Distance dependence of the effective coupling parameters through conjugated ligands of the polyene type

ChemicalPhysics 147 (1990) 131-141 North-Holland

Distance dependence of the effective coupling parameters through conjugated ligands of the polyene type C. Joachim

‘, J.P. Launay ’ and S. Woitellier

Laboratoire de Chimie des MPtaux de Transition. Universitk Pierre et Marie Curie, 4 place Jussieu, 75252 Paris Cedex 05, France Received 9 October 1989; in final form 18 May 1990

Experimental results on the decay of the effective coupling parameter in mixed valence ruthenium complexes bridged by bipyridylpolyenes are discus&. Absolute values of the coupl@s am satisfactorily reproduced by extended Httckel calculations. An exponential decmase of the effective coupling is found with a particularly small exponent, 0.078 A-‘, corresponding to a decrease by a factor of 2 for each 9 A in the metal-metal distance. The results are also discussed in the frame of a simplified valence bond taking into account the main structuralfeatures of the system. It is concluded that, in the present compounds, the transmission of electronic couplings over large distances is limited by the Peierls distortion of the polyene chain.

1. Introduction

Vab=VLexp(-7W.

Intramolecular electron transfer between a donor and an acceptor group is a fundamental process with considerable implications in transition metal chemistry [ 11, biology [ 2 1, solid state chemistry [ 3 ] and now in the emerging field of molecular electronics [4]. A particularly important point is the distance dependence of electron transfer rates which is itself related to the distance dependence of the electronic coupling parameter #I. This question has been discussed by several authors, among others Larsson [ 5 1, Beratanetal. [6,7],Hush [8], Schipper [9],Miller [ lo], and one of us [ 111. It is frequently assumed that the magnitude of the coupling V,, between donor and acceptor sites linked by a bridging ligand decreases according to an exponential law, i.e. ’ Present addresu Molecular FilectronicsGroup, CEMES/LGE, 29 rue Jeanne Marvig, 31055 Toulouse Cedex, France *I The electronic coupling parameter is diversely denoted in the literature as e, V, Hrs, C,, ... and qualitied as “electronic matrix element”, %mnelinB matrix element”, “electronic interaction energy”, “electronic coupling element”, etc. Hespite the great variety in theoretical treatments, the basic principle has always been to replace the complex molecular system by a two-site model system which can be character&d by a single number. Hem the notation P.,, will be used, keep@ in mind that it actually designates and e@ctiue coupling parameter. 0301-0104/90/$03.50

(!I

In fact, there is no general agreement about this mathematical formula, which is largely empirical and comes from early tunnel calculations. Recently, other laws (for instance inverse power laws) have been proposed [ 8 1, usually from a fitting of experimental or theoretical data. Despite this difficulty, the y and V!& coefficients can be used as a rough guide to characterize the efficiency of electronic interaction through a bridging system. From an experimental point of view, the y values are usually found near 0.5 A - ’ (table 1) . They correspond to a relatively rapid rate of decrease, so that the electron transfer process becomes generally undetectable when the donor to acceptor distance exceeds lo- 15 A. Recently the synthesis and properties of mixed valence compounds of the general type [ (NH&Ru-py-(CH=CH).-py-Ru

(py=pyridyl)

(NH,),1

5+

9

with n=2, 3,4 have been described [ 181. Here the two redox sites in oxidation states 2..+ (donor) and 3+ (acceptor) are linked by a conjugated pathway and the electronic coupling can be determined from the analysis of the intervalence band corresponding

0 1990 - Elsevier Science Publishers B.V. (North-Holland)

132

C. Joachim et al. / Eff‘ectiw couplingparameter in conjugatedligands

Table 1 Examples of experimental y values reported in the literature System

Y (A-‘)

Ref.

Zn/Ru modified myoglobins Run-spiroalkane-Rum R&conjugated bridge_Ru” Cu(en):+ +trapped epyrene+ or biphenyl + +TMPD dimethoxynapthalenc-saturated bridge-dicyancethylene

0.45 0.37 0.50 0.45

iI21

0.58

1161

0.5

[ 171 b’

16,131 [ 141 l)

iI51

l)

Computed from ref. [ 141 by taking the two extreme bridges, i.e. cyanogen and 2,6-dicyanonaphth&ne. b)Fromref. [17]byassumingratc3propottionalto V&Note that these experiments correspond actually to a photoinduced electron transfer.

to an optical electron transfer process. As could be anticipated from the conjugated nature of the bridging ligand, these systems show a marked tendency to relay electronic effects over large distances. This is shown by both theoretical and experimental results. Although this behaviour agrees well with intuitive concepts based on resonance and conjugation, it is only recently that it has been formulated in a quantitative way [ 5,111. In the present paper we first present experimental results giving the effective coupling between pentamineruthenium sites linked by an a-w bipyridylpolyene bridge. A very slow rate of decay is observed, which will be .discussed in the frame of two models: (i ) an extended Hiickel molecular orbital (EHMO ) model which reproduces satisfactorily the experimental V,, values but does not provide a physical insight into the process, and (ii ) a valence bond model [ 11,19 ] which takes into account the gross structural features of the system and allows a more physical discussion.

2. Experimental results The experimental values obtained by Sutton and Taube [ 201 for n= 0 and 1, and by Woitellier et al. [ 181 for,n=2, 3, and 4 are reported in table 2 and fig. 1 in logarithmic scale. These experimental values are obtained from the intensities of the so-called intervalence transitions, following a treatment origi-

nally proposed by Hush [ 2 I]. There is some discontinuity between the values for n = 0, 1 and for n = 2, 3,4. On could think that this is due to the use of different experimental conditions, the spectra of the compounds with n = 0 and 1 having been recorded in D20 while those for n = 2,3,4 have been recorded in nitrobenzene. However, we have also recorded the spectrum of the compound n = 2 in DzO and found that the band intensity is virtually unchanged. But there are two experimental difficulties which suggest that V, could be overestimated for the longer compounds: (i) The role of the comproportionation constant. To extract the effective coupling from the intervalence spectrum, it is necessary to know the true extinction coefficient of the mixed valence form. As it is in equilibrium with the homovalent forms, one has to make an assumption about &, the equilibrium constant for the process II-II + III-III*2

II-III .

The values given for the compounds n = 2, 3, and 4 [ 181 have been obtained assuming &=4, which is the statistical limit when n+oo. This is certainly an underestimation; recently we have undertaken the measurement of comproportionation constants and found, for n = 2, K, near 8 [ 22 1, which is intermediate between the value for .h= 1 (1y,= 14) and the statistical limit (&= 4 ) . With this more realistic value, V, drops somewhat (see fig. 1) . (ii) The largest uncertainty comes, however, from the extensive overlap between the intervalence band and the tail of the nearby metal-to&and charge transfer band. The situation is complicated by the fact that in the mixed valence form, the metal-to-ligand charge transfer does not exhibit a Gaussian profile, in particular on the low energy side [ 22,231. In any case, when using the simple method of ref. [ 181, this leads also to an overestimation in V,. More experimental work is currently performed to increase the accuracy. At the present time, one can say, however, that the general trend is a particularly slow rate of decay, which is in fact the slowest reported so far. Curiously, the series of compounds reported by Richardson and Taube [ 141, although conjugated like the present ones, showed a relatively fast rate of decay when the effective interaction was

C. Joachim et al. / Effktiw couplingparameter in conjugated&and

133

Table2

Effective coupling parameters l) for the series of complexes [ (NH& Ru-py-( CH-CH I.-py-Ru ( NH3) 5]‘+. Experimentalvaluesfrom intervalence band intensities and theoretical values from extended Hitckel molecular orbital (EHMO) method n

M-M

(A) 0 1 2

11.1 13.4 15.8

3 4

18.1 20.6

a

30.3

Experimental V.b

Theoretical (EHMO ) V, ml,b)

C) Vmiucmsn

0.047 0.039 0.042 0.039 0.032 0.029

0.053 0.044 0.036

0.053

d’ *’ C’ f, =) =’

0.03 1 0.025

0.056

0.012

0.057

‘) All V, values are in eV. Theoretical values are computed for the planar geometry.

b, With standard geometry (alternating ‘double” and “single” bonds in the polyene chain). Cl With all bond lengths equal in the polyene chain. d)Ref. [20]. e)Ref. [la] assuming&=C ‘)Assuming11;=8. Vab A

f log

scale 1

0.03

R,i 10

15

0

20’ *

1

2

3

4

n

Fii 1. Decayof the V, coupling with distance: (a ) experimental values (see text for discussion), ( + ) theoretical values from extended HUckel molecular orbital (EHMO) calculation. The straight line is drawn through theoretical values. plotted

as a function

of the number

of conjugated

atoms.

3. Extended Hikkel molecular orbital model 3.1. Computation Extended formed

details

Hiickel

according

calculations

to a procedure

have

been

per-

already described

[ 24 1. The program was the ICON 8 version written by Hoffmann. The Ru(I1) parameters were taken

from Tatsumi and Hoffmann [ 25 ] and used without charge iteration. These values have been optimized for Ru ( NH3 ) a+ and they lead to a satisfactory position of the Ru 4d levels approximately in the middle of the rr+ gap of the ligand. The geometry of bipyridylpolyenes was built by an extension of the geometry of shorter compounds [24]. For the longest systems, the essentially single bonds were attributed a 1.47 A length, and the essentially double bonds 1.35 A. For all compounds investigated, the geometry of the organic part was taken planar, although in reality the terminal pyridyl groups are probably slightly twisted with respect to the central polyene skeleton. But since the exact (experimental) angle of twist is unknown, and since we are primarily interested in Yariations of the effective couplings, a planar geometry was the best choice (in a more detailed study devoted to bipyridylbutadiene [ 19 1, we have found that twisting the terminal pyridine groups by + 25’ decreases the effective coupling by 20%). Following an earlier calculation [ 24 1, it can be shown that the main coupling occurs between ruthenium orbitals of IC symmetry with respect to the adjacent pyridine rings. Since we use a planar geometry, effective couplings can be taken as half the splitting between the orbitals with this ICsymmetry exhibiting high weights on the ruthenium atoms (see appendix for the theoretical principles of calculation).

C. Joachim et al. / E_@ctiw couplingparameter in conjugatedligands

134

3.2. Efective couplings LUMO (U)

Effective couplings computed by the EHMO method are given in table 2 and are displayed in fig. 1. It can be noticed that theoretical and experimental values are of the same order of magnitude, which was beyond expectations owing to the experimental difficulties mentioned above and the only qualitative character of the EHMO method #*.Anyway, as far as theoretical values are concerned, the decay of V, with the metal-metal distance R can be accurately fitted by an exponential law of the form: Vab= Vz,, exp( -0.078R)

,

(a) HOMO (g)

(b)

(2)

where R is the through space metal-metal distance in A. This is much less than the values usually reported (see table 1). To tix ideas, the present rate of decay would correspond to a decrease by only a factor of 2 for each 9 A increase in the metal-metal distance. The most intuitive way to explain this result in the frame of the EHMO model is to start from the prop erties of the fragments constituting the complex, i.e. the free ligand and the metal sites. The molecular orbitals can be characterized by their symmetry (g or u) with respect to the inversion center of the ligand or complex. The symmetry adapted linear combination (SALC) of ruthenium x orbit& can be written as:

and @EALC =2-“*(dA -dB) . In such a system where the long metal-metal distance precludes direct orbital overlap, the orbital splitting results from the different interactions of the SALC with the ligand orbitals, particularly the frontier orbitals. In the case of long systems (say for instance n=4) the frontier orbitals bear some resemH2In a forthcoming publication, Hush [ 26 ] has performed theoretical calculations of the efffxtive couplings for the same systems.Theligand&ctronicstructureawcrcc&ulatedatthe CNDO level, but there still remains an empirical parameterization of the di&rence between ruthenium and l&and orbital energies. The obtained values are slightly lower than ours for the longer compounds. At the present time, it appears impossible to say if this method is quantitatively superior to the EHMO approach for the calculation of the e&ctive couplings.

\

h-+

\

HOMO

‘-N--=Fig. 2. (a) Shape ofthe HOMO and LUMO in py-(CH=CH),py. (b) Interaction scheme showing the additivity of the HOMO and LUMO influences.

blance with the polyacetylene frontier orbitals. Thus the LUMO (u symmetry in this case) and HOMO (g symmetry) are depicted in fig. 2. If we restrict to the HOMO and LUMO inthrences, it can be seen that one of the ruthenium SALC will interact with the LUMO and the other one with the HOMO. As pointed out by Larsson [ 5 1, both effects add up to determine the main part of the splitting, because one of the ligand orbitals is full and the other empty (see fig. 2). When the length of the conjugated system increases, the analysis becomes more complicated because the energy levels are closer and closer, so that one has to take into account not only the HOMO and LUMO but also higher vacant orbitals and lower occupied orbitals, the contribution of which to the splitting having alternating signs. Thus other methods to represent the electronic interaction in a pictorial way are needed. 3.3. Diabatic orbitals and their decay In a one-electron molecular orbital model, the electronic coupling can be obtained from the splitting be-

C. Joachim et al. / Eflatiw couplingparameter in conjugated ligandr

tween two molecular orbitals exhibiting high weights on the terminal metal sites (see appendix). As mentioned above, owing to the symmetry of the problem, one of them will exhibit g symmetry and the other one u. Thus if we perform the following combinations:

and @&2-“*(@*-@“),

Vab=(@:wI%>

a

0.2 0.1

9

(4)

where H is the complete Hamiltonian of the system. Thus in perfect analogy with a true two-site system (say H,+ ) , the electronic coupling can be considered to follow the variation of the overlap between orbitals ofthe type @b or 06. The shape of W: and @; gives thus an idea of the ability of a given bridging ligand to relay the electronic interaction. This is shown in fig. 3 for the case n = 4. The curve showing the variation in atomic coefficients along the conjugated chain has the shape of a damped sinusoid. High coefficients (in absolute values) are obtained for each two carbon atoms. This is in agreement with the alternance rule.of Richardson and Taube [ 14 1, according to which the presence of a coordinated d6 ruthenium(I1) atom increases the electron density every two carbon atoms. The general decrease in the absolute values of coefficients along the chain provides a

_

-

&ii .

0 , -0.1

-

-0.2

t -py--)t_chaln----_)CPY-

(3)

where 4 and @, are the molecular orbitals with g and u symmetries, this will give two partially localized wavefunctions because the atomic contributions tend to add on one side of the molecule and to cancel on the other side. Each of these wavefunctions exhibits now a high weight on one of the metal atoms, with some “tails” expanding at relatively long distances on the bridging &and. Such orbitals can be called “diabatic” orbitals because they correspond to the wavefunctions the system would present in the crossing region of the Hush-Marcus diagrams if the electronic interaction could be “switched off ‘. These diabatic orbitals are identical to the “orthogonal localized molecular orbitals” [ 271 or “orthogonal magnetic orbitals” [ 281 which are used to rationalize the magnetic interaction in exchange coupled dimers. It can be easily shown that the electronic coupling Vabin a symmetrical system is then simply:

135

C 0.2 0.1

II

b

.

0 -0.1 -0.2

I

-PY_)C---

chaindtpy

-

Fig. 3. Coeficients of the one of the diabatic functions in [ (NH,) 5 Ru-~~-(CH-CH),-~~-RU(NH~),]~+: (a) with the real alternant geometry, (b) with the hypothetical nonaltemant geometry.

picture of an interaction which is usually called “through bond coupling”, by analogy with the wellknown through space coupling. Incidentally, the decay law shown by fig. 3 is reminiscent of the decay law of the probability for spin alternation in other polyenes (seeref. [29], fig. la). Finally, if we plot the absolute value of the @2:, coefficients against distance (retaining only the high values every two atoms and eliminating the values for the terminal pyridyl cycle), again an exponential decrease is observed with a 0.078 A-’ exponent.

4. Stmetmal parameters controlling the effective coupling

Although the extended Hifkel molecular orbital method gives realistic values for the effective coupling parameter, it has the disadvantage to mask the physics behind the distance dependence of the effective coupling. It is in fact a “black box” in which one

136

C. Joachim et al. / Eff&iw couplingparameter in conjugatedligands

enters a large number of structural parameters and which provides the effective coupling without explanations. It is more instructive to explain the electronic coupling by a simple model using as few parameters as possible, while retaining the basic characteristics of the structure, i.e. the repetitive nature of the chain and the existence of localization sites on both ends. Such a model has been proposed recently [ 111. It is actually a simplified valence bond model based on the positions and energetic couplings of the different electronic states of the complex (fig. 4). Of course this model cannot give quantitative values for the electronic coupling through a given ligand, but it can give the trends when some structural parameters are varied, and help in the design of efficient bridging ligands. In this model, the parameters are: a and a’, the intrachain couplings, @the coupling between the last atom of the ligand and the metal site, and a the energy difference between metal localized states and ligand localized states. It is an extension of the model already described [ 111, in which the possibility of bond length altemance has been included. The number of electronic configurations associated with an excitation of the chain is denoted N. For such a system, the effective coupling between initial I@.) and final I @,,) states can be numerically computed by any of the three methods explained in the appendix. But before looking at the results, it is instructive to determine which are the true control parameters. In fact, it can be shown that Vabdoes not depend directly on a, a’, /I, etc., but of combinations, such as a/a, p/a

t#b> Fig. 4. The simplified valence bond model. It corresponds to a chain with topology a-1-2-3...-N-b, where a and b are terminal sites and l-2-... designate the identical chain sites. The 10.) designates a configuration with an extra electron (or hole) localized on site a and 10,) an excited confguration with an extra electron (or hole) localized on site i. a and (Y’ are intrachain couplings while B is the terminal coupling.

and a’ /a. This results from the following considerations: The evolution from I @J (initial state) to 1%) (final state) via the bridge radical states I Ol ) ... I 4&) can be followed by calculating the variation in time of the I Gj,) state occupation probability [ 111. Its analytical expression depends only on the Hamiltonian eigenvalues and on the coupling product B2am+‘o’m9 where m is the integer part of N/2 [ 301. Therefore, the effective coupling is controlled by the parameters determining the eigenvalue repartition of the Hamiltonian H corresponding to the system of fig. 4. Because these eigenvalues are solutions of the secular equation I H-El I =O, /l/a and a/a determine the position of the eigenvalues, like in the case a=a’ [ 111. If a#(~‘, a simple factorisation of I H -El I shows that a’ /(I! also controls the H eigenvalues, although the secular equation is transcendental for N> 2 in this case. Thus for a given N, there are three parameters controlling the time evolution of the 1%) state occupation probability: a/a, B/a, a’ /a and also a time scaling parameter a. It is thus found that the ratio V,,/a can be expressed as a function of the three above parameters. They can be given relatively simple physical meanings. For example, /3/a describes the chemical character of the charge transfer, i.e. the amount of delocalization from the metal states to the ligand radical states compared to the intrachain delocalization. a’ /a models the single-double bond alternance in the conjugated polyene bridge. Finally a/a with a’ /a both determine the gap between the bridge LUMO and the metal sites HOMO. The results of the Vat,calculations (more precisely the ratio V&/a) are thus presented in figs. 5-7 as functions of these adimensional parameters. We have only considered systems for which the number of ligand localized states is even. For the numerical computation, we have used method 2 (see appendix), i.e. the diagonalization of the Hamiltonian corresponding to fig. 4, written in matrix form. For the present problem, this method proved to be the most practical. In the case a = a’, which has been discussed in a previous paper [ 111, a/a and /I/a control V,. If one wants to have a minimal attenuation of V’,,with distance, a/a must be high enough such that at least the LUMO of the bridging ligand be close to the HOMO redox site energy. /3/a must be small enough, in order

C. Joachim et al. / Efffpctiwcoupling parameter in conjugated ligands

(4

HOMO and LUMO even for large values of N. As a consequence, with respect to &=a’, the density of states increases more rapidly as a function of N near the band edge. Always according to eq. (A.2 ) , a strong decrease of Vabresults. This is shown in fig. 5, where the case cr = a’ has been recalled for comparison. To discuss in more detail these variations, one can try to express Vabas an exponential function of N, like in (2): V,,= V$,exp( -y’N)

N

Fig. 5. Decay of V,/a with N for /?/a=05 (a) Nonaltemant system. (b) Ahemant system with cu’/cy=O.5.

that the dominant process is a metal-to-metal transfer through the bridge and not a simple metal-to-ligand charge transfer, which would occur if j? was too large. When N increases, the density of bridge states near the redox sites energy increases too. Therefore, according to eq. (A.2 ) of the appendix, Vnbdecreases as a function of N. Other harmonics in the time evolution of the I dj,) occupation can help to fight against this decrease by tuning a/a and /3/a in such a way that one of the harmonics reaches the amplitudes of the fundamental and becomes, to its turn, the fundamental of the evolution. If now a # a’, a gap persists between the bridge

137

.

(5)

This simple law is not valid for all the a/a, B/a, and a’/a values. But in most cases, it is reasonably possible to lit the actual law by an exponential and determine y’ which can be represented in a contour plot like a “phase diagram”. The case a = a’ has been already discussed [ 111, on a large parameter scale, i.e. 0
C. Joachim et al. / Eflpectiwcouplingparameter in conjugated ligands

138

8

&’

2

0-l 0

e/,l

Fig. 6. “Phase diagram” giving the y’ values as function ofp/(Y and a(/a. The value on the contour corresponds to the quantity y’ log,0 e. The hatched zone corresponds to constant y’ areas. The dotted zone corresponds to the area where at least two y’ values are necessary to describe the variation (see fig. Sb). (a) Nonaltemant system. (b) Altemant system with (r’/(~=O.5.

to reduce the coupling altemance, i.e. to realize a’/ cyx 1, then increase a/a, and finally reduce B/a. But V$,/a decreases when B/a, decreases as shown in fig. 7. Therefore it is not possible to win at the same time on y’ and I’$, because the best B/o values to get a small y’ are also the worse from the point of view of I’$,. For the a/a parameter, the situation is more favorable because the high values which are required to get a small y’ also lead to a strong Vtb),.

5. Role of bond length alteniance It is well known that infinite conjugated polyenes are subject to the Peierls distortion, as any onedimensional system with a half tilled band [ 3 11. The main effect of this distortion is to reduce electron delocalization, with the consequence that polyacetylene for instance is not a metal but a semiconductor. In the present systems, X-ray crystallography clearly

shows the existence of unequal bond lengths [ 321, so that although our systems are molecules with a finite lengths, we may expect some analogous consequences. We have thus investigated the role of bond length altemance in the frame of the two models used ‘here. In the extended Hiickel molecular orbital model, it is easy to “suppress” the bond length altemance by assigning a common average value to every bond length in the conjugated chain. It is then found that I’*,,remains practically constant between II= 0 (4,4’bipyridine ) and n = 8 (see table 2 ) . Correlatively, the diabatic functions do not exhibit a decay with distance, at least in the central polyene region (see fig. 3b). Thus it is clear that bond length altemance (i.e. Peierls distortion) is the main factor responsible for the decay of I& with distance. This explains also why the earlier calculation performed by Larsson ( 1982) [ 5 ] using equidistant sites gave a constant Vab. In the valence bond model, the altemance destroys

C. Joachimet ai. / E@-ctiw couplingparameterin conjug@?ed iigands

139

the wire length does not increase too much the wire resistance. In the present case, it is found that what limits the electronic transmission over long distances is the alternant character of the central polyene bridge (Peierls distortion). Althou~ our systems are molecular, there is clearly a relation with the problem of the conductivity in conjugated polymers. How could we try to beat the Peierls distortion? By analogy with the case of polymers, one could try to play with the filling of the bands. Since intrinsic dop ing is not possible with our systems, auto doping by grafting donor or acceptor groups seems to be an attractive solution, but the corresponding chemistry is certainly not trivial. Another possibility would be to replace the central polyene chain by a conjugated chain which would be intrinsically stable with respect to the Peierls distortion. Fused ring systems, such as the ones described by Kenny and Miller [ 34 1 could be a first step in this direction. Fig. 7. Variation of Vzb,lawith /3/cufor CY’ /are 1.

the perfect order of a repetitive sequence of couplings. This introduces a weak disorder, which is responsible of the fast decrease of V,, with N because in this case, the manifold of bridge states 1d+ ) ... 1cpiv} acts as a spatial filter on 14) and 1c&,). Then the penetration of these two states in the bridge decreases. This situation is reminiscent of the Anderson localization process [ 33 1.

6. Conchsion Our bipy~dyl~lyene bridges present the smallest y reported so far, i.e. y=O.O78 A-’ from the extended Hilckel calculation. The detection of the electronic interaction in the longest systems (n = 4 for instance) is not limited by the weakness of the intervalence transition, but rather by its overlap by the nearby charge transfer transition. We are thus looking for other types of terminal sites which would give more resolved intervalence transitions and allow the detection of weak couplings. However, it is also crucial to design bridges with the highest possible Vzt, and the lowest y’_The search for such bridges finds its origin in the analogy with macroscopic conducting wires where an increase in

Appendix. The Merent ways to cmnpute the effective coupiing

For large and complicated molecular systems, it is often useful to characterize the electron transfer ability by a small set of numbers. This requires a reduction of the info~ation contained in the corresponding molecular Hamiltonian with respect to a complete description of the transfer process. As mentioned above, this reduction corresponds to modeling of the complete system by a simpler equivalent system using a small number of effective states. In practice, a two-state equivalent system is used most of the time. From a practical point a view, i.e. for computation purposes, there are three methods to extract this information #3. A. 1. The Fourier method The system evolution from the state Run-L-Run’ to the state Rum-L-Run through the states RunL+ - Run (or Rum-L--Rum) is computed by solving the time-dependent Schrtiinger equation. This x3 We consider here frozen nuclear degrees of freedom, i.e. we are interested only in the electronic factors involved in eleo tron transfer.

C. Joachim et al. / E&ctiue couplingparameterin conjugatedligands

140

gives the occupation probability amplitudes for the final Run’-L-Ru” state as a function of time. It corresponds usually to a complicated function which emerges from the calculation directly as a series of imaginary exponent&. In the Fourier spectrum of this series, the frequency v of the component with the largest amplitude gives the effective coupling by the relation: V..,=hv/2,

(A.11

where h is the Planck constant. This is in fact the rigorous method to define an effective coupling because it starts from the complete dynamic process and extracts the secular variation of this dynamics. This Fourier analysis has led to a better description of complex systems in which several channels compete for electron transfer [ 241, and also to a better definition of the relaxation transfer rate in photoinduced electron transfer [ 351. When the ligand is short enough, it is easy to extract the fundamental component of the evolution from all the harmonics. But for large N, the method is more tedious, because it can be shown that the number of components of the Fourier spectrum increases like (N+2)(N+3)/2. Thus alternate methods are preferable. A.2. The diagonalization method

l/2 3

A.3. The efective Hamiltonian method As a consequence of the equivalence between methods 1 and 2, an effective Hamiltonian can be constructed to model the Ru”-L-Run’ to Ru”‘-LRu” charge transfer process by its secular variation only. In this case, the N+2 level system is replaced by a 2 level system. Vabcan be calculated from this effective Hamiltonian Ha and is given by [ 24 ] : 1 V lib= 1+stb

x [H,ff,-s,,(H,ff,+H,,)/21

Here, the Hamiltonian corresponding to the system is first diagonalized to give eigenvalues and vectors (depending on the problem, the eigenvectors can be either monoelectronic wavefunctions or total multielectronic wavefunctions [ 19 ] ) . Then, as discussed in ref. [ 241, it is necessary to select the two eigenvectors which contribute the most to the evolution. Their energy difference gives the effective coupling by Pab= I&-&

sen, then method 2 is perfectly equivalent to method 1. Notice that the equivalence between the Fourier analysis and the diagonalization method has obscured the Fourier origin of I’*,,.But it is necessary to keep the memory of this origin since for certain ligands, and particularly when N is large, more than two molecular orbitals may be needed to describe the through bridge charge transfer. This comes from the fact that two or more components of the time evolution Fourier spectrum can have an equal or nearly equal amplitudes. In such a case, V,, becomes a discontinuous function of the control parameter [ 241. Methods exist to smooth these discontinuities, but they are still under development [ 30 1.

(A.21

where ES and E, are the two eigenvalues, which have different symmetries (for example g and u). This method is referred as the “dimer splitting” method 1241. A delicate step in this method is the selection of the eigenvectors. As discussed in ref. [ 241, one can conceive two criteria, called “Bloch criterion” and “Fourier criterion”. If the Fourier criterion is cho-

>

(A.3)

where &b is the overlap between the diabatic states 1Oa ) = Ru”-L-Ru”’

and I‘& ) = Run’-L-Ru”

.

Compared to (A. 1) or (A.2 ) , the calculation of Hew offers the great advantage to produce not only Vsbbut also the energy difference between the two redox sites dressed by their interaction with the bridge. This is interesting because when the molecule is not symmetric, the charge transfer process is controlled not only by V,, but also by ‘thedifference IH,, - Heffbb).

References [ 1] C. Creutz, Prop. Inorg. Chem. 30 ( 1983) 1.

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