Radiative and nonradiative electron transfer in contact radical-ion pairs

Chemical Physics 176 (1993) 439-456 Non-Hood

Radiative and nonradiative electron transfer in contact radical-ion pairs Ian R. Gould a,b, Dimitri Noukalcis a,‘, Luis Gomez-Jahn a+, Ralph H. Young d, Joshua L. Goodman ‘pcand Samir Farid a,b * Centerfor Photoinduced Charge Transfer, University OfRochester, Hutchison Hall, Rochester, NY 14627, USA b Corporate Research Laboratories, Eastman Kodak Company, Rochester, NY 14650-2109, USA c Department of Chemistry University of Rochester, Rochester8NY, 14627, USA d @j&e Imaging Research and Technology D#elopment, Eastman Kodak Company, R~hester~ NY 14650-2129, USA Received 15April 1993

The relationship between radiative and nonradiative electron transfer is explored for return electron transfer processes in the contact radical-ion pairs formed by excitation of ground state CT complexes. Using a conventional nonadiabatic theory of electron transfer, absolute rate constants for normative return electron transfer, varying over more than two orders of magnitude, can be predicted from information obtained from analyses of the corresponding radiative processes. The effects of solvent polarity, driving force and molecular dimension on the rates of nonradiative return electron transfer are studied.

1. Introduction With the shift in emphasis from second-order to first-order reactions, many of the long-standing predictions of electron transfer theories have now been realized, such as the presence of an inverted region, and the effects of changing solvent and the distance between reactants (see for example refs. [ l-7 ] ) . One consequence of this recent progress has been a revival of interest in the relationship between the rates of thermal electron transfer reactions and information obtained from corresponding optical transitions. The chemical systems most commonly associated with such optical electron transfer processes are binuclear transition metal complexes exhibiting intervalence absorption bands [ 8 1. A considerable body of evidence concerning the factors which influence electron transfer reorganization parameters has been accumulated from analyses of intervalence band shapes and energies [S-10]. Furthermore, information concerning the magnitudes of the electronic couplings in such systems has been obtained from analyses of the strengths of the absorption bands [&IO]. Despite extensive spectroscopic work in this area, however, absolute rate measurements of electron

transfer processes in intervalence compounds have been made only recently [ 111. Nevertheless, using data from a combination of steady-state absorption and resonance Raman spectroscopies, the experimental rates of inter- and mixed-valence electron transfer have been successfully simulated [ 1l- 13 1. Numerous measurements have been made, however, of the rates of intramolecular charge transfer (CT) in the excited states of Mets-to-li~nd chargetransfer (MLCT ) complexes. Furthermore, considerable progress has been made in relating the rates of nonradiative (CT) decay in these species, to information obtained from spectral fitting of the corresponding radiative processes, i.e. the LMCT emission spectra [ 14,15 1. Related approaches have been followed in studies of intr~olec~ar electron transfer reactions in rigidly linked organic donor/accep tor systems. Electronic coupling matrix elements and electron transfer reorganization energies have been estimated from studies of intramolecular electronic CT transitions [ 16-l 81. In general, however, the charge-transfer abso~tion or emission processes which correspond to such intramolecular electron transfer reactions have been described in only a few systems. As a result, the number of studies in which

0301-0104/93/% 06.00 0 1993 Elsevier Science Publishers B.V. All rights reserved.

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I.R. Gouldet al. /Chemical PhysicsI76 (1993) 439-456

electron transfer rates have been related to spectroscopic measurements in intramolecular electron transfer is quite small. The relationship between radiative and non-radiative electron transfer in intermolecular electron transfer processes was discussed by Marcus, with particular emphasis on electron transfer processes in ground state CT complexes [ 19 1. The return electron transfer reactions in the contact radical-ion pairs which are formed upon excitation of such complexes exhibit many of the properties of reactions in rigidly linked donor/acceptor systems. Although bimolecular, the reactions are first order, and in low polarity solvents, separation of the contact radical-ion pairs into looser structures (solvent-separated radical-ion pairs) does not occur [ 201. Charge-transfer absorption spectra are (by definition) available and the corresponding charge-transfer emission spectra have been observed in several cases [ 2 11. The rates of return electron transfer in the radical-ion pairs formed by excitation of CT complexes have also been measured for many systems (see for example refs. [ 6,2229] ). It is surprising, therefore, that there have been very few quantitative studies in which the rates of electron transfer in such species have been related to information from spectral fitting of the corresponding CT absorption or emission spectra [ 61. In this work we describe this relationship in detail for the return electron transfer reactions in the contact radical-ion pairs (CRIP) formed upon excitation of such ground state CT complexes. From an analysis of the CRIP charge-transfer emission spectra, information concerning the reorganization parameters for the electron transfer process can be obtained. In addition, estimates of the electronic coupling matrix elements for non-radiative electron transfer in the CRIP can be obtained from the CT radiative rates. Using these data, predictions can be made for the rate constants for nonradiative electron transfer in the CRIP, that are remarkably close to the experimentally measured values. In this way the molecular properties which influence the rates of nonradiative return electron transfer can be studied in detail.

2. Experimental The acceptors used in the present study are

2,6,9,10-tetracyanoanthracene (TCA) and 1,2,4,5tetracyanobenzene (TCB). TCA and the methylsubstituted benzene donors g-xylene (p_Xy), 1,2,4-trimethylbenzene (TMB ) , durene (Dur ) , pentamethylbenzene (PMB) and hexamethylbenzene (HMB) were available from previous work [ 6,7]. TCB (Frinton Laboratories) was recrystallized three times from chloroform. Hexafluorobenzene, trichloroethylene and diethyl carbonate were obtained from Aldrich. Cyclohexane, carbon tetrachloride, chloroform, butyl acetate and and 1,2-dichloroethane were all HPLC grade from Aldrich. All solvents were freshly distilled prior to use. Experiments were performed in 1 cm2 cuvettes equipped with stopcocks for purging. All solutions were purged with argon for at least 10 min to remove dissolved oxygen. Absorption spectra were recorded using a Perkin Elmer Lambda 9 spectrometer. Steady-state emission and excitation spectra were recorded using a Spex Fluorolog 2-l-2 spectrometer. Corrected spectra were obtained by using the procedure recommended by the manufacturer. In some experiments, excitation was performed at wavelengths where the uncomplexed acceptors absorbed. In these cases, any emissions from unquenched acceptor singlet states were subtracted. Absolute emission quantum yields, @r, were obtained by using 9,1 O-dicyanoanthracene in air saturated acetonitrile as a reference. The emission quantum yield for this system was determined to be 0.80 f 0.06 by measurements relative to quinine sulfate in 1N sulfuric acid (@r=O.SS f 0.03) and fluorescein in 0. 1N sodium hydroxide (@r= 0.92 + 0.03 ) [ 301. For the emission experiments, the concentrations of the. TCB acceptor ranged from % 1OM3M to < 10d4 M, depending upon its solubility in the different solvents. The concentration of the TCA acceptor was x 1Om4M. The concentrations of the donors were typically 0.01 M. At this concentration, complications due to the formation of 1: 2 complexes were minimized [ 3 11. Using TCB as the acceptor, excitation was performed both in the CT bands of the complexes (A> 330 nm) and also in the absorption bands of TCB (2~ 320 nm). In the latter case, excited CT complex emission is observed after diffusive encounter of the donor with excited TCB, as reported previously [ 3 11. At the donor concentrations used, only a few percent of the TCB is complexed. In addition, the extinction coefficient of the uncomplexed TCB at

I.R. Gouldet al. /Chemical Physics176 (1993) 439-456

its absorption maximum ( x 3 16 nm) is higher than that of the complex so that most of the excitation light is absorbed by the uncomplexed TCB. The lifetime of the singlet excited state of TCB, measured by single photon counting, is 4.3 + 0.1 ns in chloroform. Thus, a donor concentration of 0.01 M quenches = 40% of the TCB singlet states. For all but three of the systems discussed here, the emission spectra were identical for the two excitation conditions. For the TCB/HMB complex in chloroform, butyl acetate and dichloroethane, the maximum of the spectrum obtained with excitation of the uncomplexed TCB occurred at lower energy than with excitation in the CT band. These observations are attributed to the involvement of two excited CX species, as discussed further below. For TCA as the acceptor, the emission spectrum was identical at all excitation wavelengths. The emission spectra were measured as photons per unit time per unit wavelength (In). The spectra were converted to photons per unit spectral energy, Zr,by dividing Z, by v* [ 323 and then normalizing to unity. A plot of Z~/Yas a function of v can be made and is called the “reduced emission spectrum”, see below. An average emission frequency, v,, (defined below), can be obtained from the measured (IA) or the reduced ( Zr) emission spectra. Alternatively, v,, can be calculated from the electron transfer parameters used to simulate these spectra, as shown in section 3. This latter procedure was used when a subs~nti~ part of the emission spectrum could not be measured, i.e. when the emissions extended significantly beyond 800 nm, the detection limit of our spectrometer. For the CRIP included in the present study, the maxima of the reduced spectra (Zylv) are between 0.5 and 1.1 kK lower in energy than those of the measured speo tra (Z,). The energies of the average emission frequencies (hv,,) are between 1.0 and 1.5 kK lower than those of the ZAspectral maxima. In both cases these differences increase with increasing reorganization energies. Emission lifetimes were measured using the technique of single photon counting, using an apparatus which has been described previously [ 33 1. Typically the excitation wavelength was 378 nm. The emissions were monitored with a bandpass of 4 nm from 450 to 800 nm, depending upon the donor, the acceptor and the solvent. For most of the CT complexes, the emission decays could be resolved into a fast

441

component and a slow component. This behavior is attributed to the involvement of two excited CT complexes, as mentioned above. For some of the longer lived components, the observed lifetimes were found to depend upon the donor concentration. For those systems, the lifetimes were extrapolated to zero donor concentration. For the example of the TCB/pxylene complex in chloroform, varying the donor concentration in the range between 0.003 and 0.48 M decreased the measured lifetime from 8.0 to 6.6 ns. From a plot of decay rate versus donor concentration, a value for the lifetime extrapolated to zero donor concentration of 8.3 ns is obtained, and the selfquenching rate constant is determined to be (6If:1)X10*M-‘s-‘. Intersystem crossing quantum yields were estimated using pulsed photoacoustic calorimetry, using an apparatus which has been described previously [ 34,351. The excitation wavelength was 355 nm. The optical densities of the sample and calibration solutions (2-hydroxybe~ophenone, Aldrich) were adjusted to be within 1% of each other. The TCB/p-xylene complex in chloroform was studied in detail. Irradiation of TCB ( = 10m4M) in the presence of pxylene (0.2 M, OD 0.15 at 355 nm) in argon-degassed chloroform resulted in the conversion of >95% of the photon energy into heat with a time constant of x 10 ns. Assuming the residual energy (5%) is due to TCB triplets formed upon intersystem crossing in the TCB/pxylene CRIP, and taking the triplet energy of TCB to be 2.8 eV (see below), the yield of TCB triplets is thus determined to be < 10%. Similar experiments were performed using 1,2,4-trimethylbenzene and durene as donors. In each case the yield of TCB triplets was < 10%. A similar procedure was followed for TCA as the acceptor except that in addition, transient absorption spectroscopy was employed in this case:. Further details of the experiments on the intersystem crossing in the CRIP of TCA will be published elsewhere [ 36 1.

3. Theoretical background Most current theories of electron transfer cast the rate in the form of a golden rule expression such as

I.R. Gouldet al. /Chemical Physics176 (1993) 439-456

442

V*FC(g) ,

k- et - 7 g=

quency Y, the rate constant Z, (per unit spectral energy) can be written in the form,

(lb)

AG-et ,

FC(g)=

64x4 z,= 3h~3 n3y3M: Fe(g) , g=AG_,,+hv,

f Fj(4x;l,kBT)-“2

(2b)

JCO

x exp Fp

e -5YJ -,

J?

M,==z,

(jZ2V”+g+&)Z 4&k,T

>’

(lc)

&! S’hv,’

i.e. as the product of an electronic coupling matrix element squared, V2, and a Franck-Condon weighted density of states, FC(g) [ 37-391. The Franck-Condon term is a function ofAG_,, the free energy change associated with the electron transfer reaction, eq. ( lb). In a typical description, FC(g) also depends on a reorganization energy, A,, associated with solvent and other low-frequency modes, and a reorganization energy, 1,, associated with a single averaged high frequency (skeletal) mode of frequency v, [ 1,2 1. The other symbols in eq. ( 1) are h which is Plan&s constant, kB which is Boltzmann’s constant and j, which is the number of quanta of the v, mode. It is commonly assumed that vv is the same for the reactants and products and, at room temperature, only the lowest level of this high frequency mode is populated in the reactants, whereas several levels of this mode are produced in the products to varying extents, depending upon AG_,. Under these conditions, the Franck-Condon term is given by eq. ( 1c) . A plot of the FC(g) (and thus k_,) as a function of AC_, shows an initial increase with decreasing AG_, for endothermic reactions, which continues as AG_,, becomes negative, i.e. as the reaction becomes exothermic, When AG_, is close to &+A, however, FC(g) reaches a maximum, and as AG_et becomes yet more negative, FC(g) then decreases. This decrease in FC(g) (and Zc_,) for -AG_,>A,+A, corresponds to the well known Marcus “inverted region”. When an electron transfer reaction occurs in the “inverted region”, the return electron transfer reaction connecting AD and A.-D’+ can also occur radiatively, i.e. a charge-transfer (CT) emission band may be observed [ 191. For the radiative electron transfer process represented by emission at fre-

which is closely related to that of the nonradiative process, eq. ( 1) [ 19,401. Here n is the solvent refractive index, Iw, is the electronic transition moment, A,# is the difference in dipole moment between the CRIP and the neutral AD, and c is the speed of light. The dependence of FC(g) on energy is indicated in eq. (2b). Z, depends upon the emission frequency in three ways (1) the well-known v3 dependence for emission processes (eq. (2a) ), (2) the dependence included in the Franck-Condon term, FC(g) (eq. (2b) ), and (3) the dependence included in the transition moment, M, (eq. (2~) ) . The appearance of a ~e~e~~-~e~e~~en~ transition moment is discussed in Appendix A, where a derivation of eq. (2) is given. Rewriting eq. (2) to obtain a quantity that depends on frequency only via FC (g) , If/v, results in the following equation: 64x4 -4 = ~rz3V2A~*FC(g). V

(3)

A plot of Zr/v versus v is termed a ‘“reduced emission spectrum”. Therefore, the Franck-Condon term which determines the dependence of the nonradiative electron transfer rate on AG_, also determines the probability of emission as a function of emission energy (via the quantity (AG_,, + h v) ), i.e. the spectral distribution of the emission band. Thus, an analysis of a reduced CT emission spectrum can provide estimates of the relevant reorganization parameters for the ~~esponding nonradiative electron transfer reaction. The extinction coefficient, E,for the corresponding CT absorption spectrum can be written analogously, 8Nn3 nV* Ah2FC(g) , ev= 3000 h*cln 10 g=AG_,,--hv,

(4b)

in which N is Avogadso’s number (see Appendix A).

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I.R. Gouldet al. /Chemical PhysicsI76 (1993) 439-456

ASand A,, eq. (5d). Clearly, I, and 4 are required only as relative intensities. The absolute rate scale enters with h. The derivation of eq. (5a)- (5~) is given in Appendix B. Expressions similar to eqs. (5a) and ( 5d) have been given previously [ 17,19 1, in which the frequency dependence of the transition moment is neglected and the emission maxima appear instead of the average frequencies.

The quantity e v is a “reduced absorption spectrum”. In some of the experiments discussed here, CT excitation spectra are recorded. In this case the reduced excitation spectrum is given by I&J, where Z, is the relative intensity in the excitation spectrum. The reduced absorption and emission spectra should exhibit an exact mirror image relationship. The midpoint energy of these two spectra gives the free energy for the return electron transfer process, -AG_, [ 191. The electronic coupling matrix element V can also be estimated from CT emission data, using the relation

4. Results and discussion 4.1. Determination of return electron transfer rate constants.

n3v,V2 Ap2,

The structures and redox potentials of the cyanoaromatic acceptors (A) 2,6,9,1 O-tetracyanoanthracene and 1,2,4,5_tetracyanobenzene and the methyl-substituted benzene donors (D ) used in the present study are given in scheme 1. Ground state CT complexes (AD) are formed when the acceptors and donors are mixed in homogeneous solution. The excited states of such ground state CT complexes are essentially contact radical-ion pairs (A--D’+, CRIP ) [ 42 1. The common deactivation mechanisms for the CRIP are summarized in scheme 2. These are nonradiative (k,) and radiative (k) return electron transfer to reform the ground state CT complex, and intersystern crossing to form a locally excited triplet state, ki,. Intersystem crossing is likely to be efficient only when the energy of the CRIP is greater than that of the lo-

v-‘Irdv,

Vav=

j.v-21~dv/~v-31~dvj

hv,=-AG_,,-A,-1,,

(5d)

in which kr is the radiative rate constant. In eq. ( 5a), the average frequency of the reduced spectrum, vaV, is given by eq. (5b) or eq. (5c), for Zror ZLexpressed in photons per unit time per unit spectral energy or wavelength, respectively (see section 2 ) . Altematively, from eqs. ( 1 )- ( 3 ) , it can readily be shown that the average emission energy, h v, is related to AG_,,,

..&

CN

CN

E

red

NC

A

q

TMB

P-XY

TCA

-0.44

\

CN

NC xx ’

CN

Eox D = 2.06

n

TCB EredA = -9.64 Scheme 1. Cyanoaromatic acceptors and methyl-substituted dation potentials, EE, are in Vversus SCE [ 7,411.

1.92

&

Dur EoxD = 1.78

$

PMB

HMB

1.71

1.59

benzene donors used in this work. The reduction potentials, Ep,

and oxi-

LR. Gouldet al. /Chemical Physics176 (I 993) 43%456

444

~

A._D’+

/

+-

>

AD + heat

nonra~a~e return electron transfer

AD+ hv

radiative return electron transfer

3A’D

intersystem crossing

A-(S)D’+

formation of a solvent-separated radical-ion pair

\ x

Scheme2. tally excited triplet state of the acceptor. For the present systems, the energies of the triplet states of the cyanoaromatic acceptors are lower than those of the donors so that 3A*, rather than 3D*, is the expected product of intersystem crossing. The energy of the triplet state of TCB, 2.8 eV [ 43 1, is higher than the energies of all of the TCB CRXP studied here (see below) except for the cases of pxylene and 1,2,4-t& methylbe~ene as donors in chloroform. As discussed in section 2, pulsed photoacoustic experiments revealed that the triplet yield is still very small ( < 10%) even for these systems. Therefore, the contributions of intersystem crossing to the decay rates of the CRIP of TCB were ignored. For the TCA/HMB in trichloroethylene CRIP, however, the intersystem crossing quantum yield (0.18) is significant and therefore is taken into account in determining k_, from the measured fluorescence lifetime. The tluorescence quantum yields for most of the CRIP included in the present study are very small ( < 1%) and therefore the contributions of emission to the CRIP decay rates are negligible. Only for TCB/HMB in cyclohexane and carbon tetrachloride, and for TCA/ HMB in trichlorethylene are the fluorescence quantum yields higher. In these cases the contribution of the radiative decay to the measured lifetime is taken into account in dete~ining k_, (see below). In polar solvents, CRIP solvation to form a solvent separated radical-ion pair (SSRIP, A’-(S)D”), k,olvrmay also occur [ZO]. The rate of solvation depends upon the energy difference between the CRIP and the SSRIP, and in low polarity solvents, the formation of SSRIP is energetically unfavorable. The kinetic studies described here are limited to solvents of low to moderate polarity, so that the contribution of

the solvation reaction to the decay processes of the CRIP can be ignored. Accordingly, the lifetime of the CRIP is dominated by nonradiative return electron transfer, with a few exceptions where intersystem crossing and fluorescence account for Q 20% of the decay rates. Excitation of the complexes studied here results in emission which can be observed in various solvents. A typical case is illustrated in fig. 1. The CT absorption and emission spectra, when plotted in the reduced form, exhibit an approximate mirror image relationship. Either spectrum may be analyzed for the reorganization parameters, and AG_,, may be obtained from the crossing point (midpoint) of the two reduced spectra, whenever both spectra represent the same GRIP species, and whenever the mirror image

26

24

22

20

18

16

14

12

Wavenumber, kK Fig. 1. Reduced excitation and emission spectra (defined in the text ) for the excited CT complex of 1,2,4,5-tetracyanobenzene/ hexamethylbenzene in carbon tetmcbioride at room temperature.

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I.R. Gouldet al. /Chemical Physics176 (1993) 439-456

relationship is exact. However, close examination of the reduced spectra reveals that they are not exact mirror images. Furthermore, there is considerable evidence that for the CT complexes of TCB, the CRIP species which is reached in CT absorption relaxes rapidly to a second CRIP species. This is the conclusion from previous studies on the emission spectra of CT complexes of TCB in various media [ 43-47 1, and in which time-dependent absorption spectra of the excited CT state were observed in transient absorption experiments [ 48 1. The results of the present work are also consistent with this conclusion. As mentioned in section 2, time-resolved emission experiments on the TCB complexes clearly indicated the presence of two components. A fast component was observed as a decay when the emission was monitored on the high energy side of the spectra, and as a growth when monitored on the low energy edge of the spectra. At intermediate energies, the fast component could not be clearly distinguished. Excitation of the CT complex presumably leads to a primary excited charge-separated species ((A.-D’+)*) that subsequently relaxes to a second species ( (A’-De+),,), which emits at lower energy. The fast component observed in the time-resolved emission experiments is consistent with the process (A’-D’+)r-t (A.-D’+),*. The slow component would then correspond to the decay of the (A--D’+ )I1 species. The observed lifetimes for the slow component varied over a wide range, from x 150 ps to z 50 ns, depending upon the acceptor, the donor and the solvent. The corresponding variation in the fast component, however, was only from x 30 ps to x 60 ps. The lifetime resolution of our apparatus is = 30 ps, and so these lifetimes, and their dependence on excited CT complex structure and solvent, could not be determined accurately. As discussed in detail below, the lifetime of the slow component varies in a manner which is consistent with a return electron transfer transition from an A’-D’+ state to an AD ground state, k_,,. The relative invariance of the lifetime of the fast component is more consistent with the relaxation between the two CRIP species. Both A.-D’+ species should, in principle, contribute to the total steady-state emission spectrum. However, assuming that they have similar radiative rates, the contribution to the total emission spectrum from an A.-D’+ species with a lifetime of z 50 ps will be less than 1O%,

if the lifetime of the longer lived A.-D’+ species is greater than 500 ps. In fact the measured lifetimes of the longer lived components are considerably greater than this for most of the systems studied here, and so the contribution of the short lived component to the steady-state spectrum is negligible in most cases. In support of this idea are experiments in which the emission spectra for excitation of the AD complex and of the uncomplexed acceptor are compared. Excitation of the uncomplexed acceptor, followed by electron transfer quenching upon diffusive encounter with D, also results in excited CT complex (CRIP) emission (see section 2 for further details). The emission spectra obtained from these two excitation methods are very similar when the lifetime of the longlived component is greater than x 500 ps (i.e. the majority of the cases studied here). When the lifetime is shorter than this, however, the emission spectrum obtained via excitation of the CT complex has a maximum at higher energy than that for excitation of the uncomplexed acceptor. This behavior was observed for three of the systems discussed in the present work, i.e. for the TCB/HMB complex in chloroform, butyl acetate and dichloroethane solvents. The most likely explanation of these observations is that the CT excitation route (a, scheme 3) results in the formation of (A.-D’+ )r and subsequently (A.-D’+),,, whereas the diffusive quenching route (b, scheme 3), results only in the formation of (A.-D’+ )rr. When the (A’-D’+)rr state is long lived, the emission from the (A’-D’+)r state makes a negligible contribution to the total. When the (A.-D’+ )rr state is short lived, the (A.-D’+), state contributes significantly to the steady-state spectrum for excitation route a, and an overall higher energy emission is observed. Although we have no independent proof for scheme 3, the results described here are most reasonably explained on this basis. The important return electron transfer reactions, therefore, occur in (A’-D’+)u. This species is re-

AD + hv

a

A* + D

-

(A--D’+),

b

Scheme 3.

(A“D’+),,

I.R. Gould et al. /Chemical Physics I76 (1993) 439-456

446

ferred to subsequently in this paper simply as the CRIP. The lifetime of this species is readily obtained as that of the long lived component in the time-resolved emission studies, The emission spectrum of this species is obtained by excitation of either the AD complex or the uncomplexed A when the lifetime of the GRIP is long, or by excitation of A with diffusive quenching when its lifetime is short. The reorganization parameters for the return electron transfer reaction are estimated from fitting these emission spectra only. The crossing point of the reduced emission and absorption spectra is used only as a first estimate of -AG_,, which is further refined in the spectral fitting. 4.2. Eflect of changing solvent As indicated above, excitation of the complexes results in emission which can be observed in solvents with a wide range of polarity, from cyclohexane to dichlor~thane. The solvents studied are indicated in table 1. These spectra were examined in detail for the TCB/HMB system. As expected from previous studies of this and related systems [43-471, the emissions occur at longer wavelength and with decreasing intensity as the solvent polarity increases. In addition, the spectral width increases with increasing solvent polarity, which indicates that the reorganization

energies for the return electron transfer reactions increase [ 19,40 1. This was confirmed by emission spectral fitting studies, The CT emission spectra were fitted using eq. (2). The v, was fixed at 1400 em-‘, which is a typical value for a skeletal stretching mode of the ring systems involved [ 6 1. In preliminary fitting, A,, 1, and -AC_, were all adjusted. The range of 1, values which gave acceptable fits to the spectra for all of the solvents was found to be rather narrow, i.e. 0.3 I &0.03 eV. To refine the fitting procedure, A, was fixed and -AG_, and ;L, were then varied to give the best fit. It was found that when -AG_,, was increased, 1, also had to be increased by the same amount in order to fit the measured spectra. The range of -AC_,, and Iz,, which gave acceptable fits for each fixed value of A,, was x kO.02 eV. The average of 0.31 eV for 1, was chosen as a fixed value for the final fitting, both to reduce the number of variable parameters, and because it is unlikely that jt, would depend strongly upon the solvent polarity. Typical fits to the spectra for the average value for J., of 0.3 1 eV are shown in fig. 2, and the fitting parameters for each solvent are summarized in table 1. The effect on -AG_,, and 1, of fixing& at 0.3 I eV is iIlustrated by the examples given in table 2. In table 2, the optimum values of these parameters are given for the values of il, which give the best fits to the spectra in carbon tetrachloride and

Table 1 Calculated and measured rate constants for nonradiative return electron transfer in the contact radical-ion pairs of 1,2,4,5-tetracyanobenzene and hexamethyibenzene in different solvents Solvent

- AG_, a) (eV1

&a’ (W

(L&U b, (IO’S-1)

(L,),, (10’S’)

cyclohexane carbon tetrachloride hexatluorobenzene trichloroethyiene diethyt carbonate chIoroform butyl acetate 1,;l-dichloroethane

2.57 2.50 2.48 2.52 2.46 2.43 2.47 2.45

0.14 0.16 0.26 0.35 0.36 0.44 0.43 0.53

0.64 2.1 11 26 2:: 0 130 540

% 1.7 d) 3.8 e) 19 35 130 200 310 770

=)

*) Optimum value for spectral fitting with 1,=0.31 eV and v,= 1400 cm-‘. ‘) Calculated using eq. ( 1), with 0.3 1eV for 2, and 1400 cm-’ for v, and 860 cm- ’ for K ‘) The contact radid-ion pair lifetimes are the reciprocais of these numbers, with the exceptions noted where correction was made for the GRIP fluorescence. ‘) The contact radical-ion pair lifetime is 49.5 ns. The emission quantum yield is zzs0.14. Cl The contact radical-ion pair lifetime is 24.4 ns. The emission quantum yield is 0.068. f~Thecalculatedvalueis195~107s”‘if800cm-’isusedforV(table4).

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I.R. Gouldet al. /Chemical PhysicsI 76 (1993) 439-456

-AG_,

22

fkKf

CCI,

CGF,

BuAc.

DCE

20.16

20.00

19.92

t 9.76

0.16

0.26

20

18 Wavenumber,

sents transfer of an electron over a distance of w 3 A, and is reasonably consistent with literature estimates for related systems [ 49 1. The calculated values for Y are summarized in table 3. We assume that the average, 860 + 30 cm- ‘, is appropriate for the TCB/HMB CRIP in all of the different solvents studied. Using this average V and the reorganization parameters given in table 1, the rates of return electron transfer in the TCB/HMB CRIP in the various solvents can be calculated, (k_et)dc, using eq. (1). These are summarized in table 1. The energies of these CRIP are lower than that of the triplet state of TCB, and in all but two cases the emission quantum yields are small. As discussed above, therefore, the rates of decay of these CRIP are dominated by nonradiative return electron transfer. The reciprocals of the measured lifetimes, therefore, can be equated with the actual rates of return electron transfer, (k_, ),,, with appropriate correction for the radiative decay where necessary, table 1. The calculated return electron transfer rate constants agree to a remarkable degree with the measured values, over a range of more than two orders of magnitude, providing support for the present theoretical approach (see further below). Furthermore, the results of the spectral fitting appear to be at least qualitatively reasonable. The - AG_,, increase and the a, decrease with decreasing solvent polarity, as expected. The solvent reorganization energies in the lower polarity solvents such as cyclohexane, are larger than would be predicted using, for example, a dielectric continuum model, however. This illustrates the fact that R, contains contributions from rearranged low-frequency modes which are not accounted for by

0.

16

14

12

kK

Fig, 2. Reduced emission spectra (defined in the text) for the excited CT complex of 1,2,4,Stetracyanobenzene/hexamethylbenzene in various sdvents at room temperature, and tits to the spectra (dashed curves) calculated using eq. (2) with the parameters given in the text and table 1.

trichloroethylene, i e. 0.29 and 0.34 eV, respectively. The electronic coupling matrix elements for return electron transfer were estimated by use of eq. (5a). The radiative rate constant, i&,is related to the lifetime, r, and the emission quantum yield, @, of the CRIP according to

Values for @rand r were determined in carbon tetrachloride, diethyl carbonate, and chloroform (table 3 ) . A value of 15 D was assumed for Ap, which repre-

Table 2 Different sets of fitting parameters for the emission spectra of the contact radical-ion pairs of 1,2,4,Stetracyanobenzene methylbenxene in two different solvents, and the corresponding effect on the calculated return electron transfer rates Solvent

.X,8) ieV)

iz, (eV)

-AG_, teV)

carbon tetrachloride

0.31 0.29

0.16 0.21

2.50 2.53

trichloroethylene

0.31 0.34

0.35 0.29

2.52 2.49

and hexa-

2.1 1.5 26 41

a1 0.31 eV is the average value of 1, for the different solvents; the other values give somewhat better fits to the spectra in the specific solvents. bt Calculated using eq. (1 ), with 1400 cm-’ for v, and 860 cm-’ for V.

448

I.R. Gouldet al. /Chemical Physics 176 (1993) 439-456

Table 3 Fluorescence quantum yields and lifetimes of the contact radical-ion pairs of 1,2,4,5-tetracyanobenzene and hexamethylbenzene in different solvents, the corresponding average emission frequencies (v,), and the electron transfer matrix elements ( V’) Solvent

@r

T (0s)

k/n’ (105s’)

V.” W)

Y*’ (cm-‘)

carbon tetrachloride diethyl carbonate chloroform

6.8x 10-Z 1.7x lo-’ 9.3x lo-’

24.4 0.78 0.49

9.0 8.1 6.3

16.4 14.5 13.85

880 890 800

‘) Calculatedin cm-’ using V=56.47 (A,u)-’ (kyln3)“* (v.,)-“’ (fromeq. (5a)) where Apis in debye (hereassumed tobe 15 D), kr is in s-t, n is the solvent refractive index, and Y, is in 10”cm-‘.

such models, for example, librational motions of the AD pair. A detailed quantitative analyses of the trends for - AG_, and 1, would be premature, however, due to the uncertainties in the absolute values, as discussed above. The uncertainties in these parameters have surprisingly little effect on (k_,),,, however. Shown in table 2 are examples of (k_,),, for two solvents, determined using the reorganization parameters appropriate for 1, = 0.3 1 eV, and also those corresponding to the 1, which give the best fits to the spectra ( see above ) . The decrease in - AG_, and the increase in L, with increasing solvent polarity both cause k_, to increase. As a consequence, k_,, depends very strongly on solvent polarity (table 1). In fact, the few reports of emission from CT complexes are almost exclusively in very non-polar solvents [ 2 11, presumably because k_, is so large in more polar solvents. In fact, the emissions from the TCB/HMB complexes in solvents more polar than those in table 1 are very weak, although separation to form a SSRIP may also contribute to the nonradiative decay processes of the CRIP under these conditions. 4.3. Effect of changing driving force A critical test of any electron transfer theory is whether the dependence of the reaction rate on the driving force can be predicted. This dependence for several SSRIP and CRIP systems, in particular the decrease in rate when -AG._,>&+& (the inverted region), has been successfully explained with the nonadiabatic theory of eq. (1) [6,7,33,X)-53]. Notable exceptions, however, are provided by the work of Mataga et al. on the dependence of k_, on driving force for the CRIP of TCB and related acceptors with

a variety of donor systems [22-24,54-561. It was found that the driving force dependencies were linear, and it was suggested that they were inconsistent with conventional electron-transfer theories [ 541. These findings appear to contradict the results described above on the solvent dependence of k_,, for the TCB/HMB CRIP, and so we have investigated the driving force dependence for a series of related systems in detail. The ground state CT complexes of TCB with the methyl-substituted donors shown in scheme 1 are well known [ 2 11. Using these donors, a series of A/D systems can be obtained in which the driving force for return electron transfer increases, the absorption maximum shifts to higher energy, and the equilibrium constant decreases with decreasing substitution, i.e. with increasing donor oxidation potential (scheme 1). Emission can be observed for all of the TCB/donor systems in all of the solvents mentioned in the previous section. The emission maxima of the excited complexes also shift to higher energy with increasing oxidation potential of the donor, but less rapidly than the absorption maxima. That is, the apparent Stokes shifts become larger, as illustrated for pxylene and hexamethylbenzene in chloroform in fig. 3. Simultaneously, the widths of the emission spectra increase, rig. 4, indicating an increase in the reorganization energy for electron transfer [ 19,401 with decreasing number of methyl groups on the donor. The suggestion that the reorganization energies are not constant for the different, but closely related, complexes was somewhat unexpected, but was confirmed by fitting the reduced emission spectra, as described above. Again, 1, could be varied over a fairly narrow range for each donor ( = 0.03 eV), and an average value of 0.3 1 eV was found to give acceptable

I.R. Gould et al. /Chemical Physics 176 (1993) 439-456

449

1.0 E 2 Q, E -0 $ .z? fs s 5 0.0

z lz

1 0.0

-AG,

P

c

’

J

(kK)

h, (ev)

TMB

PMB

23.08

21 .OO

0.66

0.63

1.0.

Wavenumber (i), kK Fig. 3. Reduced absorption and emission spectra (defmed in the text) for the excited CT complexes of 1,2,4,5_tetracyanobenzene with pxylene (dashed curves) and hexamethylbenzene (solid curves) as donors in chloroform at room temperature. The pxylene spectra have been shifted to lower energy by 3.4 kK, to illustrate the difference in the apparent Stokes shifts for the two complexes. There is significant overlap of the CT absorptions of the pxylene complex and the local absorptions of the TCB.

_I 22

20

18

16

14

12

Wavenumber, kK

fits in each case. Therefore, as before, 1, was fixed at 0.3 1 eV for each CRIP. The values of -AC_, and A, which then gave optimum fits to the spectra are listed in table 4. The electronic coupling matrix elements, V, were determined from the radiative rate constants of the radical-ion pairs, as described above (table 4). A small dependence (z 17Oh) on the number of methyl groups on the donor was found for V, the rea-

Fig. 4. Reduced emission spectra (defined in the text) for excited CT complexes of 1,2,4,5-tetracyanobenzene with various methyl-substituted benzenes as donors, in chloroform at room temperature, and fits to the spectra (&shed curves) calculated using eq. (2) with the parameters given in the text and table 4.

sons for which are not clear. (There is no a priori reason that V should be constant. ) Using the reorganization parameters and the electronic coupling matrix

Table 4 Electron-transfer parameters and calculated and measured rate constants for nomadiative return electron transfer in the contact radicalion pairs of 1,2,4,5-tetracyanobenzene and methyl-substituted benzenes in chloroform Donor

V.’ (cm-‘)

L, b’ (eV)

- AG_, b’ (eV)

(L&kc) (10’s_‘)

U-et), (10’s_‘)

pxylene 1,2+trimethylbenzene durene pentamethylbenzene hexamethylbenzene

660 730 740 760 800

0.70 0.65 0.59 0.53 0.45

3.01 2.86 2.70 2.60 2.44

9.6 28 67 93 195

12 27 59 100 200

d,

‘) Calculated in cm-’ using V=56.47 (a~)-’ (/~/PI’)‘/~ (v.,)-‘I2 (from eq. (5)) where Ap is in debye (here assumed to be 15 D), /q is in s-r, n is the solvent refractive index, and a~.,is in lO’cm-‘. b~OptimumvalueforspectralfittingwithL,=0.3leVandv,=l4OOcm-’. OrCalculatedusingeq. (l),with0.31 eVforl,and 14OOcn-‘ forv,. d, The contact radical-ion pair lifetimes are the reciprocals of these numbers.

I.R. Gouldet al. / ChemrcalPhysicsI76 (1993) 439-456

4.50

element data summarized in table 4, values of the return electron transfer rate constants could be calcuas before (table 4). lated, (LJdc, Ex~~mental values for the return electron transfer rate constants, (k_,),,,, were obtained from the emission lifetimes as l/r (table 4). Again, for these systems, fluorescence and intersystem crossing contributed less than 10% to the overall decay of the CRIP. The agreement between the measured and calculated values is again remarkably good, albeit the data cover a smaller rate range than those of the TCB/ HMB CRIP in the different solvents described above. The results further support the conclusion from the solvent effect experiments that eq. ( 1) accurately predicts the return electron transfer rate constants. When the log(k_,),, are plotted against -AG_,, the slope of the approximately linear plot is = - 2.2 eV- ’ (dotted line, fig. 5), which is similar to those previously reported by Mataga et al. for related GRIP [ 54-561. This dependence of (k~&,,~~~ on -AG_, is in fact shallower than would be predicted using eq. 10.0

2.4

2.6 -A&,,

2.8 eV

Fig. 5. Plot of the logarithm of the measured rate of nonradiative return electron transfer, log k_,, versus driving force, -AG_,,, for the excited CT complexes of 1,2,4,Stetracyanobenzene as the acceptor and (A) hexamethylbenzene, (B) pentamethylbenzene, (C) durene, (D) 1,2,4-trimethylbenzene, and (E) p-xylene as donors. n represents the number of methyl substituents on the benzene ring of the donor. The solid lines represent the predicted driving force dependencies for the reorganization parameters which characterize each excited CT complex. The dashed line represents the apparent driving force dependence, ignoring the dependence of the reorganization energies on donor structure.

( 1), if the reorganization parameters were constant. However, the CT emission spectra clearly indicate they are not. The predicted driving force dependencies for each set of reorga~zation parameters used to fit the spectra are also shown in fig. 5 (solid lines). These plots, which are approximately linear for the narrow range of -AG_,, studied here, are significantly steeper z - 4.6 eV- ’ ). The driving force dependencies reported previously for the related CRIP [ 54-561, therefore, are likely to be a consequence of a reorganization energy that varies with driving force in the same manner as that observed for the present CRIP. When this variation is taken into account, the nonadiabatic theory appears to predict the measured driving force dependence very well. In the analyses of the spectra described above it was assumed that A, is constant for the different CRIP, and that the changes in total reorg~ization energy are entirely due to changes in 2,. Although the CRIP emission spectra require that the total reorganization energies change with donor in the manner indicated, from the spectral fitting alone it is not possible to determine the extent that this is due solely to changes in A,, since, as indicated above, a range of parameters was found to give acceptable tits. However, the values of the calculated rate constants (k_et)calc are not very sensitive to small variations in J., and &. Indeed, the same trend in the calculated return electron transfer rate constants is obtained if in addition to A,, 1, is allowed to increase with increasing oxidation potential, within the range which gives acceptable fits to the spectra. That 1, changes with donor structure is, however, consistent with the observation that a plot of oxidation potential in polar solvents versus gas phase ionization potential for the methyl-substituted benzenes has a slope which is considerably less than unity (0.7), i.e. the ionization potential increases faster than the oxidation potential [ 5 7 1. This indicates that the solvent stabilization of the radical cations decreases with increasing methyl substitution. It is reasonable that a reduction in the solvent stabilization of a radical cation will result in a corresponding decrease in AS. 4.4. Efsecf of changing acceptor The donors discussed in this work also form ground state CT complexes with 2,6,9,1 O-tetracy~o~thra-

451

I.R. Gouldet al. /Chemical Physics176 (1993) 439-456

cene (TCA) [ 6 ]. Excited CT emission can be observed with TCA as the acceptor just as with TCB. The reduced emission spectrum of the TCA/HMB CRIP in trichloroethylene is shown in fig. 6, together with that of the corresponding TCB system. The emission maxima occur at approximately the same energies (fig. 6 ) although the reduction potential of TCA is -0.44 V versus SCE, compared to -0.64 V versus SCE for TCB (scheme 1) . Thus the energy of the TCA CRIP with respect to the neutrals, and hence its emission m~imum, might have been expected to be lower than those of the TCB CRIP by approximately 0.2 eV. However, it is also clear from fig. 6 that the emission spectrum of the TCA CRIP is considerably narrower than that of the corresponding TCB CRIP. This indicates that the reorganization energy with TCA is considerably smaller than that with TCB, presumably as a consequence of the larger molecular dimensions of the TCA compared to TCB (see below). This is also the clear result from the spectral fitting procedure, the results of which are shown in fig. 6. The optimum values obtained for -AG_, and 1, are 2.23 and 0.21 eV, respectively, taking a, and v, to be fixed at 0.20 eV and 1400 cm - *, respectively [ 61. Thus, the emission maximum of the -AG-et -

xx::

2.52

TCA CRIP is almost the same as that for the TCB GRIP, despite the smaller -AG_,,, due to the much smaller total reorganization energy. The energies of the average emission frequencies, h v,,, of these two CRIP are also similar for the same reason, as illustrated in fig. 7. The emission and intersystem crossing quantum yields determined for the TCA/HMB CRIP in trichloroethylene are 0.026 and 0.18 respectively. From these data and the emission lifetime, 12.2 ns, a value of 6.5 x 1O7s- ’ was dete~ed. Using for (Lfthe reorganization parameters from the spectral fitting and a value of 750 cm-’ for the electronic coupling matrix element [ 581, the (k_,),, obtained for this CRIP is 1.1 x 10’ s-‘. The calculated value is again reasonably close to the corresponding measured value, although the agreement is not as good as that obtained for the TCB CRIP. Interestingly, the measured return electron transfer rate for the TCA CRIP is smafler than that for the TCB CRIP (6.5~10~ s-i for TCAversus 3.5x108 s- ’ for TCB ) , despite the fact that the return electron

A.,, 0.31

0.35

L‘ 1.0 2

P

I

ii

I

-AG+,t=2.23

-AG.et=2.52 I

jj

t 0.c

18

16

14

Wavenumber, kK Fig. 6. Reduced emission spectra (defmed in the text) for excited CT complexes of 1,2,4,5-tetracyanobenzene (thin curve) and 2,6,9,10-tetracyanoanthracene (thick curve) as the acceptors and hexamethylbenzene as the donor, in trichloroethylene at room temperature, and tits to the spectra (dashed curves) calculated using eq. (2 ) with the parameters given in the tigure and using 1400 cm-’ for v,.

TC~H~B

TCB/H~B

Fig. 7. Relationship between the average emission energy, hv., reorganization energies A,, A,, and free energy for return electron transfer, AG-,, (all in eV) for the 2,6,9,1O_tetracyanoanthracene (TCA) /hexamethylbenzene and 1,2,4,5-tetracyanobenzene (TCB)/hexamethylbenzene ORIP in trichloroethylene. The parabolas represent free energies [ 19 ] for states with 0 quanta in the v, mode, as a function of a solvent coordinate. Subtracting 1, Tom the most probable 0,O transition energy ( -AG_,-A.) gives the most probable vertical transition energy, hv,, (eq. (5d)).

452

I.R. Gouldet al. /Chemical Physrcs176 (1993) 439-456

transfer reaction is less exothermic (2.23 eV for TCA and 2.52 eV for TCB). Superficially the comparison would suggest Marcus “normal” behavior, i.e., the more exothermic reaction is faster. However, it is instead a consequence of the significantly smaller reorganization energy for the TCA CRIP (1, +I, = 0.4 1 eV for TCA versus 0.66 eV for TCB, fig. 7). In effect, this is an extreme example of the situation described above for the series of donors. The driving force dependence is meaningful only for constant reorganization parameters. When this is recognized, the conventional nonadiabatic theory again appears to be able to predict the measured rate constant. The smaller reorganization energy for the TCA CRIP compared to the TCB CRIP is most reasonably explained in terms of the relative molecular dimensions of the acceptors [ 71. The significantly larger TCA should be associated with a smaller reorganization energy as a result of the higher charge delocalization. The effects of changing the acceptor appear to be the same as those of changing the donor, magnified by the larger change in molecular dimension. 4.5. Validity of the theoretical model The remarkable agreement found here between the calculated and measured return electron transfer rate constants argues for the validity of the theory described in section 3. However, the large values of V (e.g. VB k,T) raise the question of whether the nonadiabatic, golden-rule treatment is really appropriate. Eq. ( 1) is based on a conventional model in which the CRIP thermalizes to a pure ionic state (A.-D’+) with no quanta in the vv mode, which subsequently undergoes transitions (return electron transfer) to a pure neutral state (AD) with various numbers of quanta in the v, mode. The transition yielding j such quanta is governed by a vibronic matrix element of magnitude Fi1/2V. The systems studied here are deep in the inverted region (i.e. -AG_,,%I,+J,), and therefore the most important final states have largej (eq. ( 1) ) and quite small vibronic matrix elements (see ref. [ 61 for more discussion and a numerical example). Hence, a goldenrule treatment such as eq. ( 1) should, in fact, be valid as long as the assumption [ 19,38,39,59,60 ] that the initial and final states are purely ionic and purely neutral is also valid.

This assumption is itself problematic, however. A priori, it seems more reasonable to assume states of mixed character, the degree of mixing being governed by V. In fact, the conventional rationalization of the transition matrix element (M,), which we have adapted to evaluate V from measured values of h, is based on this mixing [ 42,60,61]. In this alternative scheme, Vis incorporated in the electronic Hamiltonian, and nonradiative return electron transfer transitions are governed by matrix elements of the nuclear kinetic energy operators, rather than of V [62,63]. At the moment, it is not obvious that this mixed-state starting point should reproduce the results (eq. ( 1) ) of the pure-state assumption. Nevertheless, the pure state theory, albeit with additional simplifying assumptions, agrees with experiment. Further theoretical work is required to resolve this issue. Another issue related to the validity of both the nonradiative and the radiative theories is the common [ 19,38,39,60] but questionable [42,59] assumption that the pure electronic states are orthogonal. The consequences of relaxing this assumption also need to be investigated. Assuming the validity of the pure-state assumptions, however, the agreement between calculated and measured rate constants does appear to support the use of a single “representative” mode (v,) in place of the many vibrational modes [ 641 that must be actually involved. In fact, the spectral fitting has the sole purpose of allowing an extrapolation of FC(AG_,+hv), eq. (2), to FC(AG_,), eq. ( 1). It appears that the single-mode model is adequate for this purpose.

5. Conclusions The rates of nonradiative return electron transfer in the contact radical-ion pairs of tetracyanobenzene and tetracyanoanthracene, with alkylkbenzenes as donors, in solvents of varying polarity can be predicted from the corresponding emission (radiative return electron transfer) data. From analyses of the emission spectral shapes, information concerning the reorganization parameters for the corresponding nonradiative process is obtained. Estimates of the electronic coupling matrix elements for the nonradiative processes are obtained from the radiative

I.R. Gouldet al. /Chemical Physics176 (I 993) 439-456

rates. The agreement between the measured return electron transfer rate constants and values calculated from the emission data is remarkably good. The agreement is found in studies in which the solvent polarity and the structures of the donor and the acceptor are varied. The results illustrate the utility of the nonadiabatic theory used here for reactions in which - AG_,, is significantly larger than the total reorganization energy, even with values of V which are significantly larger than are normally associated with nonadiabatic electron transfer theories. The results also illustrate that even closely related systems may have significantly different reorganization energies, which must be taken into account in studies of driving force dependence. The studies described here further illustrate the quantitative relationship between nonradiative and radiative return electron transfer, and complement other studies in which spectroscopic data are related to the absolute rate constants for electron transfer reactions.

The authors are grateful to the National Science Foundation for a Science and Technology Center grant (CHE-88 10024), DN is also grateful to the Swiss National Science Foundation for financial support. The authors also thank S. Mukamel (University of Rochester) and M. Zimmt (Brown University) for helpful discussions, D. Foster (Eastman Kodak Company) for the measurement of the emission quantum yield of the reference compound, 9,1 Odicyanoanthracene, and W. Herkstroeter (University of Rochester) for measuring the singlet excited state lifetime of 1,2,4,5tetracyanobenzene.

Appendix A. Derivation of eq. (2)

64x4 m

?13V3 1

MO, = Fjl’=M,, ,

I

M$jPj)

[AG_,,+&+h(v+jv,)]= 4&k, T

).

(A. la) (A.lb)

(‘4.1~)

It has a simple interpretation [ 19,38 1. We treat the coordinates qs which correspond to solvent orientation and other slow motions classically. Each individual CRIP has a stick spectrum at energies determined by the instantaneous value of qs, representing O+j transitions from a CRIP state with zero vibrational quanta to neutral states with j quanta. Emission at frequency v is a composite of 0-t j transitions from CRIP in various environments such that, for some the O+O transition has this frequency, for others the O-+1 transition does, for yet others the O-+2 transition does, etc. Those CRIP whose O+j transitions have this frequency have O-+0 lines at frequency v+jv,. Their relative probability, per unit frequency, is P,. Their transition moments are the MO, Marcus [ 19,381 has derived this formula on the assumption that the electronic transition moment is constant. In our case, M, depends on qs (see below). It seems clear that eq. ( 1) still holds, as long as the proper expression is used for Moj( qs ) . To derive such an expression, we assume that the CRIP state and the ground state have the Born-Oppenheimer form, VonrP(rlqv, qs)XcRIP,O(qv), Vo(rlq,, qs)XoJ(qV) , (A.21

where r and qv represent the sets of electronic and vibrational coordinates. The electronic wavefunctions wcRlpand vo depend parametrically on the nuclear coordinates qv, qs. The vibrational wavefunctions XcRIp,oand XoJ represent the lowest sublevel for the CRIP and the jth sublevel for the ground state. The transition moment can be expressed as Mo, (4s) = (XeiuP,o(&) IM,(q,, 45) kGJ(qv)

We rewrite the desired eq. (2 ) in the form (the symbols are defined in the text) If(v)=

x exp -

453

>

,

(A.3a) M,(q,,q,)=(wcRIPI-Cerly/G)

>

(A.3b)

with -Cer representing the electronic dipole moment operator. The integrations in eqs. (A.3a) and (A.3b) are over qv and r, respectively. We now assume that vCRIPand vo are linear com-

I.R. Gouldet al. /Chemical PhysicsI76 (1993) 439-456

454

binations of a pure ionic and a pure neutral state, v/A.-D.+and ly,~. We assume that the latter states are orthogonal and that they are coupled by an electrontransfer matrix element I’. To first order

Neglecting terms of at least second order in I’, AU(q,, 4s) =

&RIP(&

AU(q,, &)=hv,(q,)

48) -

uG(&,

+&dsv,

%)

,

&)-hG(&

(A.9) 9s),

(A. 10a) (A-4) where Ap is the difference between the static dipole moments of the two pure electronic states and AU( 4, qs) is the difference between their energies. As is customary, we neglect the “direct” contribution (VA/A.-D.+ 1- Eerl v/AD). We assume that V is positive, as can be arranged by a suitable choice of phase factors for vcR1pand v/G. In eqs. (A.3) and (A.4), M,(q,, qs) is the electronic transition moment when the electronic energy difference is A U( qv, qs) . The M, appearing in eq. ( 2 ) is this quantity when AU= h vv. The approximations that yield eq. (2) are MO, = (XCRIP,O 1-

A;:q,)

IxGsJ) ’

Moja-z, J

(A.5)

(‘4.6)

S

where hv, is the energy difference between the vibronic states vcRIp&up,o and vo XGj (see eq. (A. 1Ob) below), i.e. the energy of a photon that could be emitted in a transition between these states (i.e., h v in eq. (2 ) ). This approximation results from the following considerations. The vibrational wavefunctions are solutions of [K +

&RIP(&,

= &RIP,0

[T,+uo(G,

%) ]XCRIP,O

(A.7a)

( 4s )XCRIP,O 3 %)lxG,j=EGJ(&;)xG,j

Y

(A.7b)

where T, is the kinetic energy associated with qv, UC,, and uo are the electronic energies associated with wcRIpand ~0, and EcRIp,o and EGJ are the BomOppenheimer total energies of the vibronic states v/cRIpxcR1p,oand vo &. We assume that the dependencies on qs and qv are separable, i.e. uG(&

&)=~G(~v)+'%!?(~s)

,

(A.8)

and similarly for UcRp, so that xcRIp,oand xoJ do not depend on qs and the spacing of energy levels EGJ is independent of qs.

h%(G) =&RiP,o(qs) -&J(G) &RIP(&,k)

= &RIp(qv,

(A.lOb)

7

4s) -EcRIp,o(d

(A.lOc) aG(&, %)=uG(&,

&)-zGj(qs)

(A.lOd)

.

Substituting eq. (A. 1Oa ) into eq. (A. 5 ) and expanding (hVj+&p-dG)-' in powers of (&qp-60) gives eq. (A.6 ) at zeroth order and a first-order correction proportional to (XCRIP.0

=

1&RIP


- sG

1XG,J>

I ( Z + &UP

-&UP,•

1

IXG,j)

-(T,+UG-EG,,)

=o

(A.ll)

by virtue of eqs. (A.7). Thus, eq. (A.6) is good through first order in the 6s. We have tacitly neglected any dependence of V or Ap on qs. A numerical test of approximation (A.6 ) was performed as follows. The vibrational potential energies were represented as harmonic oscillator functions &RIP(&)

= bh?:

(A.12a)

>

(A.12b)

U;;(G) = &(4” -&“)2 9

with force constants and effective mass such that h v, = 1200 cm-’ and with displacement 6q, such that A,= j/&q: varied from 0 to 0.37 eV. The solvent contributions U,&, and U.$ were chosen so that the energy of the O-+0 transition, hug, varied between 14 and 24 kK. All values of j were considered such that the emission energy h Vi= h ( v. -jv,) exceeds 12.5 kK and C,F, > 0.99. The quantities 1(XCRIP.01

LAW&,

&)I-‘iXG,j>

I*

and

WJ-*

(A.13) differed by no more than 5%, and the larger differences occurred for relatively low frequency vi and small Fr Eq. (A.6) is a remarkably good approximation. Marcus’ derivation of eq. (A. 1)) for constant electronic transition moment, is based on semiclassical

I.R. Gould et al. /Chemical Physics I76 (1993) 439-456

wavefunctions of the qs variables, a Boltzmann probability distribution among these wavefunctions, and a golden-rule expression for the state-to-state transition probabilities. It is easily adapted to the present case by use of eq. (A.61 at the stationary phase points. The result is again eq. (2), now with the electronic transition moment given by eq. (2~). It is not necessary to make the one-mode assumption embodied in eq. (1~). The treatment of charge-transfer absorption leading to eq. (4) is essentially the same as for emission. Approximation (A.6 ) is again quite accurate.

Appendix B. Derivation of eq. (5) The derivation of eq. (5) is patterned after ref. [65]. Integrating eq. (3) and noting that ZJ&=l gives

I

$dp=64x4n3V2A$ 3h3c3

-

(B-1)

On the other hand J+= j- Ird(hv)

.

(B-2)

Combining eqs. (B. 1) and (B.2 ) gives eq. (5).

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