6788
J. Phys. Chem. B 2007, 111, 6788-6797
Three-Centers Models for Electron Transfer through a Bridge. 1. Potential Energy Surfaces† Jean-Pierre Launay,* Christophe Coudret, and Cedric Hortholary CEMES-CNRS, 29 rue Jeanne MarVig, 31055 Toulouse CEDEX, France ReceiVed: January 2, 2007; In Final Form: March 5, 2007
Three centers models adapted to the description of electron transfer through a bridge are discussed, with a special emphasis on potential energy surfaces. A short historical review of the available models is given, with a particular interest on the Bersuker-Borshch-Chibotaru model (1989) and the Lambert-No¨ll-Schelter model (2002). We propose our own model, inspired from the Bersuker-Borshch-Chibotaru model, but with a more physical discussion of the parameters and coordinates. The diabatic surfaces, before the intervention of electronic couplings between external site and bridge, consist of three revolution paraboloids of equal radii. The bottoms of the paraboloids do not form in general an equilateral triangle; they form an isosceles one. At this stage, the basic parameters are the ones describing the position of the third paraboloid (corresponding to a redox process on the bridge) with respect to the other two. We define in particular an energy shift parameter (∆) and a depth parameter (d), the latter corresponding to the position of this paraboloid in the third dimension, i.e., along a coordinate of reaction perpendicular to the usual reaction coordinate. The topology of diabatic and adiabatic surfaces is discussed. As an application, we explain the contrasted behavior of two mixed valence systems bridged by anthracene and dimethoxybenzodithiophene, which differ by the value of the d parameter.
Introduction The experimental and theoretical aspects of electron transfer in complex systems containing more than two sites have led to a considerable amount of results.1,2 The variety of systems is very large because they can be made of different moieties pertaining to coordination chemistry, organometallic chemistry, or even organic chemistry.2 The case of two-state systems is now broadly understood, thanks to the simple representation introduced in the 1960s by Marcus, Hush, Sutin, and others, based on potential energy curves having the simple shape of parabolas.3 This representation was found extremely efficient as it allows a qualitative (and frequently quantitative) description based on the simple geometrical properties of these curves and their relative disposition. It is however a trivial statement to say that interesting systems are those bearing more than just two redox centers. Many systems containing three, four, etc. sites are known, but even when only two redox sites are identified, there is frequently a bridging ligand connecting them.4 The efficiency of the bridging ligand in mediating the electronic interaction is related to its ability to be oxidized or reduced at a potential not too far from the one of the redox sites. Then, the energy levels of the different subunits can mix, and the electronic interaction is increased by the super-exchange effect. Thus, in some respect, even in the relatively simple metal-ligand-metal case, the bridging ligand plays the role of a third redox site. Then, as soon as three sites are present, in an A-B-C geometry (two external sites and a central one), a frequently evoked problem is the possible coexistence of two types of mechanisms: a one-step mechanism with direct electron transfer between the external sites (super exchange) and a two-step mechanism made of two consecutive electron transfers (hopping). The coexistence of theses two †
Part of the special issue “Norman Sutin Festschrift”.
mechanisms and the need for a unified description are the subject of much current interest.5 Three-centers models began to appear in the 1980s following the particularly pedagogical approach of the PKS model for twocenter systems.6 This model gave a simple interpretation of the nuclear coordinate of reaction, as the out-of-phase combination of breathing motions on each subunits. Because in three-center systems there is an additional degree of freedom, the logical way to treat these systems was to introduce a second nuclear coordinate and establish potential energy surfaces, i.e., the threedimensional analogues of the famous Marcus-Hush parabolas. In this first paper of a series, we come back to the way of putting into equations the description of a three-center system. What we call a three-center system can encompass a wide variety of chemical structures, as shown in Figure 1. It can be a system with three metal sites, with for instance ruthenium atoms linked by a bridge (Figure 1a), in which there is some indirect (through-bridge) electronic interaction between the central metal atom and the outer ones.7 The role of the bridge is not explicitly considered, but the central ruthenium atom could play a role in the interaction between the outer ones. Another large class of systems is made of bimetallic systems bridged by a ligand that is “non-innocent”, i.e., can be easily oxidized or reduced (Figure 1b) at about the same potential as the metal. In such a system, the question arises to know if the oxidation or reduction bears on the outer metals or the bridge,8 so that again three sites have to be considered. Finally, there is the case of purely organic systems (Figure 1c) where the outer units and the bridge are all organic moieties.2 We symbolize the system under consideration of the present paper by the general topology of Figure 2 with two external sites (labeled 1 and 3 or A and C) and a central, bridging one (labeled 2 or B). We consider the case where the two external sites are chemically equivalent.
10.1021/jp070018x CCC: $37.00 © 2007 American Chemical Society Published on Web 04/19/2007
Three-Centers Models for Electron Transfer (1)
J. Phys. Chem. B, Vol. 111, No. 24, 2007 6789 When the oxidation state is varied, one assumes that the main change bears on the equilibrium distances, rather than on the force constants. There are two ways to introduce this change in eq 1: (i) adding a linear term of the form lQ, where l is called an electron-phonon coupling constant, and also an energy shift term ∆. This method comes from the solid-state literature.9 Equation 1 then becomes
E ) (1/2)kQ2 + lQ + ∆
(2)
It is easy to show that the minimum in E occurs now for Q ) -l/k and its value is
Emin ) -(l2/2k) + ∆
(3)
or (ii) shifting the curve along the Q-axis and also adding an energy shift term according to
E ) (1/2)k(Q - Q°)2 + ∆′
(4)
The minimum then occurs for Q ) Q°, and its value is directly ∆′. In eq 4, k has the dimension of a force constant and Q of a displacement (length change). A further improvement of this method is to use adimensional coordinates q and write the energy term as λ(q - q°)2, where λ has the dimension of an energy. Let us compare the two approaches. Expanding eq 4 gives Figure 1. Examples of three-center systems: (a) trimetallic complex with cyanide bridges,7b (b) bimetallic complex bridged by a noninnocent ligand,8 (c) purely organic system.2e
E ) (1/2)kQ2 - kQ°Q + (1/2)kQ°2 + ∆′ and comparing with eq 2 shows that
l ) -kQ° and ∆ ) (1/2)kQ°2 + ∆′
Figure 2. General structure of the trinuclear mixed valence systems considered in this paper. The three units can be labeled either as 1-2-3 or A-B-C. In the present paper, we consider the case where the two external sites are chemically equivalent.
We consider here only electron delocalization effects, not magnetic couplings; i.e., we envisage systems where the subunits have a simple electronic structure with either an extra electron outside a closed shell or conversely an electronic “hole”. After a short overview of the literature, we propose a simple description of diabatic potential energy surfaces in a localized basis. We will show that the influence of the third site can be characterized by just two parameters: a vertical energy shift, ∆, and a “depth” parameter, d, the latter determining the lateral displacement of the third paraboloid necessary for the description of the system. Then, introduction of the electronic interaction gives a complete picture of the system. Preliminary Considerations on Diabatic Curves and Surfaces Potential energy curves are usually written as quadratic functions of a displacement coordinate (Q) corresponding to the elongation/compression of a bond (harmonic oscillator approximation). Through a suitable definition of origins, the energy of a system for a given oxidation state is thus written as
E ) (1/2)kQ2 i.e., a parabola with a minimum in Q ) 0, E ) 0.
(1)
(5a)
(5b)
Using one or the other approach is thus a matter of convenience. Note, however, that, in the linear term approach (based on lQ), the l term influences also the vertical shift between the bottoms of parabolas (see eqs 3 and 5b); this is not the case in the shift approach based on a (Q - Q°) term. In the linear term approach, the fact that the minimum in E depends on l is not a problem when treating a system with equivalent sites, because one just has to perform a change in the origin of energies, but it introduces complications when dealing with systems containing unequivalent sites. A Short Review of the Different Models Models of trinuclear systems can be broadly divided in two categories: cyclic systems with 3-fold symmetry and systems with 2-fold symmetry of the type depicted in Figure 1. The present paper is devoted to the second category, but a short reminder of the cyclic systems is necessary for historical reasons. Table 1 summarizes the basic features of the models for systems of 3-fold symmetry. A model for a trinuclear system with 3-fold symmetry was published in 1982 by Launay and Babonneau.10 The general equations were written in the same spirit as in the PKS model. The 3-fold symmetry allowed a unique combination of vibrational coordinates into three symmetry combinations. The totally symmetrical one could be eliminated, and finally, potential energy surfaces were obtained by plotting energy as a function of the other two combinations. At about the same time, a model by Borshch, Kotov, and Bersuker yielded essentially the same results,11 and shortly after, an analogous treatment was also given by Cannon et al.12 All these models were based on a 3 × 3 energy matrix corresponding to the electronic interaction
6790 J. Phys. Chem. B, Vol. 111, No. 24, 2007
Launay et al.
TABLE 1: Overview of the Treatments for a System with 3-Fold Symmetry (Equilateral Triangle)a authors, date
equations
e-n couplings
electronic couplings
Q coordinates
electronic basis
mathematical treatment
main results
localized
numerical computation of the matrix eigenvalues
potential energy surfaces analytical solutions for slice cuts potential energy surfaces, slices
Launay, Babonneau,10 1982
PKS type
lQ term
1-2, 2-3, and 1-3
Borshch, Kotov, Bersuker,11 1982 Cannon,12 1984
Jahn-Teller analysis
VQ term
1-2, 2-3, and 1-3
polar transformation in r and θ
localized
expression of secular eq
PKS type
aQ term
1-2, 2-3, and 1-3
polar transformation in r and θ
localized
analytical solution of third degree eq
a
remarks role of sign of the electronic coupling
Sites are denoted 1, 2, and 3.
TABLE 2: Overview of the Treatments for a System with 2-Fold Symmetry (Isosceles Triangle with Two External Sites 1 and 3, and a Bridging Site 2) authors, date Launay,13
equations PKS type
1983-1987 Ondrechen, 198414
e-n electronic couplings couplings lQ linear term
second AQ linear quantification term on sites 1 and 3 only
1-2 and 2-3 1-2 and 2-3
Ondrechen, second AQ linear 1985, 198715 quantification term on all sites
1-2 and 2-3
Bersuker, Borshch, Chibotaru, 198916
PKS type
VQ linear term
1-2 and 2-3
Brunschwig, Creutz, Sutin, 200217
direct expressions of slices of paraboloids (Q - Q˚) direct shift expressions term of paraboloids with quartic term
1-2 and 2-3
Lambert, No¨ll, Schelter, 200218
1-2, 2-3, and 1-3
treatment of Q coordinates
electronic basis
mathematical treatment
main results
potential numerical energy computation surfaces of the matrix analytical delocalized analytical expressions (allyl type) solutions of all states for eigenvalues, adiabatic same then second potential energya order PT energy surfaces combined as analytical delocalized analytical Q1 ( Q3, expressions (allyl type) solutions of energy for eigenvalues, Q2 apart; adiabatic shift R then second then use of potential between order PT Q1 + Q3 energy states and Q2 surfaces, 1, 3, and absorption 2a band profile slices localized establishing recombined by in potential the 3 × 3 allylic transf, energy matrix then interaction surfaces, mode analysis of extrema defined as localized analytical slice cuts symmetrical solutions and in intersection antisymmetrical regions
combined as for 3-fold symmetry combined as Q 1 ( Q3
denoted x and y; no explicit definition
remarks
localized
localized
numerical computation of the matrix
potential energy surfaces
approximate expressions adapted to strongly delocalized systems valid for strongly delocalized system (class III)
rigorous treatment of the interaction between modes four-state model evoked paraboloids with different sizes
a Because the electron-nuclear coupling is introduced by an lQ term, the difference in energy between the bottoms of the paraboloids is not equal to 0 or R (see eqs 3 and 5b).
between diabatic (localized) states. Playing with the electronic interaction factor (Vab) allowed the smooth passage from localized (class II) to delocalized (class III) systems. Note that, in the Launay and Babonneau model, the influence of the sign of Vab was stressed, as the results are not the same for a cyclic system when Vab is 0.10 For noncyclic systems with 2-fold symmetry, the main treatments are summarized in Table 2. A new difficulty arises, due to the lower symmetry. Calling the sites 1, 2, and 3 (see Figure 1), we find that a vibrational mode localized on 2
(denoted Q2) has the same symmetry as the combination (Q1 + Q3). Thus, a mixing may occur, and there is no simple rule to define the right combinations. As for electronic interaction, most models, except one, considered only an interaction between nearest neighbors, i.e., 1-2 and 2-3 but not 1-3. In a 1983 treatment (published in 1987), Launay proposed potential energy surfaces using the same symmetry combinations of vibrational modes as for the cyclic system.13 This was of course an approximation but allowed the qualitative discussion of the two- vs one-step electron-transfer mechanism.
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J. Phys. Chem. B, Vol. 111, No. 24, 2007 6791
Shortly after, the first treatment of Ondrechen et al. considered the case of a M-L-M system, where the central site does not undergo a structural rearrangement upon oxidation or reduction, hence a system for which there is no Q2 coordinate.14 The vibrational coordinates were thus simply Q1 + Q3 and Q1 Q3. This first model was however too limited, as it is unlikely that a redox process on a given site has no structural consequence. A second model was thus elaborated by Ondrechen et al. in which the role of the central site was fully taken into account by assigning it a different relaxation and a different energy.15 The combinations of vibrational coordinates were written as Q2 on one side and (Q1 + Q3) and (Q1 - Q3) on the other side. Then, the model was applied to strongly delocalized systems with the case of the Creutz-Taube ion in mind. In this case, the (Q1 - Q3) coordinate was eliminated and the potential energy surfaces could be written as functions of the two symmetrical coordinates Q2 and (Q1 + Q3). A full analysis tackling the problem of the interaction between coordinates of the same symmetry, namely Q2 and (Q1 + Q3), was given by Bersuker et al.16 and is commented on in more detail below. A three-state model was briefly discussed by Brunschwig, Creutz, and Sutin.17 Finally, a model by Lambert et al. has been developed recently (from 2002)18 and will also be detailed.
The above energy matrix describes three paraboloids with equal radius and whose minima are located at
x ) 1/2, y ) x3/2 with radius λ2 x ) 1, y ) 0 with radius λ1 i.e., at the summits of an equilateral triangle and corresponding to situations where an extra hole is localized either on one of the triarylamine sites or on the central bridge. The original model thus uses six parameters: λ1, λ2, C, ∆G°, V12, and V13, and the simplified model (without quartic terms) uses five parameters. Although the model is appealing, we question two points, which are outlined below. The Radius of the Paraboloids. The λ1 and λ2 parameters are ultimately linked to force constants on the redox sites and to the displacements that occur upon oxidation. Thus, strictly speaking, the energy of each diabatic state depends on all force
V13
V12
[( ) (
λ2
V13
V12
q+ ) 0
x ) 0, y ) 0 with radius λ1
Note that here no direct interaction is introduced between outer sites 1 and 3.
V12
q- ) +A
Thus, eq 8 describes three paraboloids of revolution, located, respectively, at
(6)
)
q+ ) -B
In a 2002 paper, Lambert, No¨ll, and Schelter proposed a model on the basis of three diabatic potentials.18 The system upon investigation was made of two triarylamine sites connected by various conjugated bridges, so that, for chemical reasons, oxidation processes were considered. The corresponding secular determinant was directly written as eq 7. Here, x and y are vibrational ccordinates. In this expression, they used the quartic augmentation function proposed by Nelsen2c and characterized by the C parameter (this quartic augmentation is an empirical modification of the potentials, which was found useful for spectral simulations). If, for sake of simplicity, we make C ) 0, the determinant reduces to eq 8.
1 2 0 (q + q2+) + Aq- W 2 1 2 (q + q2+) + ∆ + Bq+ W W 2 1 2 (q + q2+) - Aq0 W 2 -
x2 + Cx4 + y2 + Cy4 -E 1+C
q- ) 0
The Lambert-No1 ll-Schelter Model
This model was proposed in 1989. It is based on a rigorous analysis, starting from the effect of charge localization (described by the addition of a linear term of the form Viqi, where Vi is a vibronic constant).16 The energy of the central site differs from the one of the outer sites, and this is introduced by an energy shift δ. After several coordinates transformations, including scale transformations and shift transformations, two pertinent coordinates, q- and q+, are defined, and the following matrix is obtained:
(
q+ ) 0
Thus, in general, the coordinates of the three minima do not form an equilateral triangle, except when B ) x3A. For the general case, if one wishes to have an equilateral triangle, it is possible to perform a scale transformation in the q+ direction, but then, the paraboloids are no longer of revolution. The model thus uses four parameters: A, B, W, ∆.
The Bersuker-Borshch-Chibotaru Model
λ1
q- ) -A
2 x3 1 -x + -y 2 2
)] 2
+ ∆G° - E
V12
(
λ1
λ1(x2 + y2) - E
V12
V12
λ2
V13
V12
[( ) (
2 x3 1 -x + -y 2 2
(7)
)]
)
(1 - x)2 + C(1 - x)4 + y2 + Cy4 -E 1+C
V13
2
+ ∆G° - E
V12 λ1((1 - x)2 + y2) - E
(8)
6792 J. Phys. Chem. B, Vol. 111, No. 24, 2007 constants, because the potential energy is the total potential energy of the system, not just the energy of one site. It is thus incorrect to associate different paraboloid radii to the different diabatic states. V12 vs V13: Do We Really Need the Two? V12 and V13 are electronic coupling elements that take into account the interaction between the different sites. Although their correct definition could be quite complicated, they depend ultimately on overlaps between monoelectronic wave functions. Thus, one could question the necessity to introduce two parameters. In principle, given the large distance between the external sites, the V13 parameter should be extremely small and could be neglected. The electronic interaction in the system is dominated by V12, which is enough to generate an indirect interaction between the external sites. This can be shown easily by the following argument: consider a series of three sites, labeled 1, 2, and 3 (with 1 and 3 as outer sites and 2 the bridge), and involving just one orbital each. We assume that the orbital energies are the same for 1 and 3 and is higher for 2. Taking into account the symmetry of the problem, one has to first build symmetryadapted linear combinations (SALC) of φ1 and φ3 orbitals, namely 2-1/2 (φ1 + φ3) and 2-1/2 (φ1 - φ3), and then allow them to interact with φ2. After this interaction, the two SALC are separated in energy, because one of them can mix with φ2 and the other cannot and remains nonbonding. It is equivalent to say that the energies of 2-1/2 (φ1 + φ3) and 2-1/2 (φ1 - φ3) have been split by some effective interaction given by V122/ ∆E, where ∆E is the energy difference between φ1, φ3, and φ2. This simple explanation is the basis of the superexchange mechanism. In the original treatment of Lambert, No¨ll, and Schelter, the introduction of a V13 term comes from the special treatment in which the adiabatic transition dipole moment matrix is diagonalized.18 This defines a unitary transformation that is then applied to the adiabatic energy matrix. The transformation generates diabatic (localized) states coupled by different interactions, including V13. It is not sure however that the generated states are fully localized on a given site. These states are not easy to visualize, at variance with usual localized wavefunctions. This is why, in our system exposed below, we avoid the use of a V13 coupling. Our Model Our model is inspired from the Bersuker-Borshch treatment, and the general starting equations are the same. In particular, it seemed to us necessary for a practical point of view to use the minimum number of parameters as in their model. The expression of diabatic energies were however written in the same style as Lambert et al., i.e., using a shift of the curve and defining reduced adimensional coordinates, thus with expressions of the style λ(q - q°)2. We consider a system made of three sites, A, B, and C, where A and C are chemically equivalent external sites and B is the central bridging site. In the most typical case, A and C are metallic fragments, and B is an organic part, and thus, it could involve a different number of bonds of different chemical nature than those of A and C. In a localized basis, we evaluate the energies of the diabatic states for which oxidation or reduction bears on site A, B, or C. They are given, respectively, by
EA ) (1/2)k(QA - Q0)2 + (1/2)kQB2 + (1/2)kQC2 (9a) EB ) (1/2)kQA2 + (1/2)k(QB - xQ0)2 + (1/2)kQC2 + ∆ (9b)
Launay et al.
EC ) (1/2)kQA2 + (1/2)kQB2 + (1/2)k(QC - Q0)2 (9c) Note that we use the same force constant k for all sites, although the central site is chemically different from the others. The motivation is that the equations are much more easily handled when identical force constants are used, but this choice can nevertheless be justified by noting that a change in k can be compensated by a change in the Q coordinate by writing for the deformation energy on a given site
E ) (1/2)ktrueQtrue2 ) (1/2)kQ2
(10)
because the important quantity is the energy not the individual k and Q values, which are not easily accessible anyway. Thus, for the B site, the Q parameter is a pseudo-coordinate, related to the true Q displacement (noted Qtrue) by the above scale transformation, with
Q ) (ktrue/k)1/2Qtrue
(11)
An additional advantage of this transformation is to allow handling chemically different sites, for instance containing a different number of bonds, which is the case when the terminal sites are metal centers, and the central one is organic. For the site on which oxidation or reduction has occurred, a Q0 term has been introduced, which modifies its equilibrium geometry and energy, but the force constant is kept the same. (Following many others,3,6 we use this approximation, which simplifies greatly the calculations). Q0 is usually positive for a reduction and negative for an oxidation. To take into account the different chemical nature of the B site, the Q0 parameter is however replaced in eq 9b by xQ0, where x is a dimensionless parameter. Thus, if the bridge experiences large changes in bond lengths and energies upon oxidation, x is large. Finally, a vertical shift in the EB energy is provided by the ∆ parameter. The calculation of potential energy surfaces is detailed in the Supporting Information. In the calculation, it is necessary to define new symmetry coordinates. Two of them are active for the electron-transfer process and are defined by the following equations:
Q2 )
1
x1 + R
2
[
-RQB +
Q3 )
1 (QA + QC) x2
1 (QA - QC) x2
]
(12) (13)
where R is a parameter related to x as shown in the Supporting Information. They are further transformed by adimensionalization, through
q2 )
Q2 Q0x2
and q3 )
Q3 Q0x2
(14)
The q2 and q3 coordinates allow describing the modification of the nuclear geometry during an electron exchange. Their signification is illustrated in Figure 3 for the case of a trinuclear system involving three metal sites. By a proper combination of motions along q2 and q3, it is possible to generate the geometries adapted to localization on either A, B, or C sites. The following step is then to compute the expression of diabatic surfaces, i.e., without any electronic interaction, corresponding to an extra electron (or hole) perfectly localized on either site A, B, or C. The calculation gives three paraboloids of revolution with equal radii, located at the summits of an
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Figure 4. Disposition of the paraboloids (represented as contour curves) in the q2-q3 plane. Only for d ) 1, the triangle is equilateral. Figure 3. Pictorial interpretation of the q2 and q3 coordinates.
isosceles triangle. The potential energy matrix can thus be written as
[
(
λ q22 + q3 + Vab 0
1 2
2
)]
Vab
[(
)
x3 λ q2 - d 2 Vab
2
]
+ q32 + ∆
0 Vab
[
(
λ q22 + q3 -
1 2
2
)]
(15) Note that this expression is very similar to the one of Bersuker et al.16 We have introduced only vicinal couplings between A and B or B and C, which are taken equal by symmetry and denoted Vab. The diagonal terms of this matrix correspond to three paraboloids of revolution with equal radii whose minima are located at
q3 ) -1/2 q3 ) 0 q3 ) 1/2
q2 ) 0 q2 ) d x3/2 q2 ) 0
Thus, the minima do not form an equilateral triangle, except when d ) 1. The energies of the minima are, respectively, 0, ∆, and 0. The ∆ parameter will be taken positive in most cases, which correspond to the most interesting situation where the outer sites are the most easy to oxidize or reduce. The locations of minima are represented in the q2 - q3 plane in Figure 4. There are four basic parameters: λ, ∆, d, (defining the diabatic surfaces), and Vab. Their significance is the following: (i) λ determines the radius of the paraboloids, and thus the intervalence transition between outer sites (vertical transition between surfaces A and C at the minimum of surface A or C). The intervalence transition energy is simply ∆EAC ) λ. (ii) ∆ and d determine the position of the third paraboloid, the one corresponding to the oxidized or reduced bridge. They are related to an experimental quantity, because the charge-transfer transition energy between one of the outer sites and the bridge is given by
∆EAB ) λ
(
)
1 + 3d2 +∆ 4
(16)
Thus, for d ) 1 and ∆ ) 0 (equilateral triangle, corresponding to three equivalent sites), ∆EAB ) λ. The d parameter (“d” for “depth”, because the q2 coordinate is the depth when looking in the E vs q3 plane), together with ∆, determine the energy of this transition. ∆ has a simple interpretation: because it is the energy difference between the bottom of the paraboloids, it corresponds to the energy difference between states where either the outer sites or the bridge bears the extra charge. (iii) Finally a single parameter, Vab determines the amount of mixing between the surfaces. Let us discuss the role of the d parameter in more detail. This is the most subtle effect because it introduces really the third dimension in the treatment. The corresponding consequences have been generally overlooked in many qualitative models of trinuclear systems, based on the simple introduction of a third curve in a two-dimensional E ) f(Q) diagram. d introduces in a single parameter the effect of the nonequivalence between the B site and the others. A weak d means that there is only a small rearrangement when electron transfer occurs between one of the outer sites and the bridge. This should occur for systems in which the bridge geometry is only weakly dependent on its oxidation state. Conversely, a large d means a large rearrangement on the bridge following oxidation or reduction. For d ) 1 and ∆ ) 0, as mentioned above, we join the case of a trinuclear system with 3-fold symmetry, at least for the diabatic potentials. In principle, there is no restriction on the maximum d value, which could exceed 1. Finally, as shown in the Supporting Information, d is simply related to x (defined in eq 9b) by
d)
x
1 + 2x2 3
(17)
This simple relation shows that d cannot be lower than x1/3 ) 0.577, even when there is no reorganization associated with site B (x ) 0), and the third paraboloid is then located at dx3/2 ) 0.5. In particular, it is not possible for the paraboloid corresponding to B to sit just above the saddle point between paraboloids A and C. The explanation is that even when x ) 0, electron transfer between A and B still involves some motion in the q2 direction because the metal site undergoes a geometrical change.
6794 J. Phys. Chem. B, Vol. 111, No. 24, 2007
Launay et al.
Figure 7. Representation of the diabatic surfaces as slice cuts in the E-q2 plane. The parabolas centered on q2 ) 0 are slice cuts of A and C paraboloids, whose revolution axes are at q3 ) -1/2 and +1/2. The B paraboloid lies in an intermediate plane (q3 ) 0). Figure 5. Perspective view of the three diabatic paraboloids. The drawing corresponds to λ ) 1.0 eV, ∆ ) 0.5 eV, and d ) 1.
Figure 6. Representation of the diabatic surfaces as slice cuts in the E-q3 plane. For the central parabola, the plain curve represents the profile for q2 ) 0, i.e., in the same plane as the parabolas corresponding to A and C, and the dotted curve is drawn at q2 ) dx3/2, i.e., a plane of symmetry containing the revolution axis of the paraboloid.
Representation of Diabatic Surfaces The paraboloids of diabatic surfaces play the same role as the usual parabolas in the two-site Hush-Marcus curves. There are many ways to represent them: using contour curves, slice cuts, perspective drawings, etc. They can be easily drawn with software such as Mathematica.19 In our laboratory, we found it useful to generate 3D objects using a CAD software (Catia v5)20 to treat the output files of Mathematica. The corresponding files in the .stl format can be visualized on a screen and even fabricated with a 3D printer. Some subtleties in the disposition of paraboloids were more easily discovered by manipulating real models. A perspective view of the three diabatic paraboloids is given in Figure 5. Drawings in the E-q3 Plane. If we perform a drawing as a projection in the E-q3 plane, we get a diagram with two parabolas, completed by a third curve corresponding to the oxidation or reduction of the bridge. However, the situation is markedly dependent on the d parameter, because, when we plot the third curve, it is actually a slice cut corresponding either to the q2 ) 0 plane or to the plane containing the revolution axis of paraboloid B (see Figure 6). Because d cannot be small or
zero, the difference is generally important. This problem is frequently ignored in many qualitative discussions found in the literature. Drawings in the E-q2 Plane. The drawing in the E-q2 plane shows the influence of the two fundamental parameters determining the position of the third paraboloid associated to B (see Figure 7). One can raise the following question: in the limit of small Vab, what is the condition to have the minimum of paraboloid B included inside paraboloids A and C? Under these conditions, B cannot exist or is extremely rapidly converted into A or C. The answer is not directly obtained from Figure 7, because the drawn parabola are not in the same plane. Taking into account the actual tridimensional nature of the problem, one finds the condition:
∆>λ
(
)
1 + 3d2 4
(18)
Vertical Transition Energies (in the Limit of Small Vab). Starting from one of the main wells in q2 ) 0, q3 ) ( 1/2, one finds there are two possible transitions: an intervalence A-C transition, with energy λ; a charge-transfer A-B transition involving the bridge with energy ∆ + λ(1 + 3d2)/4 (eq 16). The charge-transfer transition involving the bridge becomes the lowest in energy when
3 ∆ < λ (1 - d2) 4
(19)
Effective Interaction between A and C. Although there is no Vac term in the energy matrix, an effective interaction occurs as the result of the Vab coupling, giving rise to an avoided crossing (splitting) in the region where EA and EC are equal. It is easy to show that when the coupling is small, the splitting is 2 2V 2ab/∆E, corresponding to an effective coupling V eff ac ) V ab/ ∆E, where ∆E is the energy difference between EA,C and EB in this area. (Note that perturbation expressions of this type have been developed in much detail in ref 17.) From the energy expressions of the diabatic curves at the q2 ) q3 ) 0 point,
V eff ac )
4V2ab λ(3d2 - 1) + 4∆
(20)
This formula, based on a perturbation expression, is of course valid only if ∆ and/or d are large enough.
Three-Centers Models for Electron Transfer (1)
Figure 8. Evolution of the lowest energy surface for the following values of the parameters: λ ) 1.0 eV; ∆ ) 0.5 eV; d ) 1. Vab (eV) ) 0.05 (a); 0.2 (b); 0.5 (c); 0.8 (d).
J. Phys. Chem. B, Vol. 111, No. 24, 2007 6795
Figure 9. Lowest part of the potential energy surface as contour curves for the following values: λ ) 1.0 eV; ∆ ) 0.5 eV; d ) 1; Vab ) 0.45 eV. The dotted lines show the difference between the traditional slice cut analysis at q2 ) 0 and the true reaction path.
Link with the First Ondrechen Model (1984).14 Because in this model the central site does not play a role, it corresponds to x ) 0, and thus d ) 0.577 in the present model. But even with no Q2 coordinate, the basis localized states are made of three paraboloids, with equal radii. However, due to the use of the lQ convention in the building of diabatic states, the B state had its minimum higher than the ones of the A and C states, located at ∆ ) 0.5 with our notations. Adiabatic Surfaces Adiabatic surfaces are obtained from diabatic ones by diagonalizing the potential energy matrix, eq 15. We have explored a few typical cases, for which ∆ is positive (the central sites are more difficult to reduce or oxidize) looking at the lowest energy surface with the following questions in mind: what is the influence of the location of the third paraboloid on the electronic interaction between terminal sites and when does the class II/III transition occurs when Vab is varied? We have chosen the following values of the parameters, which are typical of many complexes: λ ) 1.0 eV, ∆ ) 0.5 eV, so that there are two low-energy states corresponding to localization on the external sites, and the third site is represented by a higher energy paraboloid. For d, we investigated d ) 0.577 () 3-1/2) corresponding to the case where no rearrangement occurs on the central site, and d ) 1 corresponding to a full rearrangement with a central site of the same nature as the external ones. In each case, we have increased the Vab coupling between the external and central site from a very small value (0.05 eV) to a large one (1.0 eV). The results for d ) 1 are displayed in Figure 8. For d ) 1 and Vab ) 0.05 (a very small initial value), there are clearly two low-energy wells separated by a ridge when looking along q3 at q2 ∼ 0, and the third paraboloid manifests as a small hollow on the right of the surface in the q2 ) 0.51.5 range (see Figure 8). When the electronic coupling increases, the surface “softens” and the hollow grows at the expense of the ridge. The barrier between the low-energy wells disappears for Vab at about 0.55, after a single minimum is obtained. This
Figure 10. Structure of the two ruthenium compounds where the bridge contains an anthracene unit (1) or a dimethoxybenzodithiophene (dmbdt) unit (2).
corresponds to the passage of a class II (localized) to a class III (delocalized) system. The same study performed with d ) 0.577 gives a critical condition near 0.47, a slightly lower value. This evolution could be anticipated from eq 20: when d decreases, the effective coupling between terminal units Veff ac increases, and thus, it is logical that the class II to III transition occurs earlier. Finally, it is worth noticing that the definition of class II or III is only possible through a careful inspection of the lowest surfaces. In the example given in Figure 9 as contour curves, the lowest part of the surface exhibits a crescent shape with two minimums, and a simple analysis based on slice cuts at q2 ) 0 would overestimate the energy barrier. Application to Binuclear Ruthenium Complexes Bridged by Anthracene and by Dimethoxybenzodithiophene (dmbdt) As a part of a systematic study of mixed valence ruthenium complexes bridged by various moieties, we have investigated and compared the two complexes represented in Figure 10. Complex 1, where the bridging unit contains an anthracene moiety, has been already described.21 For compound 2, the pertinent information about the synthesis and properties is given as Supporting Information. The two compounds present intervalence transitions in the same spectral range, ca. 1 eV (thus, λ
6796 J. Phys. Chem. B, Vol. 111, No. 24, 2007
Figure 11. Three-dimensional adiabatic lowest energy surface obtained with parameters : λ ) 1.0 eV, d ) 0.8, ∆ ) 0.5 eV, Vab ) 0.1 eV.
∼ 1 eV). They present similar electrochemical properties, showing in particular that the bridge can be oxidized at a potential more positive than the one of ruthenium sites by about 0.5 V (thus ∆ ∼ 0.5 eV), but the coupling is much stronger in eff the case of 1 (Veff ac = 0.055 eV) than in 2 (Vac ) 0.022 eV). This difference is not explainable if we restrict the discussion to the role of the ∆ parameter. We believe it comes mainly from the d parameter: the third paraboloid is more or less displaced in the third dimension (role of “depth”). Knowing ∆ and λ, it can be determined from the charge-transfer transition energy using eq 16. A preliminary calculation with λ ) 1.0 eV, ∆ ) 0.5 eV, ∆EAB (charge transfer) ) 1.14 eV for 1 and 1.35 eV for 2 gave d ) 0.72 for 1 and 0.90 for 2. From eq 20, it is then normal that the coupling is lower in 2 than in 1, and thus, the variation of d is in the right direction. However, the magnitude of the effect seems greater than expected. A likely explanation involves the chemical difference between the bridges in 1 and 2. The detailed mechanism of through-bridge coupling involves the overlap of metal orbitals with some frontier orbitals of the ligand, in this case the HOMO, and they have probably different weights at the junction with the terminal site. Thus, more theoretical work is necessary to explain the large difference in metal-metal couplings between 1 and 2. Finally, a simulation of the lowest potential energy surface is represented in Figure 11 for the following values of the parameters, which are typical of the two compounds: λ ) 1.0 eV, d ) 0.8, ∆ ) 0.5 eV, Vab ) 0.1 eV The surface exhibits a secondary minimum for the “B state”, the one for which the bridge has been oxidized. It shows the possible coexistence of the two mechanisms for electron transfer, although the hopping mechanism would have a higher activation energy. Conclusion The present model allows a simple description of the electrontransfer process in a trinuclear system, starting from diabatic surfaces constituted by three revolution paraboloids of the same size. The model is intended to use as few parameters as possible. Thus, for weak interactions, the basic features of the system
Launay et al. are described by the two parameters determining the position of the third paraboloid (corresponding to the redox process on the bridge): its vertical shift ∆ and its lateral position “d” in the q2 direction. The physical meaning of the “d” parameter has been established as linked to the amount of rearrangement of the bridging site upon oxidation or reduction. However, one has to recall that a given paraboloid is not related to a specific site but represents the total energy of the system when a given site has been oxidized or reduced. As a result, the d parameter cannot be zero and must be greater than 3-1/2 ) 0.577. Starting from these localized surfaces, the introduction of the electronic interaction can then generate adiabatic surfaces and eigenvectors, thus allowing the junction with delocalized systems and the calculation of pertinent experimental observables. Future papers will be devoted to the exploitation of the model, with the calculation of transition dipole moments and the link with the Generalized Mulliken-Hush model (GMH) model.22 In particular, it will be interesting to apply the GMH concept to the present system where the influence of the third site is explicitly taken into account. Another important point is the way to put numerical values to the parameters, in particular the “d” parameter. This can be achieved through complete quantum chemical calculations, in which the role of nuclear relaxation (i.e., geometry optimization) is explicitly taken into account. Acknowledgment. The help of Christine Viala is gratefully acknowledged for the synthesis of compound 2. Supporting Information Available: Details of the calculation of diabatic potential energy surfaces and details of the chemical preparation and properties of complexes 1 and 2. This material is available free of charge via the Internet at http:// pub.acs.org. References and Notes (1) Electron Transfer in Chemistry; Balzani, V., Ed.; Wiley-VCH: Weinheim, Germany, 2001; Vols. 1-5. Nelsen, S. F. Chem. Eur. J. 2000, 6, 581. Kaim, W.; Klein, A.; Glo¨ckle, M. Acc. Chem. Res. 2000, 33, 755. Demadis, K. D.; Hartshorn, C. M.; Meyer, T. J. Chem. ReV. 2001, 101, 2655. D’Alessandro, D. M.; Keene, F. R. Chem. ReV. 2006, 106, 2270. (2) (a) Bonvoisin, J.; Launay, J.-P.; Rovira, C.; Veciana, J. Angew. Chem., Int. Ed. Eng. 1994, 33, 2106. (b) Bonvoisin, J.; Launay, J.-P.; Van der Auweraer, M.; De Schryver, F. C. J. Phys. Chem. 1994, 98, 5052. (c) Nelsen, S. F.; Ismagilov, R. F.; Powell, D. R. J. Am. Chem. Soc. 1997, 119, 10213. (d) Nelsen, S. F.; Ismagilov, R. F.; Gentile, K. E.; Powell, D. R. J. Am. Chem. Soc. 1999, 121, 7108. (e) Lambert, C.; No¨ll, G. J. Am. Chem. Soc. 1999, 121, 8434. (3) Marcus, R. A. Annu. ReV. Phys. Chem. 1964, 15, 155. Hush, N. S. Trans. Faraday Soc. 1961, 57, 557. Brunschwig, B. S.; Logan, J.; Newton, M. D.; Sutin, N. J. Am. Chem. Soc. 1980, 102, 5798. Sutin, N. Prog. Inorg. Chem. 1983, 30, 441. Sutin, N. Acc. Chem. Res. 1982, 15, 275. Creutz, C. Prog. Inorg. Chem. 1983, 30, 1. Marcus, R. A. Angew. Chem., Int. Ed. Eng. 1993, 32, 1111. (4) Taube, H.; Gould, E. S. Acc. Chem. Res. 1969, 2, 321. Taube, H. Angew. Chem., Int. Ed. Eng. 1984, 23, 329. Launay, J.-P. Chem. Soc. ReV. 2001, 30, 386. Crutchley, R. J. AdV. Inorg. Chem. 1994, 41, 273. (5) Petrov, E. G.; May, V. J. Phys. Chem. A 2001, 105, 10176. Bixon, M.; Jortner, J. J. Am. Chem. Soc. 2001, 123, 12556. Sim, E. J. Phys. Chem. B 2005, 109, 11829. (6) Piepho, S. B. ; Krausz, E. R. ; Schatz, P. N. J. Am. Chem. Soc. 1978, 100, 2996. (7) (a) Von Kameke, A.; Tom, G. M.; Taube, H. Inorg. Chem. 1978, 17, 1790. (b) Bignozzi, C. A.; Paradisi, C.; Roffia, S.; Scandola, F. Inorg. Chem. 1988, 27, 408. (8) Ernst, S.; Ha¨nel, P.; Jordanov, J.; Kaim, W.; Kasack, V.; Roth, E. J. Am. Chem. Soc. 1989, 111, 1733. Kaim, W.; Kasack, V. Inorg. Chem. 1990, 29, 4696. (9) Austin, I. G.; Mott, N. F. AdV. Phys. 1969, 18, 41. (10) Launay, J.-P.; Babonneau, F. Chem. Phys. 1982, 67, 295. (11) Borshch, S. A.; Kotov, I. N.; Bersuker, I. B. Chem. Phys. Lett. 1982, 89, 381. (12) Cannon, R. D.; Montri, L.; Brown, D. B. ; Marshall, K. M.; Elliott, C. M. J. Am. Chem. Soc. 1984, 106, 2591.
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