Electron transfer in mixed-valence clusters: spin-dependent dielectric

J. Phys. Chem. 1995,99, 2656-2661

2656

Electron Transfer in Mixed-Valence Clusters: Spin-Dependent Dielectric Loss and Hamiltonian Parameters B. G. Vekhter and Mark A. Ratner" Department of Chemistry and Materials Research Laboratory, Northwestem University, Evanston, Illinois 60208 Received: September 21, 1994; In Final Form: November 21, 1994@

The dielectric relaxational processes (low-frequency electric field absorption, or dielectric losses) in mixedvalence clusters are investigated by using a binuclear cluster example. We show that study of these processes can provide valuable information about electron transfer. In particular, study of the frequency-dependent loss for a range of temperatures permits (in the R o b i f l a y classes I and 11) experimental estimation of the tunneling and electronic asymmetry parameters. The effective activation energy can also be found from the temperature dependence, and this, in turn, fixes the reorganization energy.

I. Introduction Intramolecular electron-transfer systems, in particular mixedvalence polynuclear metal complexes, can be characterized rather well by simple polaron-type models that include both local vibronic interactions and tunneling integrals.'-'* These determine the barrier height for intramolecular electron transfer, as well as the prefactor for the transfer rate in the nonadiabatic limit. Many experiments have been performed to deduce these parameters (the reorganization energy and the tunneling matrix element) on the basis of intervalence transfer intensities, resonance Raman spectra, measured electron-transfer rates, et^.'^-^ For example, the Hush relationship3 permits estimation of the tunneling integral from measurement of the intensity of the intervalence charge-transfer band. Dielectric relaxation rates are also sensitive to barrier heights for intramolecular transfer and can therefore be used to characterize these barriers (see, for example, refs 41-44). Indeed, by measuring the dielectric relaxation times and their temperature dependence, it is possible to deduce, from the measured dielectric response, the barrier heights (reorganization energy) and matrix elements for electron tunneling. This makes dielectric measurements a very attractive means for determining the electron-transfer rate parameters; however, to our knowledge this method has not been applied extensively till now to mixed-valence clusters. We concern ourselves with the mixed-valent rather than average-valent situations. That is, we assume that the systems are in the RobidDay II or I situation,' rather than 111: this means that the extra electron (or electrons) is localized on one of the equivalent cluster sites. This in turn gives a large observed dipole moment for the cluster. The large dipole moment, appearing due to the localization of the extra electron, is one of the characteristic features of mixed-valence clusters. This dipole moment has several possible orientations, corresponding to electron localization on different metal ions. Usually t ~ e s elocalized states have equal, or close, energies, and this must manifest itself in the interaction of mixed-valence systems with an applied electric field. In the present paper the dielectric relaxational processes (lowfrequency electric field absorption, or dielectric losses) in mixedvalence clusters are investigated and the information that can be obtained in this way is discussed (we restrict consideration @Abstractpublished in Advance ACS Abstracts, February 1, 1995.

to the energy regime corresponding to microwave spectroscopy, in particular to transition energies less than 10 cm-I; the socalled intervalence transfer band, which has been extensively discussed in the mixed-valence l i t e r a t ~ r e , ~ ~ ~provides -~.*~-~~ different information). There are, in the microwave, two main mechanisms for absorption of energy from an alternating electric field of frequency w . The first is the resonance absorption of a photon with energy ho due to the transition between levels spaced by energy gap AE = hw. Its intensity is proportional to the difference of initial- and final-state occupations, this difference being small for closely spaced levels (in mixed-valence clusters the energy gap between states under consideration is usually less than 10 cm-I, due mainly to the vibronic reduction of the pure electronic splitting). Strong applied fields decrease the occupation difference, causing a nonlinear effect, Le., decrease of absorption when the amplitude of the field increases. Thus the mechanism of resonance absorption between close lying levels is usually switched off at even moderate field intensities. The second mechanism of absorption is the relaxational one, caused by the coupled relaxation of level energies and occupancies in the field. The relaxation manifests itself in the appearance of a delay (phase shift) between the changes of occupations and of energy levels and this shift causes dissipation of the field energy. The relaxational absorption is essential in a wider frequency range than the resonance one, and often tums out to be more intense. For independent energy levels in an applied field, the system energy can be written

with E, and n, respectively the energy and occupation of the jth level. Then the time evolution is simply dE,,,,,,ldt

+

= C{n,(t)[d~~(t)/dt] cj(t)[dnj(t)/dtl}

The first term on the right is the relaxational one, and the second corresponds to resonant absorption or emission. Study of the relaxational part allows direct estimation of the relaxation times and activation barriers, and we consider it below in detail.

11. Absorption and Activation Barriers: General Formula A. Electronic Effects: Frequency Dependence. The pure electronic Hamiltonian of a mixed-valence cluster (the vibronic

0022-365419512099-2656$09.0010 0 1995 American Chemical Society

J. Phys. Chem., Vol. 99, No. 9, 1995 2657

Electron Transfer in Clusters interaction effects will be considered a little later) includes terms that describe the tunneling of an extra electron (HJ, the crystal field effect of surrounding ions (Her), and interactions of the electron with the electron core of metal ions (He)and with an electric field (Hf) and looks as follow^^^-^^

+ + He + Hi

H = H, H,,

(1)

A basis set of functions that describes the ion configurations, corresponding to localization of the electron on different centers, is usually used to find the eigenvalues of H. For a binuclear cluster these functions are &&,* (=and ‘IJ &*&, ,) (=YI),#i and #i* describing states of ith center without the extra electron and with the electron, respectively. In a homonuclear cluster, Le., with ions a and b being identical, electron localization causes the appearance of cluster dipole moment, d eR, where e is the electron charge and R is the distance from the center of cluster to a metal ion. In this (Yr,Yl)basis the extra electron transfer (tunneling) between the centers looks like

Ht = Pa,

o and p being a Pauli matrix and tunneling parameter. For a symmetrical case, Le., clusters with identical ions in identical environments, this term, acting alone, gives rise to the delocalized, symmetrical or antisymmetrical, states (Y, f Y])with vanishing dipole moments. In the more usual nonsymmetrical case, when the surroundings of the two metal ions are different, the energies of two ionic states are different too. The “crystal field” term of the Hamiltonian accounts for this asymmetry and has the form

H,, = 60: with 26 being the energy gap between the ionic states. The electric field E(t), applied along the cluster axis, also lifts the equivalence of states with different localization of the extra electron, causing localization of charge. The corresponding term has the form

Hf = dEaz with d the dipole moment of the localized state. The electron-electron interaction term He acting on one site (intraatomic exchange) gives rise to two states with spin SO i ] / I , SObeing the spin of the metal core. We take the spin value Si* of the state with lower energy as the ground-state spin of the ith center with an extra electron. The intersite electronelectron correlations, which are much smaller than intraatomic ones, form multiplets of total spin S (Sa Sa* > S > ISa Sa*]) with energies JS(S l), J being the intercenter exchange parameter. Using the basis set of 2(2Sa 1)(2Sa* 1) functions that accounts for the degeneracy connected with different electron localizations and with different projections of ions’ spin moments Sa and Sa*, one can diagonalize the Hamiltonian (1) and obtain the following expression for the electron energy levels7,1I .45-47

+

ccs)(t)= JS(S

+

+

+

+ 1) i @’(S + 1/2)’ -t(dE(t) + 6)2]1’2(2)

-

Note that even at negligible exchange ( J 0) the energies of states with different total spin S do not coincide because the states with different spin multiplicity are split by the tunneling in a different way (-(S l/2)). The existence of several tunnel sets with different spin and spin-dependent tunneling splitting are the special features that determine the peculiarities of dielectric behavior of mixed-valence systems. The expression

+

for the effective dipole moment of the cluster

de, = (dc/dE), = dS/cO,

+

+

co = v2(S 1/2)2 62]1/2(3)

clearly shows the dependence of deff and, as a result of dielectric properties as a whole, upon the spin multiplicity. Equation 3 illustrates also the suppression of the dipole moment by tunneling (deff p-’ at I/?/ >> 6) and the enhancement of deff by ions’ asymmetry, as already mentioned. In the simplest case when the spin of one of the ions, let it be S*, is equal to zero (as it is in low-spin Fe2+-Fe3+ clusters, for example) the total spin has single value SOand the energy spectrum consists of two states with the same spin SO,split by tunnling. The features of the relaxational absorption of an electric field E = Eo cos cot by such two-level systems is wellknown (see, for example, ref 48). The power P , absorbed during a field period, is given by the expression

-

P = (n(t)[dc(t)/dt]) = (o/2n)h2.7/Wdtn(dc/dt)

(4)

In (4) n is the occupation number of the upper level and is determined by the kinetic equation dnldt = -(n - n,)/z

(5)

where t is the relaxation time. The equilibrium occupation no

n,(t) = [l

+ exp(2c(t)/k,T)]-’

corresponds to any value of ~ ( tat) temperature T. In the linear approximation, i.e., under assumption of weak field dE 1 feel only averaged (over time t)field; in such systems the level occupations are almost time independent and the contribution to the absorption is small. As eqs 3 and 6 show, the greater the degree of ion inequivalence, described by the 6 parameter, the greater is deffand the greater (4 are * the ) losses (note that at small 6, such that dE > 6, the linear approximation fails and the dipole moment of the cluster is dominated by the induced term so the losses have to show a nonlinear behavior). Importantly, since deff is proportional to E O - ’ , the increase of tunneling parameter always leads to more symmetrical charge distribution with smaller dipole moment, and the dielectric losses fall off (-p-* at Ipi >> 6)as a result. B. Vibronic Coupling Effects: Temperature Dependence and Barriers. At low temperatures the temperature dependence of the relaxation time is determined by one-phonon transitions; however, temperature increase leads to Arrhenius behavior with 5 = A exp(E,,/kBT), where E,, is the height of the energy barrier between states with different localization of the extra electron. In mixed-valence clusters this (vibrational) barrier appears due to the electron-transfer-induced displacements of the ligand atoms surrounding the metal ion^.^^^^^^^-^^^^^^^^-^^ For linear electron-vibrational couplings, this is usually described by the antisymmetrical vibrational mode Q = (qa - @)/@)I”, where qa,b are the totally symmetrical breathing displacements of the

2658 J. Phys. Chem., Vol. 99, No. 9, 1995 200 , b

Vekhter and Ratner I

H

150

-

100

o_

50

f 1

0

i -50 -100

-150

-20

-40

0 antisy mmetrical coordinate Q

1

20

40

Figure 1. Adiabatic potential curves for states with different spin multiplicity (stars for S = '/2, circles for S = 3/2). The lower, doublewell potentials show smaller activation barriers for the state with larger S, due to dependence of the effective tunneling parameter Peff(=p*(S 1/2)) on the value S of the total spin (Q is dimensionless; V = 15, p = 30, 6 = 20 in units of h a ; J has been taken equal to zero to make the comparison of the potentials for different S more clear).

+

surrounding of ions a and b, which are most strongly coupled to the electron transfer. We use Q as a dimensionless variable, in units of [h/MQ]'/2,M and Q being the mass and frequency of the Q mode. The vibronic (electron-vibrational) interaction term in the (Yr,Yl) basis has the form Hvib=

VQo,

(74

with V being the vibronic constant. One can obtain, with account of this term, the following expression for the dependence of the electronic energies upon displacement Q (compare with eq 2) e(Q) = JS(S

+ 1/2) f

+

+

U2(S 1/2)2 (dE

+ 6 + VQ)2]1'2 (7b)

The antisymmetrical displacement Q causes the inequivalence of a and b ions in the same way as the ion's asymmetry and electric field, which is why the VQ term appears in eq 7b in linear combination with E and 6. Note that for J = 0, d = 0, 6 = 0, and S = l/2, the energies of eq 7b become E(Q$ = 0) = &[p2

+ V2Q2]'/2/2

+ JS(S + 1/2) f P2(S+ 1/2)2 + (dE+ 6 + VQ)2]112(8)

that is shown in Figure 1. We see that the lowest curve possesses two minima correspondingto two possibilities of extra electron localization. If (fil,V/hQ >> dE,6, the minima positions Qo and height Ea, of the barrier between them are given by the following expressions

Qo = [v/(hQ)2 - P2(S Ea, = V2/2hQ

+ 1/2)2]1/2/V

+ P2(S + 1/2)2hQN2 -IPl(S

111. Determination of Transfer Parameters from Experimental Data: Single Pair of Levels (a) Frequency Dependence: Estimation of the Energy Gap and the Tunneling Integral. In accordance with eq 6 the frequency dependence of the two-level absorption has a maximum at wmax= l/z, and the intensity of absorption at this frequency, P,,,, is given by the following expression: P,,, = w,,,d 2E2 62/2e,2kBTcosh2(edkBT)

(10)

At different temperatures the relaxation times are different, so that the maximal absorption must occur at different frequencies. Examining the frequency dependence of P at different temperatures TI and TI, the values of the frequencies Omaxl and ~ m a x 2 , at which the maximal absorption occurs, and the intensities of the absorption,PmUl and Pmax2.at these frequencies can be found. It follows from eq 10 that

cosh(Edk,T,)lcosh(E,,/k,T,) =

As Pma, Wmax, and Ti are known from experiment, eq 11 allows one to estimate EO, the value of the energy gap, which in tum, in accordance with eq 3, depends upon values of 6 and tunneling parameter p. In limit cases of low (kBTi < E O ) and high ( k ~ T i > E O ) temperatures, relation 11 can be simplified, and one finds

(7c)

as is familiar from the simplest two-site electron-transfer discussions.10.49 The sum of E(Q) and vibrational potential hS2Q2/2 gives the total potential energy W(Q) W(Q) = hQQ2/2

and the equilibrium displacements Qo and the barrier Ea, vanish if Ipl(S -t 1/2) L V/hQ. The dependence of QOand E,, upon the value of spin S means that potential W ( Q )for pairs of states with different values of S has different Qo and E,. As Figure 1 illustrates the barrier height essentially depends upon S, so the relaxation times in pairs with different S may differ too. In the next section we show how examination of thefrequency and temperature dependencies of dielectric losses allows estimation of the main parameters of mixed-valence clusters: the energy gap (eq 1l), the activation barrier (eq 15), tunneling parameter (eq 3), the asymmetry parameter (eq 13), reorganization energy (eq 9), and the effective electronic gap, Le., the spacing of electronic levels near the top of the barrier.

-I- 112) (9)

Equation 9 shows that since the transfer term Ht of the Hamiltonian (1) acts against the electron localization, large enough tunneling does not allow the electron to be localized,

and

eo

2kB[(TlT2)2/(T,2- T22)x 1/2

(1 - Pmax10max2/Pmax20maxl)l

9

kBTi >> €0 (12b)

The value of EO being known, one can use eq 10 to estimate the parameter of asymmetry

6 = [2Pmax~:kBT ~ o s h ~ ( ~ d k , T ) / ~ ~ , d ~ E(13) ~]"~ where, as has been mentioned, the cluster dipole moment d sz eR. After the values of 6 and EO have been found from eq 12, the transfer parameter fi can be estimated from eq 3. When vibronic coupling of the form of eq 7a is added to the purely electronic Hamiltonian of eq 1, the relevant states between which transitions occur are now at the bottoms of the double-well potential of Figure 1. To find their energies and wavefunctions it is convenient, for the RobidDay I or I1 case under consideration here, to use the diabatic basis, Le., the functions Y d r , Q ) = Yr(I)WWQ- Qord, where Y r d r )are the localized pure electronic functions defined above and (D

J. Phys. Chem., Vol. 99, No. 9, 1995 2659

Electron Transfer in Clusters are harmonic oscillator functions with displaced equilibrium positions. The displacements account for the vibronic coupling at p = 0, so Qor(l,= fV/hS2. In that basis the energies of two lowest tunneling states are given by the formulas (3), and only two important effects of the vibronic coupling occur in the loss spectra: first is the renormalization (reduction) of the pure electronic tunneling element /3 by vibrational overlap terms (like Franck-Condon factors in optical spectra, Huang-Rhys factors in color centers, or Ham factors in Jahn-Teller system^^^-^^), and second is the appearance of a vibrational activation barrier (Figure 1). The vibronic renormalization can be formally dealt with by replacing p in eq 3 by = PFo. Here FOis the overlap integral of the vibrational component of the localized vibronic states in the minima of Figure 1 computed for p = 0 (that is, Fo = - Qor ) I @ ( Q is the nuclear overlap between diabatic vibrational states; the square of FOis a Franck-Condon factor). Therefore, for our simple one-mode harmonic picture

(@(e

eo]))

p, = p e x p [ - ( ~ / h ~ ) ~ / 2 1

(14)

When vibronic coupling occurs, these renormalization effects mean that dielectric losses determine, directly, p, rather than /3. If the intervalence transfer band has been used (with the Hush formula3) to deduce p, then the parameter (V/hsZ) can be found from knowledge of pvand p (eq 14). This can, coupled with knowledge of hS2 itself, give the reorganization energy. Alternatively, and perhaps more conveniently, temperature dependence can be used to determine (V/hS2), as we now show. (b) Temperature Dependence and Estimation of Activation Barrier. Equation 6 points out two different possible origins for the temperature dependence of P. One is connected with the temperature dependence of the initial level occupancies. One can seen that at very low and very high temperatures the absorption is weak: in the former case the upper state is practically unoccupied, so the field shift of this level does not affect the state populations, while in the latter case the populations of both levels are almost the same and the field has nearly no effect. Most commonly the temperature behavior of P is determined mainly by the strong temperature dependence of the relaxation time t. As a result the temperature dependence of P shows a maximum at some T, when z-l is approximately equal to w. By measuring the temperature dependence of P at different frequencies, one obtains different values of the temperature T,,,, corresponding to the maximum of P . As the maximum of P occurs when wmaxz= 1, for Arrhenius behavior the dependence of upon Tmax looks like Omax = A-' exp(-Eac/kBTmax). Then it follows immediately that In a,,, = -EaJkBTmax - In A

microwave frequency.

single pair has been examined in the section 111. The singlepair situation is realized when one spin Sa or Sa* is equal to zero, or when the exchange parameter J is much larger than kBT and Eac, so that all the properties are determined by the lowest pair of tunnel states with the same value of S. To discuss the features of the more general situation with several pairs of tunneling states, we consider the case of Sa = '/2 and Sa* = 1 that corresponds, for example, to d'-d2 clusters with the highspin d2 configuration. In this case the energy spectrum consists of two pairs of spin multiplets, S = l/2 and 3/2, split by tunneling. Such a situation is more complicated than the simple two-level one considered above. Instead of single pair of levels with single relaxation time, now we have six pairs of states with different, in a general case, times of relaxation inside each pair. Typically S is a nearly good quantum number, so that relaxation between levels with the same spin value occurs much faster than relaxation accompanied by the change of spin multiplicity, and then only two relaxational times appear, 2(1/2) and t(3/2), inside S = l/2 and S = 3/2 pairs. Then each pair of levels with S = '/2 and S = 3/2 gives an independent contribution, P(S), to the total absorption P , that can be written as

where N(SJ is the population of the multiplet Si

(15)

so by examination of the losses at different frequencies and temperatures the activation energy E,, can be easily found (Figure 2). Then, using eq 14, one can rewrite eq 9 in the form

Once the values of E,, are known, we canfind V/hQ from eq 9a, and knowledge of ,Byand hQ. From this, the vibronic reorganization energy follows as VZ/hQ.

IV. Case of Several Pairs of Levels with Different Spins As has been discussed in section 11, the energy spectra of a cluster contains several pairs of tunnel states with different total spin S (Sa Sa* > S > IS, - Sa*[). The relaxation inside a

+

Figure 2. Typical relationship between the temperature T,,,,,, at which the maximum of dielectric losses occurs, and In w ; w is the experimental

and P(S) is given by eq 6 for the two-level absorption. The only way in which P(S) values differ for different S is the relaxation time z(S) and value of S in the formulas for €0. The relaxation times are determined by the corresponding activation barriers. If they are spin independent, Le., are the same for the pair of tunneling states with different spins, all P(S) in eq 15 show the same frequency and temperature dependencies, and the value of the barrier can be found from the frequency dependence of the losses at different temperatures in the same way as for the two single-pair cases. If the frequency dependence of P cannot be fitted by the relaxation formulas with a single activation barrier, dependence of the activation barriers on the spin value is indicated. So it is crucial to determine if t(1/2) and t(3/2) are independent parameters or there exists some ratio between them. It is clear from the preceding discussion of the dependence of potential W(Q)on the value of S that t(3/2) must in principle be shorter than

Vekhter and Ratner

2660 J. Phys. Chem., Vol. 99, No. 9, 1995

I -J

Figure 3. Electronic levels at strong exchange ( J > P.8). Energy spectra show two well-separated (-4 sets of states with different total spin S; each set is split by the tunneling, the splitting (-P,d) is larger for set with larger S. It is the lowest set that contributes chiefly the

dielectric losses. z( 1/2), since the effective tunneling parameter Peff= p(S

+

+ 112)

-

Figure 4. Electronic levels at weak exchange ( J PJ). The energy spectra arise from two overlapping sets of tunneling doublets for each spin S; tunneling for the set with higher S is stronger. Both sets contribute to the dielectric losses with different, in principle, relaxation times; the ratio of their contributions can be changed by an external

magnetic field.

that enters eq 7 is proportional to (S U2). As a result the minima of lowest potential curve are closer to each other for S = 3 / 2 than for S = '/2 and the barrier for S = 3/2 is lower than that for S = '/2 (see Figure 1). This means that both the lowtemperature tunneling relaxation and the high-temperature activational one are faster inside the S = 3/2 split doublet. Using eq 9, one finds for the difference of activation energies, 6Eac (GEac(S = 1/2) - Eac(S = 3/2)),

unusual phenomenon appears: the magnetic field essentially affects the dielectric losses, causing increase of the loss channel with z(3/2). For ferromagnetic coupling (J 2 0) the high-spin state is the ground one, so the magnetic field increases the effective gap between states with different S. Thus no new relaxation channel appears in this case, and the magnetic field does not influence the relaxation.

6Ea, = 1/31 - 3/32hQ/2V2

We have shown that the examination of dielectric losses can provide valuable information about the electron transfer in mixed-valence clusters. In particular, study of the frequencydependent loss for various temperatures permits (in the Robid Day class I or I1 cases considered here) experimental estimation of the important parameters of the effective multisite electronic Hamiltonian: p (the tunneling parameter) and the electronic asymmetry, or strain, parameter 6 (see refs 55-57 on the important role of strains for the vibronic systems). The FrankCondor factors can be found either from the temperature dependence or from comparison with intervalence transfer band intensities. The effective activation energy and therefore the reorganization energy can also be found, directly from the temperature dependence. Note that to estimate the value of the reorganization energy we use formulas of a single-vibrational model. This model is widely explored in the mixed-valence cluster theory; however, for more accurate estimations one has to account for a many-mode vibronic coupling. At the same time we emphasize that eqs 12 and 13, which allow estimation of the energy gap and asymmetry parameter, are of a general form and do not use the single-mode model. The same is true for the activation energy found from eq 15. A very simple example of a binuclear cluster has been considered. It may be expected that the peculiarities of dielectric losses will be even more pronounced in polynuclear clusters with a greater number of metal ions, higher values of individual ion spin S,,a much more complicated energy spectrum, and, as a result, a greater number of different relaxation times and activation energies. The observability of the dielectric losses discussed here clearly will differ from system to system: (1) If the local site energy differene 26 were to vanish, the treatment given here is inapplicable, since the assumption above eq 6 fails; thus generally one must choose slightly asymmetric ligation environments around the metal centers. (2) If the barrier height in Figure 1 becomes too large, the frequency umax of eq 15 may

(18)

We see that the value of 6EaCvaries from IpI (for IpI >\PI ,d) the energy spectrum contains two well-separated ( W ) tunnel doublets with different spins; the gaps between components of doublets are determined by p and 6 parameters (Figure 3). At ~ B T 6,lPI the lowest doublet is populated only, so the dielectric losses are described by equations found above for the two-level case with z = 2(1/2) or z(3/2), depending upon the sign of J. When the temperature increases and kBT becomes greater than d,l/3l the dielectric losses in the ground doublet decrease due to the almost equal population of both active states. At ~ B xT J , the second tunneling doublet with another value of S becomes populated and at a first glance the losses with corresponding z(S) might be expected. However, it is easy to see that they are small to the same extent as they are for the ground state, since the difference between occupations of two components of this excited multiplet is proportional to exp( - c / k ~ T )and this parameter is small at ~ B =TJ >> 6,lpl. At J d,lpi the split components of different spin multiplets are close one to another and even the component of the lowspin state can lie inside the components of high-spin multiplet (Figure 4). At ~ B xT 6,lpl all four states are populated, so one has to expect losses with both relaxational times z(1/2) and z(3/2). In such a case the examination of the temperature dependence of the ratio of intensities losses inside each pair with different spins can be used for estimating the J value. Note that if J is negative, the upper level has higher spin than the lower one, and under a magnetic field its lower component S, = -3/2 approaches the S = ' / 2 ground state. As a result an

-

V. Conclusions

Electron Transfer in Clusters

J. Phys. Chem., Vol. 99, No. 9, 1995 2661

be smaller than can be measured conveniently by using a standard microwave apparatus. (3) Other types of dielectric losses may also enter and might obscure the signal from the mixed-valence electron.58

Acknowledgment. This work was supported by the Chemistry Division of ONR and of NSF. It is a pleasure to dedicate this paper to Stuart Rice, whose major contributions to physical chemistry, to science, and to the University of Chicago have always been demarked by insight, understanding, imagination, and excellence. References and Notes (1) Robin, M. B.; Day, P. Adv. Inorg. Chem. Radiochem. 1967, 10, 247. (2) Hush, N. S. Chem. Phys. 1975, 10, 361. (3) Hush, N. S. Prog. Inorg. Chem. 1967, 8, 391. (4) Allen, G. C.; Hush, N. S. Prog. Inorg. Chem. 1967, 8, 357. ( 5 ) Mixed-valence compounds; Brown, D. W. Ed.; Reidel: Dordrecht, 1979. Crutchley, R. J. Adv. Inorg. Chem. 1994, 41, 273. (6) Mixed-Valence Systems: Applications in Chemistry, Physics and Biology; Prassides, K., Ed.; Kluwer: Dordrecht, 1991. (7) Blondin, G.; Girerd, J. J. Chem. Rev. 1990, 100, 1359. (8) Mikkelsen, K. V.; Ratner, M. A. Chem. Rev. 1987, 87, 113. (9) Cannon, R. D. Electron Transfer Reactions; Butterworths: London, 1980. (10) Schatz, G. C.; Ratner, M. A. Quantum Mechanics in Chemistry; Prentice-Hall: Englewood Cliffs, NJ, 1993; Chapter 10. (11) Munck, E.; Papaefthymiou, V.: Surerus, K. K.; Girerd, J. J. In Metals in Proteins; Que, L., Ed.; Washington, DC; 1988; Chapter 15, p 302. (121 Todd. M. D.: Mikkelsen. K. V.: HUDD.J. T.: Ratner. M. A. New J . Chem. 1991, 15, 97. (13) Joachim, C. Chem. Phvs. Lett. 1991, 185, 569. (14) Launay, J. P.; Tourrei-Pagis, M.; Lipskier, J. F.; Marvaud, V.; Joachim, C. Inorg. Chem. 1991, 30, 1033. (15) Launay, J. P.; Babonneau, F. Chem. Phys. 1982, 67, 295. (16) Jortner, J. J . Chem. Phys. 1976, 64, 4860. (17) Van Duyne, R. P.; Fischer, S. F. Chem. Phys. 1974, 5, 183; 1977, 26, 9. (18) The subject of polaron-type models for electron-transfer rates has been extensively reviewed. For example, compare refs 8 and 9 and Newton, M. D.; Sutin, N. Annu. Rev. Phys. Chem. 1984, 35, 437. Marcus, R. A.; Sutin, N. Biochem. Biophys. Acta 1985,811,265. DeVault, D. C. QuantumMechanical Tunneling in Biological Systems; Cambridge University: New York, 1984. Ratner, M. A. Naval Res. Rev. 1993,4,49. Ulstrup, J . Charge Transfer processes in Condensed Media; Springer: New York, 1979. (19) Salaymeh, F.; Berhane, S.; Yusof, R.; de la Rosa, R.; Fung, E. Y.; Metamoros. R.: Lau. K. W.: Zheng. 0.: Kober. E. M.: Curtis, J. C. Inorn. Chem. 1993, 32, 3895. (20) Curtis, J. C.; Sullivan, B. P.; Meyer, T. J. Inorg. Chem. 1983, 22, 224. (21) Curtis, J. C.; Roberts, J. A,; Blackboum, R. L.; Dong, Y.; Massum, M.; Johnson, C. S.; Hupp, J. T. Inorg. Chem. 1991, 30, 3856. (22) Curtis, J. C.; Meyer, T. J. Inorg. Chem. 1982, 21, 1562. (23) de la Rosa, R.; Chang, P. J.; Salaymeh, F.; Curtis, J. C. Inorg. Chem. 1985, 24, 42331. (24) Nakano, M.; Sorai, M.; Hagen, P. M.; Hendrickson, D. N. Chem. Phys. Lett. 1992, 196, 486. (25) Dong, T.-Y.; Hendrickson, D. N. Bull. Inst. Chem., Acad. Sin. 1989, 36, 73. (26) Sorai, M.; Kaji, K.; Hendrickson, D. N.; Oh, S. M. J , Am. Chem. SOC.1986, 108, 702. I

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