8863
J. Phys. Chem. 1995,99, 8863-8871
Thermodynamics of Intramolecular Electron Transfer in Alkane Solvents Joseph Morais and Matthew B. Zimmt* Department of Chemistry, Brown University, Providence, Rhode Island 02912 Received: September 22, 1994; In Final Form: January 3, 1995@
The thermodynamics of the intramolecular charge recombination, electron transfer reaction between a dimethoxynaphthalene donor and a 1,2-dicarbomethoxyethyleneacceptor, separated by a rigid three-bond spacer, have been determined in alkane solvents using picosecond optical calorimetry (POC) and fluorescence spectroscopy. The molar enthalpy and volume changes of the charge recombination are -288 f 12 kJ/mol and 14 f 4 mL/mol, respectively. The determination of these quantities from the POC data is described. The free energy change for the reaction was determined in three ways: from an analysis of the charge transfer (CT) emission band, from the calorimetric data, and from redox potentials plus Coulombic corrections. The results from the first two approaches are in reasonable agreement, AGO = -289 kJ/mol. The redox potential approach also generates similar free energy changes provided “proper guesses” of the ion radii and separation are used. Experimental determination of the entropy change upon charge recombination presents the greatest challenge. The temperature dependence of the CT emission band’s first moment may be accurately determined but is related both to the entropy change and to the temperature dependence of the reorganization energies, &/dT. Using an estimate of the entropy change, derived from the volume of reaction, and the temperature dependence of the CT band first moment produces an estimate of &ldT that is twice the magnitude of AS”. The thermodynamic values determined in this investigation are compared with predictions of simple continuum models and with results from related charge transfer systems.
SCHEME 1
I. Introduction Electron transfer reactions continue to be of considerable interest to a wide range of scientists. The important role of electron transfer reactions in biochemical’ and technological systems,2in part, motivates these efforts. An additional factor has been the existence and success of theoretical models that relate the kinetics of electron transfer reactions to the thermod y n a m i c ~ . These ~ formulations of electron transfer kinetics, developed and advanced by Marcus and other^,^ are functions of the free energy of electron transfer, AGO, the electronic coupling between the redox sites, I VI, and the magnitude of the intrinsic barriers to reaction. Kinetic studies directed toward determination of one of these quantities, e.g., efforts to understand the structural dependence of electronic coupling? require a thorough knowledge of the remaining quantities. Fortunately, the free energy of reaction may often be determined using electrochemical methods. However, for studies of electron transfer reactions in inhomogeneous or nonpolar media, it may not be possible to recreate the relevant charge distribution or medium polarity in an electrochemical measurement, resulting in considerable uncertainty in the transfer thermodynamics. In such cases, altemate methods are needed to determine the reaction thermodynamics. Thus, AGO for electron transfer reactions have been determined by direct measurements of reactant-product equilibrium constants5 and through the analysis of charge transfer absorption and emission spectrae6 In this paper, we report the results of calorimetric and spectroscopic studies directed toward determination of the enthalpy, AHO, volume, A T , entropy, AS”, and free energy of the intramolecular charge recombination, electron transfer reaction for the donor-spacer-acceptor (DSA) molecule 1 in alkane solvents. The photophysics of 1 in alkane and ether solvents has previously been investigated (Scheme l).7,8 Photoexcitation of @
Abstract published in Advance ACS Abstracts, April 15, 1995.
s1
kT
-
..._
hve,
T1
SO
1
the dimethoxynaphthalene chromophore results in transfer of an electron to the dicarbomethoxyethyleneacceptor with a time constant of 2 ps or less. The resulting charge transfer state of 1 decays by three pathways. Radiative charge recombination, with re-formation of the ground state, SO,proceeds with the rate constant kfand with an overall quantum yield @. Nonradiative charge recombination, which also re-forms SO,proceeds with the rate constant ~ N Rand with a quantum yield of a. Additionally, the CT state undergoes a combined intersystem crossing1 charge recombination process to form the excited triplet state of the dimethoxynaphthalene chromophore with a rate constant k~ and with quantum yield of p. The CT state decay rate constant, R-cT, is the sum of the rate constants for these three processes. The triplet state persists for microseconds, a time scale much longer than that investigated in the current experiments.
0022-365419512099-8863$09.00/0 0 1995 American Chemical Society
8864 J. Phys. Chem., Vol. 99, No. 21, 1995 With the aid of the quantum yields for each decay pathway of the CT state in alkane solvent^,^ AHO and AV” for charge recombination from CT to SO have been determined from picosecond optical calorimetry9 studies. Analysis of the CT state emission spectra as a function of temperature, using a semiclassical model,6 provides values of AGO, AS”, and AHO for the charge recombination reaction. The results of this analysis are very sensitive to the intrinsic barriers used within the model, leading to unacceptably large uncertainty limits in the desired thermodynamic quantities. Thus, a second analysis of the temperature-dependent CT emission spectra is advanced to obtain estimates of AS” and of the temperature dependence of the low-frequency reorganization energy. Overall, good agreement is found between the values of the thermodynamic quantities obtained from the calorimetric and spectroscopic methods and from studies of related chemical systems. Continuum models also yield similar values of the thermodynamic quantities provided one uses the “correct” values of ion radii and separations. However, use of reasonable ion radii and separations that differ from the “correct” values by f l A lead to significantly different thermodynamic estimates. 11. Experimental Section
The synthesis of 1 has been reported previously.8 Purification of 1 was carried out by chromatography on silica using 25% ethyl acetate in hexanes as eluent followed by recrystallization from ethanol. All purifications were performed in the dark to minimize photodecomposition of 1. All alkane solvents were stirred with sulfuric acid for 7-10 days, washed with saturated aqueous NaHC03, and dried over Na2S04. Each solvent was fractionally distilled from sodium under argon immediately prior to use. All solvents had absorbances 50.001 in a 1 mm cell at the excitation wavelength (283 nm). Steady state emission spectra were recorded and corrected as described p r e v i o ~ s l y . ~For ~ ’ ~ the temperature-dependent fluorescence measurements, the quartz fluorescence cell was mounted in an insulated aluminum block which was cooled using a NESLAB LT-50 low-temperature circulating bath. The temperature of the sample was monitored using a Digi-Sense thermocouple thermometer (Cole-Parmer, Model No. 8528-20) and a T-type thermocouple probe which was mounted through a Teflon cap and immersed in the solution to within 0.5 cm of the irradiated region. Each sample was allowed to thermally equilibrate for 15 min prior to data collection. The sample was maintained to within f 0 . 2 “C of the target temperature during acquisition of the emission spectrum. A fresh solution of 1 was used for each emission spectrum, and at least five independent emission spectra were recorded at each temperature. Values of AGO, A”,hw, and I S (vide infra) were extracted from the CT emission line shape6by minimizing using a nonlinear least-squares fitting program based on Powell’s method.” The picosecond optical calorimetry (POC) experiment and apparatus have been described p r e v i o u ~ l y . The ~ ~ POC signal is proportional to the time-dependent diffraction efficiency of an optically generated transient grating.9asb The grating is formed by crossing two excitation pulses at an angle 5 = 35.5’ in a 1 mm path length Suprasil flow cell. The signal produced by diffraction of the time-delayed probe pulse was spatially isolated, filtered (Hoya R-62 long pass filter, Optics for Research 633 nm interference filter), and detected using a Hamamatsu R928 phototube (525 V, 3 Mohm termination) and a lock-in amplifier (Stanford Research Systems Model SR-5 IO). The lock-in amplifier was synchronized to an optical chopper (Photon Technology Intemational Model OC-4000) inserted into the path of one of the excitation beams. The probe beam delay
x2
Morais and Zimmt
(-0.8 to +9.0 ns) was obtained with an optical delay stage (Gaertner Scientific). The temporal resolution of the delay line is 2.8 pstpoint. The theoretical equations for the calculation of picosecond optical calorimetry diffraction wave forms have been described elsewhere?%’*The expressions relevant to this work will be discussed in more detail in the next section. The method used to obtain calorimetric quantities from the experimental wave forms is described in detail below. The experimental wave forms were fit by minimizing x2 using a variation of the Levenberg-Marquardt method.” Each data set was fit starting at five or more initial guesses in parameter space, and all starting guesses converged to the same results, within the uncertainties in each parameter determined from the fitting algorithm. The results reported represent averages from at least six independent experiments in each solvent. Nine parameters were extracted from the fitting process: a wave form amplitude scale factor that is proportional to the total heat released in the first 10 ns following excitation; the ratio of (i) the fractional density change produced by decay of the CT state of 1 to (ii) the fractional density change produced by all the heat released in the first 10 ns following excitation; k-CT, the decay rate constant for the CT state; w , the grating acoustic wave frequency; k,,, the acoustic wave attenuation rate constant; ktd, the thermal diffusion rate constant; At, the delay line position at which the centers of the excitation and probe pulses overlap (t = 0); N, a contribution to the phase grating9”vb arising from the CT state absorption spectrum at 633 nm; and K, a contribution to the amplitude g r a t i ~ ~ garising ~ ” ~ from the CT state absorption spectrum at 633 nm. Independent values of the acoustic and apparatus parameters w , kac, At, and ktd were obtained from analyses of POC wave forms generated upon excitation of the inorganic complex bis(2,2,6,6-tetramethyl-3,5-heptanedionato-O,O’)copper(II) (2) in each of the alkane solvents. The complex was purchased from Lancaster Synthesis and used as received. Copper(II) complexes are useful for determining these experimental parameters in that they appear to convert the entire photon energy into heat with a time constant that is much smaller than the grating acoustic period, have negligible emission quantum yields, and undergo no phot~chemistry.’~When POC wave forms from 1 were analyzed, all nine parameters were varied. However, any data set that yielded values of w , k,, At, or ktd that differed by more than 10%from those measured in the grating experiments with 2 was ignored. Supplementary Table S 1 lists the average values of w , kac,and ktd from the best fits for the grating experiments with both 1 and 2 in each solvent. Overall, the agreement between the values measured in the experiments with 1, the experiments with 2, and those values obtained from the literature, where available, is very good. Similarly, values of k-CT for 1 in alkane solvents have been measured by timeresolved fluore~cence.~The agreement between k-CT determined by fluorescence and measured in the calorimetry experiments (Table 1) is good. The fitting parameters Nand K originate from the absorption spectrum of the CT state. The latter exhibits a broad band, centered around 540 nm, characteristic of the dimethoxynaphthalene radical cation.14 The presence of this optical transition changes both the real (dispersive) and the imaginary (absorptive) components of the sample’s index of refraction upon formation of the CT state. The two amplitude factors Nand K provide a means to estimate the molar extinction coefficient of the CT state of 1 at the probe wavelength and to determine whether the probe wavelength is to the red or to the blue of the transient absorption maximum (vide infra).
a,
J. Phys. Chem., Vol. 99, No. 21, 1995 8865
Thermodynamics of Intramolecular Electron Transfer
TABLE 1: Picosecond Optical Calorimetry Fitting Parameters Used To Determine AH(CT-So) and AV(CT-So) for 1 solvent pentane hexane
f
R
(108 s-1) 5.32 f 0.24 (5.5 f 0.2) 5.36 5 0.22 (5.7 f 0.5) 5.7 i 0.4 (5.8 f 0.4) 5.48 & 0.18 (5.4 f 0.3) k-CTO
K
N
Loh(V)"
Pb
Qb
(kJ/mol) 216 215
UX,b
(kJ/mL) 0.901 1.10 1.25
0.019 5.5 f 0.7 -11.1 k 0.5 0.47 0.596 h 0.014 32.2 f 1.3 0.621 f 0.014 32.7 f 0.5 6.2 f 0.5 -12.5 f 0.5 0.45' 0.017 0.017 214 6.2 f 0.4 -14.4 f 0.7 0.44 heptane 0.622 f 0.011 32.7 f 0.8 214 1.47 0.015 7.1 f 0.3 -17.8 f 0.6 0.42 nonane 0.642 f 0.009 , 34.0 f 0.4 a Rate constants in parentheses taken from ref 7. * Values taken from ref 7. The uncertainties in these terms are fO.01 in /3, f0.002 in @, and hO.1 kJ/mol in &h(v,,). The quantum yield of triplet formation from the CT state of 1 in hexane was extrapolated from the values in pentane and heptane.
111. Results A. Optical Calorimetry. The theory of picosecond optical calorimetry has been discussed in detail p r e v i o ~ s l y . ~A~brief '~~ review of the theory and its application to the current experiment will be presented here. A transient optical grating is generated within a sample by spatially and temporally crossing two excitation pulses. Interference between these two pulses produces a sinusoidal variation of the light intensity along the grating wave vector. The grating wave vector is a line that lies within the plane defined by the paths of the excitation pulses and which is perpendicular to the line bisecting the angle, between the excitation pulse paths. The spatial period of the light intensity oscillation is A = &/[2 sin(c/2)], where A,, is the excitation wavelength. Photophysical and photochemical processes following photon absorption convert the sinusoidally modulated light intensity into a sinusoidal modulation of the sample refractive index; Le., a grating is generated. The experimental observable in the picosecond optical calorimetry experiment is the time dependence of the light intensity diffracted from a variably delayed probe pulse which is incident on the optically generated grating. The ratio of the diffracted light intensity to the incident probe pulse intensity is defined as the grating diffraction efficiency, 7,and is expressed asI5
c,
where Dav(Apr)is the average optical density at the probe wavelength, Apr, in the region irradiated by the excitation pulses, d is the thickness of the grating generated, and 8' is the angle that the probe pulse makes with the line bisecting the angle 5. An(A,,,t) is the amplitude of the sinusoidal modulation in the real part of the sample refractive index along the grating wave vector. Ak(Apr,t)is the amplitude of the sinusoidal modulation in the imaginary part of the sample refractive index along the grating wave vector. For the grating generated by formation and decay of the CT state in 1, An(Apr,t) arises from (1) thermallyI6 and electr~strictively'~induced changes in the sample density and (2) dispersive contributions associated with the absorption spectrum of the CT state at the probe wavelength.9asb Ak(Apr,t) arises from absorption by the CT state at the probe ~ a v e l e n g t h . ~ ~ , ~ Radiationless transitions following photoexcitation of 1 are exothermic processes which raise the solvent temperature and induce corresponding reductions in the solvent density in the regions of nonzero excitation inten~ity.~ The fractional change in the solvent density resulting from the radiationless transitions, (6/@),h,is given by'*'
where 6 is the amplitude of the thermally induced density modulation, e is the ambient solvent density, a is the cubic thermal expansion coefficient of the solvent, CV is the solvent heat capacity per gram, Ac(A--+B)is the molar concentration of species A converted to B in the regions of maximum excitation intensity,'* and AW(A-+B) is the molar enthalpy change for the conversion of A to B. Species A and B may have significantly different partial molar volumes. Thus, the conversion of A to B may be accompanied by a second change in the solvent density which, expressed in terms of the fractional change in solvent density, (S/e),,l, is
= -Ac(A-B)
AV'(A-B)
where AV'(A-B) is the difference between the partial molar volumes of B and A. The combined fractional density change accompanying the conversion of A to B contributes to the modulation in the real component of the refractive index, An(t) in eq 1, in the amount12c,16
(dn/@)p is the derivative of the solvent refractive index with respect to density at constant pressure, and 9 f t : k ) is a timedependent function which depends on the value of k, the rate constant for the conversion of A to B. 9 f t : k ) varies between 0 and a theoretical upper limit of l.93'9 From the kinetic scheme pertaining to 1 in alkane solvents (Scheme l), An(& for the formation and the decay of the CT state is comprised of four such terms: An(?), =
AV"(S,-CT)
I
+
st:k=~)
where is Avogadro's number, hv,, is the excitation photon energy, ,!?(TI)is the energy of the first excited triplet state, h(vem) is the mean energy of a photon emitted by the CT state,20and ,8, R-cT, a,and @ are defined as in the Introduction. All the thermodynamic quantities are per mole. The first term in the sum, contained within the bold square brackets in eq 2, is the
Morais and Zimmt
8866 J. Phys. Chem., Vol. 99, No. 21, 1995 density modulation produced by formation of the CT state from the dimethoxynaphthalene locally excited state. The rate constant for this reaction is much faster than all the acoustic processes and, thus, may be approximated as k = m in 4 t : k).2' The second term in the sum is the density modulation produced by the conversion of the CT state to the dimethoxynaphthalene triplet state. The third and fourth terms in the sum are the density modulations produced by the nonradiative and radiative conversions, respectively, of the CT state to SO. In the analysis it is assumed that AVO(CT-So) = Av"(CT-+Tl) = -A VO( So'CT) .22 It was not possible to obtain acceptable fits to the POC wave forms in the first nanosecond following sample excitation using a model for the time-dependent diffraction efficiency which incorporated only An(&. An additional contribution to the POC wave form was clearly present at these times. Acceptable fits to the wave forms required inclusion of additional real and imaginary contributions to the refractive index modulation, both of which decayed with the rate constant k - c ~ .The assignment of the optical transition responsible for these index contributions will be described in the Discussion section. The additional terms included in An(Apr,t)and Ak(Apr,t) are23
310
r
T
I
I
/=f
,!F /---
t 270 0.0
0.3
0.6
0.9
1.2
1.5
i/Xs(kJ/mL)
Figure 1. Plot of the left-hand side of eq 6 versus l/Xs. From the intercept, AHO(CT--So) = -288 f 12 (20) kJ/mol. From the slope, AVO(CT--So) = 14 f 4 (20) mL/mol. The error bars (see the textz5 for the generation of the uncertainty limits) on the data points represent k l standard deviation from the mean. The dashed lines bracket the 68% confidence level of the linear analysis.
j
i
$
5
1000
900
800 700 600 500 400
where g(Apr)is an unknown dispersive line shape function for the CT excited state and Ac(Apf)is the difference in the molar extinction coefficients (M-I cm-') of the CT and the SO states at the probe wavelength. For 1, there is no absorption from the SOstate at the probe wavelength; thus, Ac(l,,) = €(Apr), the molar extinction coefficient of the CT state at 633 nm. The two fitting parameters Nand K (vide supra) are proportional to the time-independent components of eqs 3 and 4, respectively. The complete expression for the time-dependent diffraction efficiency of the grating generated by photoexcitation of 1 in alkane solvents is9a,b
i
300
L
200 0
100
n 0
1
2
3
4
5
6
7
8
9
7
5
9
Delay Time ( nsec ) 1000 900
a
i
800 700
600 500 400
300
200
The parameter f used in fitting the experimental POC wave forms is equal to the ratio of (i) An(t)@for the decay of the CT state (the sum of the last three terms in the bold square brackets in eq 2 ) to (ii) the sum of An(t)e for both the formation and the decay of the CT state (all four terms in eq 2). This equation may be rearranged to yield the following equation relating AH0(CT-So), and AV"(CT--So).
flohy,, + P ( 1 -J)E(T,) + -f)~oh(y,,) = -AHo(CT-So) AV'(CT-So)/X5
+
(6)
where X, = a/$" has units of L M . The right-hand side of eq 6 depends on two unknown quantities; thus, it is not possible to unambiguously determine AZP(CT-So) and AV"(CT-So) from POC experiments in a single solvent. This problem has been circumvented by performing the POC experiment in the linear alkane solvents from pentane to nonane. The dielectric constants of alkanes are weakly dependent on alkane chain length.24 As a consequence, AH0(CT-So) and AV"(CT-So) should be relatively independent of the alkane solvent used. By
100 0 0
1
2
3
4
5
6
Time (nsec)
Figure 2. Experimental (dots) and best fit calculated (solid line) POC wave forms for 1 in pentane (a, top) and nonane (b, bottom). The time between data points is 45 ps. The inset shows the first 1.5 ns after excitation on an expanded scale. The time between data points for the insets is 23 ps.
contrast, X, decreases more than 60% as the solvent is changed from pentane to nonane. All the parameters on the left-hand side of eq 6 have been determined. Thus, AH"(CT--So) and AV'(CT-So) may be established from the intercept and slope, respectively, of a plot of the left-hand side of eq 6 versus l/X5 (Figure 1). The experimental and best fit calculated POC wave forms for 1 in pentane and nonane are presented in Figure 2. The insets show the wave form shape for the first 1.5 ns on an expanded scale. The values of X,, ,8, and @ and the best fit values off and k - c ~are presented in Table 1 for each solvent. From the slope of the best fit straight line in Figure 1,
J. Phys. Chem., Vol. 99, No. 21, 1995 8867
Thermodynamics of Intramolecular Electron Transfer
TABLE 2: Results from CT Emission Analysis for 1 in Heptane at 293 K parameter set AGo(CT-So)" I s (eV)b ho (cm-'). 1,(eV)d {AI
{Bl
{CI
-2.88 -3.00 -2.90
0.18
0.46' 0.30
1350 1700 1530
0.48
0.32 0.39
Driving force for charge recombination in eV. 1 eV = 96.48 kJ1 mol. Low-frequency reorganization energy, Average quantized mode frequency. Quantized mode (high frequency) reorganization energy. e Varies from 0.50 eV at 243 K to 0.44 eV at 333 K.
1.o
0.9
2
Y
2 $ 4
5
0.8 0.7 0.6 0.5 0.4
0.3 0.2
AV'(CT+So) is equal to 14 f 4 &mol; from the intercept, AH"(CT-So) is equal to -288 f 12 kJ/mol in the alkane solvents.25 B. Emission Spectra. The dominant feature in the emission spectra of 1 in alkane solvents is a broad featureless band extending from 400 nm to beyond 800 nm, with a maximum near 515 nm. Based on its large emission stokes shifts with increasing solvent polarity or polarizability, this band has been assigned as the CT SO charge transfer emission.8b The CT emission of 1 was recorded as a function of temperature from -30 to f 6 0 "C in heptane and methylcyclohexane (MCH) in order to investigate the effect of temperature on the driving force of back electron transfer. Qualitatively, as the temperature is lowered, the spectrum narrows slightly, and the emission maximum shifts to lower energies. The CT emission bands at each temperature were analyzed using Marcus' semiclassical description of optical charge transfer transitions.6 The Franck-Condon factors for the transition depend on the free energy change associated with the CT SO electron transfer process, AG"(CT-So): a lowfrequency reorganization energy, I S ; a high-frequency vibrational reorganization energy, I,; and an averaged quantized accepting mode of frequency, hw. The analysis of the CT state emission line shape of 1 in both heptane and MCH yields three sets of values for A,, hw, and I S that generate equally good fits to the observed line shape. The values of A", h,and 1 s determined in both solvents are identical, within experimental uncertainty. Table 2 lists all three parameter sets for heptane at 293 K. The emission spectra for the highest (333 K) and the lowest (243 K) temperature in heptane, along with the best fit calculated line shapes using set {C}, are shown in Figure 3. For all three sets of fitting parameters, 1" and ho obtained from the fitting procedure are, essentially, temperature independent. However, for the two sets {A} and {B}, reported previously,' Is exhibits a slight temperature dependence whereas for set {C} it is temperature independent. The three parameter sets also yield different temperature dependences for AG"(CT+So). With parameter sets {A) and {C}, AGo(CT4So) becomes less negative as the temperature decreases, in agreement with the hypsochromic shift of the emission maximum. Parameter set {B} shows the opposite trend, with the driving force becoming more negative as the temperature decreases. These different temperature dependences for the driving force lead to qualitatively different values of AW(CT-So) and AF(CT-SO). From a plot of AG"(CT-So) versus temperature, values of AS"(CT-So) equal to 21, -22, and 32 J/(K mol) are obtained from parameter sets {A}, {B), and {C}, respectively, in both heptane and MCH. From a plot of AG"(CT+So)/T versus UT values of AW(CT-30) equal to -272, -295, and -271 kJ/mol are obtained from parameter sets {A}, {B}, and {C}, respectively, in both heptane and MCH.
-..
-
IV. Discussion
A. Enthalpy of Charge Recombination. Two different experimental approaches have been used to determine the
0.1
14
15
16
17
18 19 20 21 Energy ( k K )
22
23
24
25
Figure 3. Experimental (solid line) and best fit calculated (symbols) CT state emission spectra from 1 in heptane. The simulations were calculated using parameter set {C}. The circles denote the spectrum at the lowest temperature investigated (243 K); the diamonds denote the spectrum at the highest temperature (333 K). The inset shows a plot of the mean photon energy,35 h(v,,) (in lo3 cm-I), versus temperature (K) for 1 in heptane. The error bars, hidden by the plot symbols, represent f l standard deviation for the average of h(vem) from sets {A}, {B}, and {C}.
enthalpy of the CT state charge recombination reaction in 1. Neither approach provides a direct measure of AH"(CT--So). In the POC experiments, the reaction enthalpy and volume contributions to the parameterf (actually the left-hand side of eq 6) must be separated. In the fluorescence experiments, the reaction enthalpy and entropy contributions to AG"(CT+So) must be determined. Furthermore, in both approaches, the quantity which is analyzed is obtained by fitting the experimental observables. The unknown quantity on the left-hand side of eq 6, is obtained as one of nine variable parameters in fits of the POC wave forms.26 Four of the nine parameters, At, w , kat, and k t d , are strongly constrained by the grating experiments with 2.The CT state decay rate constant, k - c ~ is , also strongly constrained through its measurement by time-resolved fluorescence. The real and imaginary index contributions from the CT state, N and K,are weakly constrained by independent determinations of K (vide infra). Thus, only the quantity f and the overall amplitude scaling factor, @, remain as wholly unconstrained quantities. Only five parameters are required to simulate the CT fluorescence line shape. However, significant a prior constraints exist for none of these parameters. As discussed in the Results section, equally good fits of the CT emission line shape at 294 K are generated using different values of A,, hw, and I S . Unfortunately, use of the three different parameter sets leads to qualitatively different temperature dependences of AG"(CT+So) and values of AS"(CT-So) that have different signs and magnitudes. Since there is little basis to choose a particular A,, hu pair, the average of the results from the three sets provides the least biased estimates of AW and AS". This yields a value of AW(CT-So) equal to -280 zk 25 kJ/mol. The estimate of AW(CT-So) from this fluorescence line shape analysis is within experimental uncertainty of the value obtained from the POC experiments, -288 f 12 kJ/mol. B. Volume of Charge Recombination. The analysis of the POC results from 1 in alkane solvents indicates that the partial molar volume of the CT state is 14 mL/mol smaller than that of the SOstate. The sign of this volume change is in agreement with predictions based on solvent electrostriction. The dipole moment and electric field gradients associated with the CT state are much larger than the corresponding quantities in SO.
8868 . I Phys. . Chem., Vol. 99, No. 21, 1995 Consequently, electrostatic interactions between the solvent and the CT state are larger, resulting in densification of the solvent around the CT state.I7 An estimate of the difference between the partial molar volumes of the CT and SO states may be obtained using a dielectric continuum model. The free energy of solvation for a point dipole p located at the center of a spherical cavity with radius a0 in a medium of dielectric constant EO is2'
Morais and Zimmt
An estimate of the entropy change accompanying charge recombination may be obtained using the dielectric continuum model. The entropy charge attending solvation of a point dipole, ASsolv,c~, is determined from the derivative of A G s o l v ,with ~~ respect to t e m p e r a t ~ r e ~ ~
ASsolv,CT
=
-aAGsolv;CT 2 aco-VaT --3L0E aT
a; ( 2
+ 1/Eo)2
2 Eo-l
AGsolv.CT
= -LO
l 'a03 2E0 + 1
The volume change attending solvation of the dipole, AVsolv,c~, is determined from the derivative of AGsolv,c~ with respect to pressure27
"solv,CT
=
aAGsolv;CT
ap
= 3Lo
2 aE,-'/aP a; (2 +
It can be demonstrated that AVsoiv,c~ -AV(CT-SO).~~ Calculation of AVsolv,c~ within the continuum model requires estimates of p2/aO3and of the pressure dependence of the dielectric constant, aeo-'/aP. The CT state of 1 neither is a point dipole nor does it lie within a spherical cavity. Nonetheless, a value of L,&42/m3)equal to 97 Id/mol was experimentally derived from the solvent dependence of 1's CT emission band maximum in alkane and ether solvents.29 Using aco-'/aP from the l i t e r a t ~ r e ,AVsolv,c~ ~~ is estimated to be -18 mL/mol in pentane and -12 mL/mol in heptane. These continuum predictions are in reasonable agreement with AV(S0-CT) of - 14 mL/mol determined using POC. Hara et aL30 studied the effects of high pressure on the steady state fluorescence from the CT state of 9,9'-bianthryl. They reported that the charge separation reaction is accompanied by a small volume change, AV"(LE--CT) % 2 "01, in both polar and nonpolar solvents. By contrast, the pressure dependence of the LE-CT equilibrium constant, derived from transient absorption kinetics in glycerol tria~etate,~' indicates that the partial molar volume of the 9,9'bianthryl CT state is 10-20 mL/mol smaller than for the LE state. The volumes of the structurally related, zwitterionic, twisted excited singlet states of tetraarylethylene~~'~~~ are 2030 mL/mol smaller than for the planar ground states. Despite different shapes and charge distributions, these three charge transfer systems exhibit comparable volume changes upon charge separation. In the analysis of the POC data for 1, AW(CT-So) and AV(CT-So) were assumed to be independent of alkane chain length. The validity of this assumption may be checked using the continuum results for AV(CT-So) to determine AH"(CT-So) directly from the experimental value off in each solvent. This analysis yields AW(CT-So) equal to -286 kJ/ mol in pentane and -292 kJ/mol in heptane. These values are in reasonable agreement with the POC estimate of -288 f 12 Id/mol. C. Free Energy and Entropy of Charge Recombination. The analysis of the CT state emission line shape as a function of temperature yielded three sets of best fit parameters. Each of these parameter sets acceptably reproduced the CT emission line shape but led to qualitatively different AS"(CT-So) and temperature dependences of AG"(CT-So). As mentioned previously, there is little basis to choose a particular parameter set. The average of the results from the three sets yields AS"(CT+So) equal to 10 f 30 J/(K mol) for 1 in alkane solvents.
Once again, it is readily demonstrated that ASsolv,c~L -AS"(CT-SO).~~ As before, a value of 97 kJ/mol for b(p2/ ao3)and literature24 values of aeo-'/aT yield AS"(CT-So) 2 18 J/(K mol) in pentane and 16 J/(K mol) in nonane. The large uncertainty in the value of AS"(CT-So) (and of AW(CT-So)) determined from the emission line shape analysis arises because AG"(CT-So) is not unambiguously featured in the emission spectrum. There is negligible intensity in the 0-0 region of the CT emission spectrum. Thus, AG"(CT-So) is obtained by an extrapolation which depends strongly on the values of A", hw, and As. The first moment of the emission spectrum,34i.e., the mean photon energy,35is a useful quantity and is well-defined by the data. All three fitting parameter sets generate the same value of the mean photon energy in the emission spectrum, h(~,,,,).~~ A plot of h(ve& versus temperature for 1 in heptane is shown in the inset of Figure 3. The slope of the best fit straight line to this data yields 47 J/(K mol) for 1 in heptane. A similar plot for 1 in MCH yields a slope of 43 J/(K mol). The mean photon energy is equal to -AG"(CT-So) - AS The derivative of this quantity with respect to temperature, AS"(CT-So) - MsIaT - aAv/aT, contains the temperature dependences of the reorganization energies in addition to the desired reaction entropy change. Unless the temperature dependences of the reorganization energies are insignificant (vide infra), AS"(CT-So) cannot be unambiguously determined from the h(Yed data without additional information. For organic structures such as 1, the reaction entropies may be partitioned into changes associated with the high-frequency, principally intramolecular, vibrations and with the low-frequency, principally solvent, motions. To the extent that the low-frequency contribution to the reaction entropy is entirely determined by the response of the solvent, it can be independently calculated from the experimentally determined volume of reaction using the thermodynamic relationship (aS/aV)T = a/p, where p is the coefficient of compressibility and a is the cubic thermal expansion ~ o e f f i c i e n t .In~ ~this manner, the solvent contribution to AS"(CT-So) in heptane is estimated to be 13 J/(K mol). Including the lower frequency modes associated with the methoxy and ester groups, it seems unlikely that there would be sufficient reductions in the intramolecular vibrational frequencies upon reverse electron transfer to contribute more than 4-5 J/(K mol) to A S " ( C T - + S O ) . ~ With ~ ~ > ~this ~ conjecture, AS"(CT--So) lies between 13 and 18 J/(K mol) and -(aA&T aAv/aT) lies between 29 and 34 J/(K mol). This range of values for AS"(CT-So) is in reasonable agreement with the continuum estimate and the "averaged" spectroscopic result described above. There is a paucity of experimental data in the literature concerning AS"(CT-So) or (WaT aA,/aT), for electron transfer reactions between organic molecules in nonpolar solvents, with which to compare the above partitioning of the spectral shifts. Yamagishi et aL3* reported that thermal formation of radical ion pairs from neutral molecules in polar solvents is characterized by a substantial entropy decrease, AS"(So--CT) = -70 J/(K mol). Meeus et al.39reported the entropic changes for (bimolecular) exciplex Av.34335
+
+
Thermodynamics of Intramolecular Electron Transfer formation between excited methylnaphthalene and amines in ether and alkane solvents to be M -130 and M -30 J/(K mol), respectively. Leinhos et a1.5breported an entropy change for the LE to CT transition of DMABN in toluene equal to -24 J/(K mol). The latter system is most similar to 1 in alkanes, in terms of molecularity and the solvent polarity, and exhibits a similar entropy change upon formation of the CT state. However, the potentially significant differences in the solvents, Le., aromatic versus aliphatic, and charge distributions in these two systems makes any statement regarding the accuracy of the spectral dissection, effected above, premature. As for the temperature dependence of the reorganization energies, negligible thermochroism has been observed in the maxima of symmetrical intervalence charge transfer absorption bands in nitrile34band aqueous solutions:0 indicative of temperature-independent reorganization energies. From an analysis presented by Hupp and Weaver:' the temperature dependence of the solvation reorganization energy attending charge recombination (zwitterionic to neutral) in these polar solvents is expected to be a small fraction of the corresponding entropy of reaction. The temperature dependence of Av is also expected to be small. Thus, the conclusion that -(aAs/aT W a T ) is nearly twice as large as AS0(CT4So) for 1 in alkane solvents finds little precedent in the literature. Recently, CortCs et al.42a analyzed temperature dependences of the CT emission bands produced by rigid donor-acceptor molecules containing a similar acceptor as in 1. They also found a larger temperature dependence for 1s than for AGo(CT4So). However, they attributed the apparent temperature dependence of AS to variations in the Franck-Condon factors of intermediate frequency modes (400-600 cm-I) coupled to the charge recombination. Temperature dependence studies of both the CT emission band and the photoinduced charge separation kinetics of rigid donoracceptor molecules in t e t r a h y d r ~ f u r a nindicate ~ ~ ~ that -( &/ aT aL,/aT) upon charge recombination is approximately half as large as A.S"(CT-So) in this solvent. Thus, the relative magnitudes of these two quantities (ASO(CT--So)) and -(a&/ aT aAv/8T)) may vary with solvent polarity. Further study of this problem is clearly warranted. Finally, it is worthwhile to compare the values of AG"(CT4So) at 293 K obtained using the different approaches. From the fitting of the CT spectra, the average AGO is -283 f 6 kJ/mol. The combination of A W from POC with AS" derived from the volume change for charge recombination yields AGO = -294 f 14 kJ/mol. These two results are in reasonable agreement. The reaction driving force may also be calculated from the redox potentials determined in acetonitrile, Coulomb's law, and the Born equationsa The center-to-center separation of the ions and the ion radii must be known in order to calculate the electrostatic corrections. Oevering et aLSaexperimentally determined that 4.5 A is a useful value43for the radii of similar donors and acceptors in related donor-spacer-acceptor (DSA) molecules. For various definitions of the ion centers in 1, the ion separation distance ranges between 5 and 7 Accordingly, the calculated AGO for the charge recombination reaction varies from -261 to -307 kJ/m01.~~ The experimental values of AGO above certainly fall within this range. Uncertainty of f l A in the definition of the ion separation or in the ion radii translates into uncertainties in AGO of greater than 23 kJ/mol (0.22 eV) in alkane solvents. This magnitude of uncertainty in the reaction driving force can have considerable impact on the analysis of electron transfer kinetics in nonpolar solvents. However, the ease with which relatively accurate estimates of AGO are obtained from informed of the simple
+
+
+
J. Phys. Chem., Vol. 99, No. 21, 1995 8869
electrostatic formulation, even in nonpolar solvents, supports their judicious use. D. Spectroscopic Properties of the CT State of 1 from POC. The two fitting parameters N and K originate from the absorption spectrum of the CT state of 1 at the probe wavelength. These two components are present at t = 0 and decay with the CT state lifetime as shown in eqs 3 and 4. From a comparison of Ak(jlpr,t)exwith An(&, a value of the CT state molar extinction coefficient at the probe wavelength can be obtained. Similarly, from a comparison of An(Apr,t)exand An(t),, the position of the probe wavelength relative to line center for the CT state absorption spectrum can be determined. The spatial modulation of the amplitude grating, Ak(AprJ)ex, is determined by the concentrationlS of the CT state and its extinction coefficient at the probe wavelength (eq 4). The amplitude of the diffracted signal arising from Ak(Apr,t)ex also depends on instrumental settings and beam overlap factors. The fitting parameter K may be related to Ak(Apr,t)ex as
where Z is the proportionality constant determined by the experimental setup. In order to determine the CT state molar extinction coefficient at the probe wavelength from K , Z and the CT state concentration must be eliminated from this equation. This can be achieved using the modulation in the real component of the solvent refractive index generated by the formation of the CT state, An(t),(s~-cT).~~ An(t),(So-CT) is the product of the prefactors, the term within the first { } and f l t : k = m ) in eq 2. Instrumental and beam overlap factors affect the amplitude of the grating signal produced from An(t),(So-CT) in the same manner as for Ak(,Ipr,t)ex.The combination of fitting terms comprising the thermal phase grating produced by the formation of the CT state may be equated with the theoretical expression
@ l - f ) S t : k = m ) = ZAn(t),(So-CT) Taking the ratio of these two expressions, with appropriate substitutions from eqs 4 and 2, yields the following expression for the molar extinction coefficient
where (6/@)fast is given by
Substitution of e, ( d n / d ~ ) p , *and ~ the results from the POC analysis (see Table 1) into this expression yields E(CT)633 = 1800 M-I cm-I in both pentane and nonane. This estimate of the molar extinction coefficient is reasonably close to the value obtained from flash photolysis, 800 f 200 M-' cm-l.I4 The dispersive component of the grating diffraction efficiency, An(dpr,t)ex,may be related to the fitting amplitude factor N in the same manner by which Ak(lZpr,t)exand K were related. Unfortunately, quantitative interpretation of An(Apr,t)exat a single wavelength is of little value. However, the sign of An(1prJ)ex indicates whether the probe is to the red or to the blue of the absorption band's center. From the fitting procedure, it is found that the numerical signs of N and the contribution (6/@)faSt are different. As the latter reduces the real component of the refractive index in the irradiated regions, the formation of the
8870 J. Phys. Chem., Vol. 99, No. 21, 1995
CT state must increase the real part of the sample refractive index at the probe wavelength. This indicates that the probe wavelength is to the red of the CT state absorption spectrum maximum,23in agreement with the transient absorption spectrum measured by flash phot01ysis.I~
V. Conclusion The thermodynamics of the intramolecular, charge recombination reaction of 1 in alkane solvents have been determined using calorimetric and spectroscopic methods. The molar enthalpy and volume changes of charge recombination, determined by POC, are -288 f 12 (20) kJ/mol and 14 f 4 ( 2 4 mL/mol. The volume change is similar to estimates derived from continuum models and to experimental measurements for charge recombination reactions in other intramolecular charge transfer molecules. The free energy of the charge recombination reaction has been determined in three ways: from a spectroscopic analysis of the CT emission band, AGO = -283 f 6 kJ/mol; from the calorimetric data, AGO = -294 f 14 kJ/mol; and from electrochemical data, continuum models for electrostatic corrections and judiciously chosen distance parameters, AGO = -289 f 23 kJ/mol. The agreement between the experimental and calculated free energy changes demonstrates that, with carefully determined distance parameters,8a the continuum models provide reasonably accurate free energy estimates, at least for this donor-spacer-acceptor molecule. Experimental determination of the reaction entropy for the charge recombination reaction in 1 presents the greatest challenge. The temperature dependence of AGO, extracted from analysis of CT emission band shapes, varies significantly depending on the reorganization parameters used to fit the spectrum. The temperature dependence of the mean CT band emission frequency can be accurately measured. However, this quantity depends on both the reaction entropy and the temperature dependence of the reorganization energies. Spectral studies of intervalence charge transfer bands in symmetric molecules in polar organic and aqueous solvents indicate very small temperature coefficients for the reorganization energies. Whether similarly small temperature dependences pertain in nonpolar and weakly polar solvents has yet to be experimentally determined.
Acknowledgment. We gratefully acknowledge financial support from National Science Foundation Grants CHE-9206765 and CHE-8957389 and from the Camille and Henry Dreyfus Foundation. We thank Dr. I. R. Gould (E. Kodak) for the determination of ion and CT state spectra and extinction coefficients. We also thank Prof. E. A. Mason (Brown) for helpful and enjoyable discussion of thermodynamics. Supplementary Material Available: Table of average values of w , k,,, and ktd from best fits for grating experiments with 1 and 2 in alkane solvents ( 2 pages). Ordering information is given on any current masthead page. References and Notes (1) (a) Moser, C. C.; Keske, J. M.: Wamcke, K.; Farid, R. S.; Dutton, P. L. Nature 1992, 355, 796. (b) Beratan, D. N.; Onuchic, J. N.: Winkler, J. R.; Gray, H. B. Science 1992, 258, 1740. (2) (a) Lenhard, J. R.; Hein, B. R.: Muenter, A. A. J . Phys. Chem. 1993, 97, 8269. (b) Koval, C. A.: Howard, J. A. Chem. Rev. 1992, 92, 411. (3) (a) Marcus, R. A. J. Chem. Phys. 1965, 43, 679. (b) Marcus, R. A. J . Chem. Phys. 1984, 81, 4494. (c) Marcus, R. A,; Sutin, N. Biochim. Biophys. Acta 1985,811,265. (d) Sumi, H.; Marcus, R. A. J . Chem. Phys. 1986, 84, 4894. (e) Levich, V. G. Adv. Electrochem. Electrochem. Eng. 1966,4,249. (0Levich, V. G.; Dogonadze, R. R. Dokl. Phys. Chem. 1959,
Morais and Zimmt 124, 9. (9) Hush, N. S. Trans. Faraday SOC.1961, 57, 557. (h) Kestner, N. R.; Logan, J.; Jortner, J. J . Phys. Chem. 1974, 78, 2148. (4) (a) Miller, J. R.; Beitz, J. V.; Huddleston, R. K. J. Am. Chem. SOC. 1984, 106, 1285. (b) Warman, J. M.; Smit, K. J.; de Haas, M. P.; Jonker, S. A,; Paddon-Row, M. N.; Oliver, A. M.; Kroon, J.; Oevering, H.; Verhoeven, J. W. J . Phys. Chem. 1991, 95, 1979. (c) Closs, G. L.; Miller, J. R. Science 1988,240,440. (d) Closs, G. L.; Calcaterra, L. T.: Green, N. J.; Penfield, K. W.; Miller, J. R. J. Phys. Chem. 1986, 90, 3673. (5) (a) Liang, N.: Miller, J. R.; Closs, G. L. J . Am. Chem. SOC.1989, I 11, 8740. (b) Leinhos, U.; Kiihnle, W.; Zachariasse, K. A. J . Phys. Chem. 1991, 95, 2013. (c) Weller, A. In The Exciplen; Gordon, M., Ware, W. R., Eds.; Academic Press: New York, 1975; p 23. (d) Warman, J. M.; Smit, K. J.; Jonker, S. A.: Verhoeven, J. W.; Oevering, H.; Kroon, J.: PaddonRow, M. N.; Oliver, A. M. Chem. Phys. 1993, 170, 359. (6) (a) Marcus, R. A. J. Phys. Chem. 1989, 93, 3078. (b) Gould, I. R.; Young, R. H.: Moody, R. E.: Farid, S. J. Phys. Chem. 1991, 95, 2068. (7) Morais, J.: Hung, R. R.: Grabowski, J. J.; Zimmt, M. B. J. Phys. Chem. 1993, 97, 13138. (8) (a) Oevering, H.; Paddon-Row, M. N.; Heppener, M.: Oliver, A. M.; Costaris, E.; Verhoeven, J. W.: Hush, N. S. J. Am. Chem. Soc. 1987, 109,3258. (b) Oevering, H.; Verhoeven, J. W.; Paddon-Row, M. N.: Hush, N. S . ; Warman, J. M. Tetrahedron 1989, 45, 4751. (9) (a) Miller, R. J. D. Time Resolved Spectroscopy. In Advances in Spectroscopy; Clark, R. J. H., Hester, R. E., Eds.: John Wiley and Sons: New York, 1989; Vol. 18, p 1. (b) Fayer, M. D. IEEE J. Quantum Electron. 1986, QE-22, 1437. (c) Morais, J.; Ma, J.; Zimmt, M. B. J . Phys. Chem. 1992, 96, 8359. (d) Zimmt, M. B. Chem. Phys. Lett. 1989, 160, 564. (10) Zeng, Y.; Zimmt, M. B. J . Phys. Chem. 1992, 96, 8395. (11) Press, W. H.; Flannery, B. P.; Teukolsky, S. A,: Vetterling, W. T. Numerical Recipes: Cambridge University Press: Cambridge, 1988; Chapter 10. (12) (a) Kogelnik, H. Bell Syst. Technol. 1.1969,48,2909. (b) Marcuse, D. Light Transmission Optics; Van Nostrand Reinhold: New York, 1972; Chapter 2. (c) Sun, T.; Morais, J.: Diebold, G. J.; Zimmt, M. B. J . Chem. Phys. 1992, 97, 9324. (13) Phkhnyi, C.: Sturm, P.; Jeffries, A. T., 111: Pannell, K. H. J. Coord. Chem. 1981, 11, 153. (14) Private communication from I. R. Gould. (15) In the limit of ideal plane wave excitation and probe (16) (a) Peters, K. S.: Snyder, G. J. Science 1988, 241, 1053. (b) Herman, M. S.; Goodman, J. L. J. Am. Chem. SOC.1989, 111, 1849. (c) Callis, J. B.; Parson, W. W.; Gouterman, M. Biochim. Biophys. Acta 1972, 267, 348. (d) Churio, M. S.: Angermund, K. P.; Braslavsky, S. E. J. Phys. Chem. 1994, 98, 1776. (17) (a) Asano, T.; Le Noble, W. J. Chem. Rev. 1978, 78, 407. (b) Schwartz, H. A. J . Phys. Chem. 1993, 97, 12954. (18) Technically, Ac(A-B) is the diference in the concentrations of A converted to B in the regions of maximum and of minimum excitation intensity. In the idealized case discussed above, the regions of minimum light intensity are exposed to no excitation light. (19) For the case of infinitely rapid heat release, 3 t : k ) is usually given as (1 - cos(wt))/2 where w is the radial frequency of the acoustic wave generated. (20) Use of Loh(v,,) with AHo(CT-So) is not strictly correct, as the former is related to a free energy. However, this error is insignificant due to the small value of a. (21) With k = -, . x t : k ) oscillates between 0 and the theoretical upper limit of 1. (22) Photoacoustic studies of triplet states usually assume AV(SO-TI) = 0. See for example: (a) Ni, T.; Caldwell, R. A,; Melton, L. A. J . Am. Chem. Soc. 1989, 111, 457. (b) Schuster, D. I.: Heibel, G. E.; Caldwell, R. A,: Tang, W. Photochem. Photobiol. 1990, 52, 645. (23) Demtroder, W. Laser Spectroscopy; Springer-Verlag: New York, 1982; Chapter 2. (24) Bartels, J.: Bruggencate, P. T.; Hausen, H.; Hellwege, K. H.; Schafer, Kl.; Schmidt, E. Landolt-Bornstein: Zahlenwerte und Funktionen; Hellwege, K. H., Hellwege, A. M., Eds.; Springer-Verlag: Berlin, 1959; Vol. 11.6. (25) The uncertainties in AW(CT-So) and AVD(CT-So) were obtained via analyses of the propagation of random errors in each of the quantities on the left-hand side of eq 6. The uncertainties in these terms are +0.01 in p,' f 9 . 6 kJ/mol in E(Tl),' f0.002 in a,' and f O . l kJ/mol in &h(v,,).' (26) The grating excitation and probe pulse widths are also, in principle, variable parameters, as the autocorrelation technique used does not enable determination of the actual pulse shape. For the relatively small grating wave vector used in these experiments, the values offobtained from the fits are not systematically altered by the assumed values of the excitation and probe pulses. (27) Whalley, E. J. Chem. Phys. 1963, 38, 1400. (28) (a) Following Weller,28b A G s o ~ v = . ~ ~AG"(CT-SO),~~ AG"(CT-So),*where AG"(CT-So),,, is the free energy change for charge recombination in the gas phase. Thus, A V s o i v ,= ~ ~AVD(CT--SO),~~AVO(CT-So), where AV"(CT--SO),~, is the volume change upon charge recombination in the gas phase. AVO,,,(CT-SO) is arguably set to zero as the rigid hydrocarbon spacer should prevent Coulombic attraction between
J. Phys. Chem., Vol. 99, No. 21, 1995 8871
Thermodynamics of Intramolecular Electron Transfer the anion and cation from producing a significant change in the bond lengths of 1. Thus, AIP(CT-So) = -AIPsoiv,c~.(b) Weller, A. Z. Phys. Chem. (Munich) 1982, 133, 93. (29) Beens, H.; Knibbe, H.; Weller, A. J. Chem. Phys. 1967, 74, 1183. Values of E were taken from ref 24. For the alkanes, E = n2 was assumed. (30) Hara, K.; Arase, T.; Osugi, J. J. Am. Chem. SOC. 1984, 106, 1968. (31) Leuck, H.; Windsor, M. W.; Rettig, W. J . Phys. Chem. 1990, 94, 4550. (32) Ma, J.; Dutt, G. B.; Waldeck, D. H.; Zimmt, M. B. J. Am. Chem. SOC.1994, 116, 10619. (33) Following a similar derivation as for AVsoiv,~~,** A S s o l v ,= ~~ AS"(CT-So),,, - AS"(CT-So), where AS"(CT-So),,, is the entropy change upon charge recombination in the gas phase. AS"(CT-So),,, is determined by changes in vibrational frequencies attending the electron transfer reaction. As discussed later in the text, this quantity is small, but likely positive (0-5 J/(mol K)), thus yielding the relationships AS"(CT-So) = AS"(CT-Sa)gas - A S o s o i v 2 , ~ -ASosoiv,~~. ~ (34) (a) Gould, I. R.; Noukakis, D.; Gomez-Jahn, L.; Young, R. H.; Goodman, J. L.; Farid, S. Chem. Phys. 1993, 176, 439. (b) Hupp, J. T.; Neyhart, G. A,; Mayer, T. J.; Kober, E. M. J . Phys. Chem. 1992,96, 10820. (35) Actually, the mean photon energy as described by the FranckCondon factors alone, without the frequency dependence of the radiative factors and of the transition dipole moment.6b (36) Castellan, G. W. Physical Chemistry, 2nd ed.; Addison-Wesley: Reading, MA, 1971; p 217. (37) In ref 10, the vibrational reorganization energy determined for the neutral acceptor was considerably smaller than that determined for the
reduced acceptor. Although an analysis of the vibrational frequencies was not performed, the result indicates a higher average frequency (curvature), for nuclear displacements in the reduced acceptor. (38) (a) Yamagishi, A.; Iida, Y. Bull. Chem. SOC.Jpn. 1980, 53, 1340. (b) Yamagishi, A.; Watanabe, F.; Masui, T. J . Chem. SOC.,Chem. Commun. 1977, 273. (39) Meeus. F.: Van der Auveraer. M.: De Schriver. F. C. J. Am. Chem. Soc. 1980, 102, 4017. (401 Done. Y.; Hum. J. T. 1nor.e. Chem. 1992, 31, 3322 (41) Hupp, J. T.; Weaver, M. J.-J. Phys. Chem. 1984, 88, 1860. (42) (a) CortCs, J.; Heitele, H.; Jortner, J. J. Phys. Chem. 1994, 98, 2527. (b) Tepper, R. J.; Zimmt, M. B. Work in progress. (43) The average ion radius was extracted from analyses of charge transfer emission spectra and bimolecular electron transfer quenching rate constants (ref 8a). (44) The charge separation distances were calculated using molecular mechanics from the CAChe software package. The distance from the center of the first ring of the dimethoxynaphthalene donor to the center of the dicarbomethoxyethylene double bond is 4.95 A. The distance from the center of the dimethox naphthalene donor to the carbonyl groups of the acceptor esters is 7.15 These two distances represent a lower and upper limit on the charge separation distance in 1. (45) The donor oxidation potential is 1.1 eV (ref 8a). The acceptor reduction potential is -1.64 eV (ref 10). S I CT. (46) The process is actually So
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