2  d 0 -dimer as follows:



Page 18 of 30

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

  d A2  d B0 

  d 1A  d B1 

 q

       2  Hˆ   q  2     q  2    2

  d A0  d B2 

t0   q2

t0   q

0 t0   q

u

2 

2 

t0   q2

0

 q

  ,  

(11)

where the difference in the frequencies of out-of-phase vibration for d 2  d 0 and d 1  d 1 configurations is neglected. It is also assumed that the two-electron transfer matrix element connecting d A2  d B0 and d A0  d B2 configurations are negligibly small as compared with the one-electron transfer integral t0 mixing these configurations with the d 1A  d B1 - one. To find the stability criterion we will set   u  t0  0 and transform the matrix of the Hamiltonian

  d A2  d B0    d 1A  d B1    d A0  d B2   Hˆ   2

 0  2 2   2  q  2      q q      0 

 q2 0

0  q2

 q2

0

  ,  

(12)

to the diagonal form. This can be done by passing to the following new basis: 1  1   2   d 1A  d B1    d A2  d B0    d A0  d B2   ,

2 1  2   2   d 1A  d B1    d A2  d B0    d A0  d B2   , 2

3 

(13)

1   d A2  d B0    d A0  d B2   .  2

Then we find the following transformed Hamiltonian:

1  1  2  2    q1  2  q1   2   Hˆ   0    0  

2

(14)

3 0

0

2  2  2   q2  2  2  q2 

0

0

  2  2   q  2  2  q 

    .     

In Eq. (14) the following modified frequencies and rescaled normal coordinates are introduced: 18 ACS Paragon Plus Environment

Page 19 of 30


1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60



1  2   1   2 2  , q1  2  1

  

2 2



14

q .

From Eq. (15) one obtains the stability criterion   1 2 2  0.357 .

(15) It is seen that this

criterion is more rigorous than the criterion   1 2 found above for d1-d0-cluster. To illustrate the role of the bridge induced vibrational interaction (related to the parameter  ) we will fix the transfer parameter and the vibronic parameter  at t0  14 and   8.9 , respectively, and examine the evolution of the lower sheet of the adiabatic potential U gr q  as function of  for two different in-site Coulomb repulsion energies u  14.5 and u  34 .

For u  14.5 and vanishing bridge induced vibrational interaction the adiabatic potential U gr q  possesses two symmetric minima corresponding to presumable d A2  d B0 and d A0  d B2 localizations of the electronic pair (Fig. 9). Thus, in the right minimum the electronic

pair is mainly localized at the site A ( d A2  d B0 -configuration) and to much less extent it is localized at the site B ( d A0  d B2 ) and equally distributed between A and B sites ( d 1A  d B1 ). The dependences of the position of the minima, the height of the barrier between them and the electronic density distribution in the minima on the parameter   0 are shown in Figs. 9, 10. It is seen that these dependences are quite similar to those described above for d 1  d 0 -dimer providing strong  (see Fig. 4 and the case of   3.5 in Fig. 2). The only difference is that now instead of only two possible electronic distributions ( d 1A  d B0 and d A0  d B1 ) in case of oneelectron MV dimer the adiabatic wave-function in each minima is given as a superposition of three electronic distributions d A2  d B0 , d A0  d B2 and d 1A  d B1 and the increase (decrease) of the of the dominant d A2  d B0 -distribution probability in the right minimum is accompanied by simultaneous decrease (increase) of the d A0  d B2 and d 1A  d B1 -probabilities (Fig. 10). For

  0.357 the two symmetric minima disappear (instability), and simultaneously instead of maximum an excited shallow minimum appear at q  0 (see case of   0.4 in Fig. 9). In case of stronger in-site Coulomb repulsion u  34 the lower sheet of the adiabatic potential U gr q  at   0 possesses three minima: two ground symmetric minima in which the



Page 20 of 30

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

electronic pair is mainly localized on one or another site ( d A2  d B0 and d A0  d B2 configurations) and the central ( q  0 ) slightly excited d 1A  d B1 -type minimum (Fig. 11). It is to be noted that even providing strong Coulomb repulsion destabilizing the states in which the valences of the metals differ in two, the doublet state d A2  d B0 , d A0  d B2 is still the ground one while the singlet state d 1A  d B1 proves to be excited. This is evidently the result of strong vibronic stabilization caused by the local type interaction. The dependences positions of two symmetric minima and the electronic density distributions in these minima upon the parameter  are shown in Fig. 12. They are quite similar to those found for the case of weak

Fig. 9. Lower sheet U gr q  of the adiabatic potential calculated at t0  14, u  14.5 ,   8.9 and different values of  (marked curves).

(c)

(d)


Page 21 of 30


1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

(e)  Fig. 10.  -dependences (calculated at t0  14, u  14.5 and   8.9 ) of the position qmin of the

right minimum of U gr q  (a); barrier  b separating two minima (b); and occupation probabilities    cd22 d 0  qmin  (c); cd21 d1  qmin  (d) and cd20 d 2  qmin  (e) in the right minima. A

B

A

B

A

B

Fig. 11. Lower sheet U gr q  of the adiabatic potential calculated at t0  14, u  34 ,   8.9 and different values of  (marked curves)

(a)

(b)



Page 22 of 30

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

(c)

(d)

(e)  Fig. 12.  - dependences (calculated at t0  14, u  34 and   8.9 ) of the position qmin of the



 

 right minimum of U gr q  (a); difference in energies U gr 0  U g qmin  (b); and occupation    probabilities cd22 d 0  qmin  (c); cd21 d1  qmin  (d) and cd20 d 2  qmin  (e) in the right minima. A

B

A

B

A

B

Coulomb interaction shown in Figs. 9, 10. A significant difference is, however, is that at strong Coulomb repulsion the  - coupling can change the valence states in the dimer. Indeed, since the local (parameter  ) and bridge induced (  ) types interactions compete against each other, the stabilization of the charge (valence) disproportionated state d A2  d B0 , d A0 - d B2 is weakened as a result of  - coupling, and providing large enough  (cases of   0.05, 0.1 and 0.2 in Fig. 11) the central d 1A - d B1 -type minimum becomes the ground one. In the central minimum the occupation probabilities prove to be independent of  and for t0  14, u  34 ,

  8.9 theyare found to be cd22  d 0  0   cd20  d 2  0   0.097, A

B

A

B

cd21  d1  0   0.826 , that is, at A

B

q  0 the contribution of the d 1A  d B1 - dimer (equal valences) is dominating. The region of

 - values at which the central minimum proves to be the ground one is outlined by dashed 22 ACS Paragon Plus Environment

Page 23 of 30


1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

lines in Fig. 12 b. It is seen that in the left part of this region the stabilization of the energy in the central minimum U gr q  0  with respect to the energy in the two symmetric equivalent





 minima U gr q  qmin is an increasing function of  , then starting from   0.17 it starts to

decrease and at   0.27 the central minimum becomes an excited state again (case of   0.3 in Fig. 11). Finally, at   0.357 the two symmetric equivalent minima disappear giving rise to the vibronic instability (case of   0.37 in Fig. 12).

5. Concluding remarks

We have proposed that in addition to the change of the local crystal fields around the metal sites, the vibronic PKS vibrations produce a considerable effect on the indirect electron transfer matrix due to the involvement of the bridging ligands in these vibrations. This has been shown to significantly change the localization-delocalization properties of the MV clusters and modify the criteria for the assigning of MV systems to the Robin and Day classes. Finally, we have shown that the proposed extension of the PKS model is closely related to the problem of vibronic stability of MV compounds, and the stability criteria with respect to PKS vibrations have been found. The model so far developed can be applied to the wide class of MV compounds in which the bridging ligands are active in the electron transfer. It is to be noted that the model brings additional vibrational degrees of freedom (with respect to PKS model) and consequently an extension of the basis of the electron-vibrational states. Therefore one can expect that more computational resources are required when solving the dynamical vibronic problem for the high-nuclearity MV systems. These difficulties can be hopefully overcome by using the symmetry assisted approach. 75 Acknowledgements

A.P. and B.T. are grateful to COST Action CM1203 “Polyoxometalate Chemistry for molecular Nanoscience (PoCheMon)” for supporting this work. A.P. acknowledges the STCU (project N 5929), and the Supreme Council for Science and Technological Development of the Republic of Moldova (CSSDT project 15.817.02.06F) for financial support.

J.M.C-J thank

EU (ERC Advanced Grant SPINMOL), the Spanish MINECO (grant CTQ2014-52758-P), and the Generalidad Valenciana (Prometeo and ISIC Programmes of excellence) for the financial support of this work. 23 ACS Paragon Plus Environment


Page 24 of 30

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

References

1.

Creutz, C.; Taube, H. Direct Approach to Measuring the Franck-Condon Barrier to Electron Ttransfer Between Metal Ions. J. Am. Chem. Soc. 1969, 91, 3988-3989.

2.

Park, J. et al. Coulomb Blockade and the Kondo Effect in Single-Atom Ttransistors. Nature 2002, 417, 722-725.

3.

Liang, W.; Shores, M.P.; Bockrath, M.; Long, J.R.; Park, H. Kondo Resonance in a Single Molecule Transistor. Nature 2002, 417, 725-729.

4.

Soncini, A.; Mallah T.; Chibotaru, L.F. Molecular Spintronics in Mixed-Valence Magnetic

Dimers: the Double-Exchange Blockade Mechanism. J. Am. Chem. Soc.

2010, 132, 8106-8114. 5.

Palii, A.; Clemente-Juan, J.M.; Tsukerblat, B.; Coronado, E.; Electric Field Control of the Optical Properties in Magnetic Mixed-Valence Molecules. Chem. Sci. 2014, 5, 35983602.

6.

Bosch-Serrano, C.; Clemente-Juan, J. M.; Coronado, E.; Gaita-Ariño, A.; Palii, A.; Tsukerblat, B. Molecular Analog of Multiferroics: Electric and Magnetic Field Effects in Many-Electron Mixed- Valence Dimers. Phys. Rev. B 2012, 86, 024432-024443.

7.

Clemente-Juan, J.M.; Coronado, E.; Gaita-Ariño, A. Magnetic Polyoxometalates: From Molecular Magnetism to Molecular Spintronics and Quantum Computing. Chem. Soc. Rev. 2012, 41, 7464-7478.

8.

Miller, J. S.; Min, K. S. Oxidation Leading to Reduction: Redox-Induced Electron Transfer (RIET). Angew. Chem. Int. Ed. 2009, 48, 262-272.

9.

Tsukerblat, B.; Palii, A.; Clemente-Juan, J. M. Self-Trapping of Charge Polarized States in Four-Dot Molecular Quantum Cellular Automata: Bi-Electronic Tetrameric MixedValence Species. Pure Appl. Chem. 2015, 87(3), 271-282.

10.

Lent, C. S.; Tougaw, P.; Porod, W.; Bernstein, G. H. Quantum Cellular Automata. Nanotechnology 1993, 4, 49-57.

11.

Porod, W.; Lent, C. S.; Bernstein, G. H.; Orlov, A. O.; Amlani, I.; Snider, G. L.; Merz, J. L. Quantum-Dot Cellular Automata: Computing With Coupled Quantum Dots. Int. J. Electron. 1999, 86, 549-590.

12.

Lent, C. S. Molecular Electronics: Bypassing the Transistor Paradigm. Science 2000, 288, 1597-1599.

13.

Lent, C. S.; Isaksen, B.; Lieberman, M. Molecular Quantum-Dot Cellular Automata. J. Am. Chem. Soc. 2003, 125, 1056-1063. 24


Page 25 of 30


1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

14.

Malrieu, J. P.; Caballol, R.; Calzado, C.J.; De Graaf, C.;

Guihèry, N. Magnetic

Interactions in Molecules and Highly Correlated Materials: Physical Content, Analytical Derivation, and Rigorous Extraction of Magnetic Hamiltonians. Chem. Rev. 2014, 114(1), 429-492. 15.

Sutin, N. Theory of Electron Transfer Reactions: Insights and Hindsights. Prog. Inorg. Chem. 1983, 30, 441-498.

16.

Brunschwig, B.S.; Sutin, N. Energy Surfaces, Reorganization Energies, and Coupling Elements in Electron Transfer. Coord. Chem. Rev. 1999, 187, 233-254.

17.

Sutin, N. Electron Transfer Reactions in Solution: A Historical Perspective. Adv. Chem. Phys. 1999, 106, 7-33.

18.

Creutz, C.; Newton, M. D.; Sutin, N.

Matal-Ligand and Metal-Metal Coupling

Elements. J. Photochem. Photobiol. A: Chem. 1994, 82, 47-59. 19.

Hush, N.S. Intervalence-Transfer Absorption. Part 2. Theoretical Considerations and Spectroscopic Data. Prog. Inorg. Chem. 1967, 8, 391-444.

20.

Hush, N.S. Adiabatic Theory of Outer Sphere Electron-Transfer Reactions in Solution. Trans. Faraday Soc. 1961, 57, 557-580.

21.

Hush, N.S. Distance Dependence of Electron Transfer Rates. Coord. Chem. Rev. 1985, 64, 135-157.

22.

Demadis, K.D.; Hartshorn, C.M.; Meyer, T. J. The Localized-to-Delocalized Transition in Mixed-Valence Chemistry. Chem. Rev. 2001, 101, 2655-2686.

23.

Brunschwig, B.S.; Creutz, C.; Sutin, N. Optical Transitions of Symmetrical MixedValence Systems in the Class II-III Transition Regime. Chem. Soc. Rev. 2002, 31, 168184.

24.

Piepho, S. B.; Krausz, E. R.; Schatz, P.N. Vibronic Coupling for Calculation of MixedValence Absorption Profiles. J. Am. Chem. Soc. 1978, 100, 2996-3005.

25.

Prassides, K.; Schatz, P. N. Vibronic Coupling Model for Mixed-Valence Compounds. Extension to the Multimode Case. J. Phys. Chem. 1989, 93, 83-89.

26.

Wong, K.Y. Charge-Transfer-Induced IR Absorptions in Mixed-Valence Compounds. Inorg. Chem 1984, 23, 1285-1290.

27.

Wong, K. Y.; Schatz, P. N. A Dynamic Model for Mixed-Valence Compounds. Prog. Inorg. Chem. 1981, 28, 369-449.

28.

Piepho, S. B. Vibronic Coupling Model for the Calculation of Mixed-Valence Line Shapes: A New Look at the Creutz-Taube Ion. J. Am. Chem. Soc. 1990, 112, 4197-4206. 25



Page 26 of 30

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

29.

Piepho, S.B. Vibronic Coupling for Calculations of Mixed-Valence Line-Shapes-the Interdependence of Vibronic and MO Effects. J. Am. Chem. Soc. 1988, 110, 6319-6326.

30.

Hush, N. S. Parameters of Electron-Transfer Reactions. ACS Symp. Ser. 1982, 198, 301332.

31.

Reimers, J. R.; Hush, N.S. The Effects of Couplings to Symmetric and Antisymmetric Modes and Minor Asymmetry on the Spectral Properties of Mixed-Valence and Related Charge-Transfer Systems. Chem. Phys. 1996, 208, 177-193.

32.

Ondrechen, M. J.; Ko, J. ; Zhang, L.-T. A Model for the Optical Absorption Spectrum of (m-Pyrazine) Decaamminediruthenium (5+): What Hath Creutz and Taube Wrought? J. Am. Chem. Soc. 1987, 109, 1672-1676.

33.

Ferretti, A.; Lami, A.; Ondrechen, M. J.; Villani, G. The Role of Vibronic Coupling and Correlation Effects on the Optical Properties of Mixed-Valent and Monovalent Dimer Compounds: The Creutz-Taube Ion and its Monovalent Analogues. J. Phys. Chem. 1995, 99, 10484-10491.

34.

Ondrechen, M. J.; Ratner, M. A. Intramolecular Electron Transfer in Simple Model Systems. A Propagator Study. J. Chem. Phys. 1977, 66, 938-946.

35.

Root, L. J.; Ondrechen, M. J. Through-Bridge Electron Transfer: A Propagator Study of a Simple Three-Site Model. Chem. Phys. Lett. 1982, 88, 538-542.

36.

Root, L. J.; Ondrechen, M. J. Adiabatic Potentials for a Bridged Three-Site ElectronTransfer System. Chem. Phys. Lett. 1982, 93, 421-424.

37.

Ondrechen, M. J.; Ko, J.; Root, L. J. Two-Dimensional Potential Surfaces for Bridged Mixed-Valence Dimers. J. Phys. Chem. 1984, 88, 5919-5923.

38.

Ko, J.; Ondrechen, M. J. A Model for the Intervalence Transfer Band Profile of Bridged Mixed-Valence Dimers. Chem. Phys. Lett. 1984, 112, 507-512.

39.

Ko, J.; Ondrechen, M. J. Line Shape of the Intervalence Transfer Band in Bridged Mixed-Valence Dimers: The Delocalized Case. J. Am. Chem. Soc. 1985, 107, 61616167.

40.

Ko, J.; Zhang, L.-T.; Ondrechen, M. J. The Creutz-Taube Ion: A Model for the EPR g Tensor which Includes the Bridging Ligand. J. Am. Chem. Soc. 1986, 108, 1712-1713.

41.

Zhang, L.-T; Ko, J.; Ondrechen, M. J. Electronic Structure of the Creutz-Taube Ion. J. Am. Chem. Soc. 1987, 109, 1666-1671.


Page 27 of 30


1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

42.

Zhang, L.-T; Ko, J.; Ondrechen, M. J. A Non-Adiabatic Quantum Mechanical Treatment of the Absorption Lineshape of Bridged Mixed-Valence Dimers. J. Phys. Chem. 1989, 93, 3030-3034.

43.

Ondrechen, M. J.; Gozashti, S.;

Zhang, L.-T.; Zhou, F. Bridged Mixed-Valence

Systems: How Polarizable Bridging Ligands Can Lead to Interesting Spectroscopic and Conductive Properties. ACS Symposium Series 1990, 226, 225-235. 44.

Zhang, L.-T.; Ondrechen, M. J. The Electronic Structure of Pentaammine (Pyrazine) Ruthenium (II) and (III): The Metal-Ligand Pi Conjugation and Its Implications in Electron Transfer.Inorg. Chim. Acta 1994, 226, 43-51.

45.

Ferretti, A.; Lami, A.; Ondrechen, M. J.; Villani, G. The Role of Vibronic Coupling and Correlation Effects on the Optical Properties of Mixed-Valent and Monovalent Dimer Compounds: The Creutz-Taube Ion and Its Monovalent Analogues. J. Phys. Chem. 1995, 99, 10484-10491.

46.

Murga, L. F.; Ondrechen, M. J. Potential Energy Surfaces for a Mixed-Valence Dimer in an Applied Electric Field. Theor. Chim. Acta 1995, 90, 331-339.

47.

Londergan, C.H.; Kubiak, C.P. Vibronic Participation of the Bridging Ligand in Electron Transfer and Delocalization: New Application of a Three-State Model in PyrazineBridged Mixed-Valence Complexes of Trinuclear Ruthenium Cluster. J. Phys. Chem. A 2003, 107, 9301-9311.

48.

Robin, M. B; Day, P. Mixed-Valence Chemistry: a Survey and Classification. Adv. Inorg. Chem. Radiochem. 1967, 10, 247-422.

49.

Burdett, J. K. A Molecular Orbital Approach to Electron-Transfer Reactions Between Transition-Metal Ions in Solution Inorg. Chem. 1978, 17, 2537-2552.

50.

Robert, V.; Borshch, S. A.; Bigot, B. MO Model of Electron Delocalization in MixedValence Bioctahedral Clusters. Chem. Phys. Lett. 1995, 236, 491-496.

51.

Solomon, E. I.; Xie, X.; Dey, A. Mixed Valent Sites in Biological Electron Transfer. Chem. Soc. Rev. 2008, 37, 623-638.

52.

Meyer, K.; Bill, E.; Mienert, B.; Weyhermüller, T.; Wieghardt, K. Photolysis of Cis- and Trans-[FeIII(cyclam)(N3)2]+

Complexes:

Spectroscopic

Characterization

of

a

Nitridoiron(V) Species. J. Am. Chem. Soc. 1999, 121, 4859-4876. 53.

Guihéry, N. The Double Exchange Phenomenon Revisited: the [Re2OCl10]3− Compound. Theor. Chem. Acc. 2006, 116, 576–586.



Page 28 of 30

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

54.

Sadakane M.; Tsukuma D.; Dickman M. H.; Bassil B.S.; Kortz U.; Capron M.; Ueda W., Dimerization of Mono-Ruthenium Substituted a-Keggin-Type Tungstosilicate [aSiW11O39RuIII(H2O)]5− to l-Oxo-Bridged Dimer in Aqueous Solution: Synthesis, Structure, and Redox Studies, Dalton Trans. 2007, 2833-2838.

55.

Englman, R. The Jahn-Teller Effect in Molecules and Crystals, Wiley, London, 1972.

56.

Bersuker, I. B.; Polinger, V. Z. Vibronic Interactions in Molecules and Crystals, Springer-Verlag, Berlin, 1989.

57.

Bersuker, I. B. The Jahn-Teller Effect, Cambridge University Press, 2006.

58.

Bersuker, I. B. Pseudo-Jahn−Teller Effect - A Two-State Paradigm in Formation, Deformation, and Transformation of Molecular Systems and Solids. Chem. Rev. 2013, 113, 1351-1390.

59.

The smallness of the non-Condon effect for the Creutz-Taube ion was indicated by the Reviewer in his comment.

60.

Seth, P.N.A.; Underhill, A.E.; Watkins, D.M. Studies of the Mixed Valence Complexes Cs[PtBr3(NO2)(NH3)] and Cs[PtI3(NO2)(NH3)]. J. Inorg. Nucl. Chem. 1980, 42, 11511154.

61.

Wong, K.Y.; Schatz, P.N. Resonance Raman Calculations in Mixed-Valence Systems. Wolfram’s Red and the Creutz-Taube Complex. Chem. Phys. Lett. 1980, 73, 456-460.

62.

Nasu, K.; Toyozawa, Y. Charge Transfer Instability with Structural Change. II. Mean Field Theory for One-Dimensional N-Sites N-Electrons System. J. Phys. Soc. Jpn. 1982, 51, 2098-2106.

63.

Clark, R. J. H.; Franks, M. L.; Trumble, W. R. Resonance Raman Spectra and Excitation Profiles of the Mixed Valence Compound Wolffram’s Red [PtII(C2H5NH2)4] [PtIV(C2H5NH2)4Cl2]Cl4 4H2O. Chem. Phys. Lett. 1976, 41, 287-292.

64.

Clark, R. J. H.; Turtle, P. C. Electronic and Resonance Raman Spectra of OneDimensional Mixed-Valence Platinum-Ethylamine Complexes at Ca. 80 K. Inorg. Chem. 1978, 17, 2526-2531.

65.

Day, P. Mixed Valence Chemistry and Metal Chain Compounds. In: Low-Dimensional Cooperative Phenomena, H. Keller, Ed., Plenum, New York, 1975, pp. 191-214.

66.

Tanino, H.; Kobayashi, K. Relaxation of Electron-Phonon System in Optically Excited Quasi-1-D Mixed Valence Crystal Wolffram’s Red Salt. J. Phys. Soc. Jpn. 1983, 52, 1446-1456.


Page 29 of 30


1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

67.

Cahen, D.; Lester, J. E. Mixed and Partial Oxidation States. Photoelectron Spectroscopic Evidence. Chem. Phys. Lett., 1973, 18,108-111.

68.

Interrante, L. V.; Browall, K.W.; Bundy, F. B. Intermolecular Interactions in Transition Metal Complexes. IV. High Pressure Study of Some Mixed-Valence Platinum and Palladium Complexes. Inorg. Chem. 1974, 13, 1158-1162.

69.

Prassides, K.; Day, P. Temperature Variation of the Intervalence Absorption Band of Hexachloroantimonate(III,V) Ions in a Crystal Lattice. J. Chem. Soc., Faraday Trans. II 1984, 80, 85-95.

70.

Prassides, K.;

Schatz, P.N.; Wong, K.Y.; Day, P.

Vibronlc Coupling Model for

Mixed-Valence Compounds. Extension to Two-Site Two-Electron Systems. J. Phys. Chem. 1986, 90, 5588-5597. 71.

Prassides, K.; Schatz, P. N. Optical Properties of the Prototype II–IV Mixed Valence System Wolffram's Red Salt: a One Mode Treatment. Chem Phys. Lett. 1991, 178, 227-234.

72.

Voegeli,

R.

H.;

Kang,

H.

C.;

Finke,

R.

hexamethylbenzene)(ƞ4,ƞ6-[24](1,2,4,5)cyclophane)

G.; -

Boekelheide,

V.

diruthenium(0,II)

Bis(ƞ6Bis(

tetrafluoroborate). A Novel Mixed-Valence Ion (Class 11) Exhibiting a Net TwoElectron Intervalence Transfer J. Am. Chem. Soc. 1986, 108, 7010-7016. 73.

Dulebohn, J. I.; Ward, D. L.; Nocera, D. G. Luminescence from a Novel Mixed-Valence Dirhodium Fluorophosphine Complex. J. Am. Chem. Soc. 1988, 110, 4054-4056.

74.

Field, L.D.; Guest, R.W.; Turner, P. Mixed-Valence Dinitrogen-Bridged Fe(0)/Fe(II) Complex. Inorg. Chem. 2010, 49, 9086–9093.

75.

Tsukerblat, B.; Palii, A.;

Clemente-Juan, J. M.; Gaita-Ariño, A.; Coronado, E. A

Symmetry Adapted Approach to the Dynamic Jahn-Teller Problem: Application to Mixed-Valence Polyoxometalate Clusters with Keggin Structure Int. J. Quantum Chemistry, 2012, 112, 2957-2964.



Page 30 of 30

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Graphical Abstract

Here we describe a new vibronic model of mixed valence (MV) dimer inspirited by the conventional Piepho, Krausz and Schatz (PKS) approach. We attempted to partially lift the main restriction of the PKS model dealing with the vibronically independent moieties of a MV molecule. The refined version of PKS model in which the bridging ligands are included deals with the three main interactions: electron transfer (integral t0) related to the high-symmetric ligand configuration, on-site vibronic coupling (parameter  ) arising from the modulation of the crystal field on the metal sites by the breathing displacements of their nearest ligand surroundings, and inter-center vibronic coupling (parameter  ) describing the dependence of the electron transfer on ligand positions in course of their breathing movement. We apply the modified model to the analysis of the adiabatic potentials and electronic density distributions in the minima of their lower sheets for the cases of one-electron MV dimer with long and short bridges and for the two-electron MV dimer exhibiting valence disproportionation effect. The inclusion of the inter-center interaction in addition to the conventional PKS coupling is shown to produce a strong effect on the degree of localization in MV dimers, and, in particular, on the assignments to the Robin and Day classes, and on the conditions of stabilization of valence disproportionated states in the bielectron transfer systems. 30 ACS Paragon Plus Environment

LocalizationâDelocalization in Bridged Mixed-Valence Metal Clusters

Recommend Documents

LocalizationâDelocalization in Bridged Mixed-Valence Metal Clusters