J. Phys. Chem. 1993,97, 23442354
2344
Dynamical Solvation Effects on the Cis-Trans Isomerization Reaction: Photoisomerization of Merocyanine 540 in Polar Solvents+ Yavuz Onganer, Mary Yin,* David R. Bessire, and Edward L. Quitevis’ Department of Chemistry and Biochemistry, Texas Tech University, Lubbock, Texas 79409 Received: September 22, 1992; In Final Form: December 22, 1992
The fluorescence quantum yield, @f, of merocyanine 540 in n-alkyl alcohol and n-alkanenitrile solvents was measured as a function of temperature and solvent shear viscosity at ambient pressure. The Strickler-Berg equation was used to calculate the radiative rate, k,,from the absorption and fluorescence spectra. The values of @randk,were used to obtain the rate of photoisomerization, kiso. This rate is consistent with the Smoluchowski limit of the Kramers theory for activated barrier crossing. The intrinsic barrier height, which is taken to be equal to the difference between the single-solvent activation energy and the viscosity activation energy, is lower in alcohols than in nitriles. The lower barrier can be partly rationalized in terms of static solvation effects which stabilize the twisted state of the dye. Because of the dye’s zwitterionic resonance structures, hydrogen bonding plays a key role in stabilizing the twisted state in alcohols. The intrinsic barrier depends weakly on the solvent for the nitriles, but it decreases dramatically in going from lower to higher alcohols. The behavior of the intrinsic barrier in these solvents is attributed to dynamical solvation effects which couple the solvent polarization to the intramolecular motion.
1. Introduction Cis-trans isomerization reactions in polar solvents provide useful examples in which to understand the role of solute/solvent friction and solute/solvent dielectric interactions in activated barrier crossing processes.1 Dielectric interactions (i.e., solvent polarity) play a central role in chemical reactions in solution.2 The most obvious effect is static and involves the solvation of reactants, products,and transition states. Staticeffectsaredivided into specific short-range solute/solvent interactions (e.& hydrogen bonding) and nonspecific interaction^.^*^ The nonspecific interactions arise from the bulk influence of the solvent as a dipolar medium and depend on the dielectric constant and the index of refraction. The second but less well-understood effect is dynami~.5-~ During the past decade, a large effort has been made to understand this effect, especially in intramolecular chargetransfer reactions, through the use of time-resolved Stokes shift techniques.*-I0 Briefly, if the reaction involves the generation or redistribution of charges, the surrounding solvent dipoles will respond in some time-dependent way. If solvation is rapid compared to the reactive motion, the transition state will be near the equilibrium solvated activation barrier. However, if the solvation is slow compared to the reactive motion, the reaction will proceed through nonequilibrium states of the solute/solvent complex. The barriers in this case will result from nonequilibrium solvation. The effect of solvent polarity on reactions will be complex, if solvent dynamics are strongly coupled to solute dynamics. Evidence for dynamical solvation effects has been recently found in the photoisomerization of stilbene and its analoguesin polar solvents.~ Studies of the photoisomerization of 4,4’-dimethoxystilbene and 4,4’-dihydroxystilbene have shown12 that static and dynamical solvation effects can be probed experimentally by comparing the rate of photoisomerization in nitrile solvents (nonassociating solvents), which have a rapid dielectric response, to the rate in alcohol solvents (associating solvents), which have a dispersive dielectric response. To shed further light on these dynamical solvation effects, we have begun a program in our laboratory to study the cis-trans * To whom correspondence should be addressed.
’ This work is dedicated to Dudley R. Herschbach in honor of his 60th
birthday.
1 1991 Welch Foundation Summer Scholar and 1992 Clark Foundation Summer Scholar.
photoisomerization of merocyanine dyes. Merocyanine dyes are interesting systems, because their spectral properties are very sensitive to both specific and nonspecific solute/solvent interacti~ns.l~-’~ The chromophore in merocyanine dyes is a resonance hybrid of an uncharged and a zwitterionic (dipolar) structure:
The zwitterion is stabilized by protic solvents through hydrogenbonding interactions with the negatively charged oxygen atom. In aprotic solvents, the zwitterion is not as stable. Previous studies of the merocyanine dye, stilbazolium betaine, have shown that solute/solvent interactions can also dramatically influence cistrans isomerization.16 Protonation of the oxygen atom either directly or through hydrogen bonding in protic solvents greatly increases the rate of isomerization in this molecule. In this article, we report a study of the photoisomerizationof merocyanine 540 (MC 540). MC 540 is an anionic lipophilic polymethinedye(Figure 1) which binds to biological and synthetic membrane~.I’-~~ The excited-state properties of MC 540 are sensitive to electrical potential, and its fluorescence has been used to probe the transmembrane potential of many cell and organelle membra ne^.^^-^^ The observation that leukemia cells stained with MC 540 are reduced by 5 orders of magnitude upon exposure to light has heightened the interest in its photophysical and photochemical proper tie^.^^ The mechanism for the cytotoxic behavior of MC 540, however, is controversial and remains an area of active research.2*-30 The photophysics and photochemistry of MC 540 have been investigatedby a number of groups over the last severalyear~.3’-’~ Its photophysics are characterized by a normal state N and a photoisomer state P.32Resonance Raman data34Pindicate that N correspondsto the molecule in the trans conformation and that P corresponds to the molecule in the cis conformation. Because of the large barrier which exists between the trans and cis forms in the electronic ground state, the dye is predominantly in the trans state. Optical excitation reduces the bond order in one of the double bonds along the polymethinechain, thereby allowing the molecule to twist during its excited-state lifetime. In the twisted conformation, the molecule rapidly undergoes internal
0022-3654/93/2091-2344904.00~08 1993 American Chemical Society
Effects on the Cis-Trans Isomerization
The Journal of Physical Chemistry, Vol. 97, No. 10, 1993 2345
a (trans)
a (cis)
. . ,T
N, Rz I
RZ
1 q
.
RZ
i
R
o
Rl
y s
z
y
s
-
q + l N , R Z R,
-0
RZ
b
' A s RZ
C
Figure 1. Resonance structures of merocyanine 540 in the all-trans conformation and a possible cis conformation; R I = (CH&SOj-Na+, Rz = (CH2)jCHj.
conversion to the electronic ground state. From the twisted conformation, the molecule can relax to either N or P. Because P is higher in energy than N (39 kcal mol-' in 95%ethanol),32a P will convert back to N. At high excitation levels, such as those obtained under intense laser irradiation, the concentration of P can be high enough that absorption and fluorescencedue to both N and P must be considered. The structure of MC 540 is a mixture of an uncharged resonance structure (resonance structure a) and two zwitterionic resonance structures (resonance structures b and c). In the zwitterionic forms of MC 540, the negative charge resides on the oxygen atoms of the thiobarbituric acid moiety. A sulfonate group is attached to the chromophoreby a propyl chain on the benzoxazole subunit. Because the negative charge on the sulfonate group remains localized, solute/solvent interactions at this site play less of a role in influencing the electronicpropertiesof the chromophore than do solute/solvent interactions at the carbonyl groups. The cis structure that results from isomerization about the central C-C bond in the polymethine chain is shown in Figure 1. This example illustrates the relationship of the structures that are important in cis-trans isomerization. The central C-C double bond breaks and becomes a single bond during the isomerization process. Clearly, the twisted intermediatestateof MC 540should have a high degree of zwitterionic character and should be very polar. Solute/solvent interactions that stabilize the zwitterion will therefore have the largest impact in modulating the isomerization rate. Although the static aspects of solvation have been previously established in the cis-trans photoisomerizationof MC 540,30b934b the dynamicaspects have not. This article describes a comparative study which probes these dynamical effects in n-alkyl alcohol and n-alkanenitrile solvents. The article is organized as follows. In section 2, we describe how the photoisomerization rates were inferred from fluorescencequantum yield and steady-statespectral measurements. In section 3, the photophysical parameters and single-solvent activation energies are presented and compared to values previously reported by other workers. In section 4, photoisomerization rates are analyzed by using the hydrodynamic Kramers equation in both the intermediate- and high-friction limits. We find that a more consistent interpretation of the data is obtained in the high-friction limit, if we allow the intrinsic barrier to vary with solvent. The behavior of the intrinsic barrier height in these solvents can be rationalized in terms of dynamical solvation effects which couple the solvent dynamics to the solute dynamics. 2. Experimental Details
Merocyanine540 (Molecular Probes or Sigma) and rhodamine 101 (Exciton, laser grade) showed single spots on a thin-layer chromatographyplateand wereused without further purification. Solvents of the highest purity available were used. The solvents
were further dried over molecular sieves and purged with nitrogen. This procedure ensured greater reproducibility in the results. For example, anomalously high values of the activation energy were obtained for I-octanol, unless the solutions were prepared with 99.99+%anhydrous solvent under nitrogen. MC 540 was stored in the dark as a concentrated stock solution (1 mM) in a 1:l volume mixture of ethanol and carbon tetrachloride. Samples were prepared by evaporating 12 pL of the stock solution and then redissolving with 4 mL of solvent. The final concentration was * 3 pM. Solvent shear viscosities were taken from the literature.'2b,cValues of the solvent polarity parameter, &(30), were obtained from the literaturea2 They were also determined from the energy corresponding to the peak of the absorption spectrum of Reichardt's dye, pyridium-N-phenoxidebetaine, in the solvent of interest. The absorption spectra at room temperature were recorded on a Shimadzu 265 UV-vis spectrophotometer. Theemissionspectra were recorded on an SLM Aminco 4800C fluorometer. The samples in the fluorometer were contained in a 1-cm cuvette that was maintained to f l "C with a temperature-controlled water circulator (Neslab R T E J ) . Fluorescencequantum yields from corrected fluorescence spectra were calculated by using the equation35
where D is the integrated area under the corrected fluorescence spectrum, n is the refractive index of the solution, and OD is the optical density at the excitation wavelength ( L= 520 nm). The subscriptss and r refer, respectively, to the unknown and reference solutions. The values of the OD were 0.10-0.15. The values of D were obtained by numerically integrating the areas under the corrected fluorescence spectra. Rhodamine 101 (Rh 101) in ethanol was used as the reference. The fluorescence quantum yield of Rh 101 (@r = 1.00) is relatively independent of temperature in alcoholicsolvents.3G3*In principle, thedependence of n with temperature must be taken into account. However, this dependence is weak in the temperature range ( 4 - 8 0 "C) of our measurements. For example,because the temperature coefficient of the refractive index, dnldt, lies within the range (3.74.0) X 1W K-'for n-alkyl alcohols,39nz varies by no more than 3%over the temperature range of our measurements, which is less than the 10% error in the measured values of @ f a Because of the low excitation intensities in our measurements, the fluorescence is assumed to occur only from the N state. The fluorescencequantum yield is related to the radiativerateconstant, k,,and to the nonradiative rateconstant, k,,,by the photophysical equation where In this equation, kic, kist, and ki, are the rates for internal conversion, intersystem crossing, and photoisomerization. Thus, information about photoisomerization is contained in @f. Rearranging eq 2a yields
(3) This equation implies that the measured values of arcan be directly converted to values of k,,, if the values of k, are known. The values of k,are obtained in this work by using the Strickler-Berg (SB)eq~ation.~O4'This method of calculating k,, was used by Hochstrasser and co-workers42 in their studies of the photoisomerization of "stiff" stilbene. The SB equation is a modification of Einstein's fundamental relationship between the probability of absorption and emission, which is applicable to polyatomic molecules in solution. Through this equation, k,can be obtained
2346
Onganer et al.
The Journal of Physical Chemistry. Vol. 97, No. 10, 1993
TABLE I: Comparison of Solvent Properties and Photophysical Parameters for Merocyanine 540 in &Alkyl Alcohols and RAlkaneniMles at 25 O C solvent E ~ ( 3 0 ) , kcal " mol-' q,b CP @f kr,dns-' k,,,' ns-I
- - Methanol
400
450
500
550
600
Wavelength (nm)
methanol ethanol 1-propanol 1 -butanol 1-pentanol 1-hexanol 1-heptanol 1-octanol acetonitrile propionitrile butanenitrile pentanenitrile hexanenitrile heptanenitrile octanenitrile nonanenitrile
55.13 52.00 50.35 49.53 49.12 48.80 48.50 48.30 46.00 43.70 43.10 42.70 42.00 41.85 41.10 40.30
0.55 1.14 1.95 2.60 3.57 4.54 5.81 7.32 0.348 0.416 0.542 0.678 0.917 1.19 1.55 1.95
0.13 0.20 0.23 0.28 0.31 0.35 0.42 0.58 0.20 0.25 0.29 0.30 0.42 0.48 0.57 0.62
0.48 0.45 0.43 0.45 0.38 0.53 0.50 0.48 0.48 0.46 0.53 0.53 0.50 0.50 0.48 0.45
3.2 1.8 1.4 1.2 1.0 0.98 0.69 0.35 1.9 1.4 1.2 1.2 0.69 0.54 0.36 0.28
"Solvent polarity parameter, &(30), obtained from ref 2 or by measuring the absorption maximum of Reichardt's dye. Solvent shear viscosity, 7 , obtained from refs 12b,c. Fluorescence quantum yield, @f, obtained from eq I ; uncertainty &lo%. Radiative rate constant, k,, calculated by using thestrickler-Berg formula (eq4); uncertainty 120%. Nonradiative rateconstant, k,,,calculated from eq 3; unccrtaintyf3046.
*
TABLE II: Arrhenius Parameters for Solvent Viscosity and Nonradiative Rates in &Alkyl Alcohols and bAlkanenitriles solvent 70,'' lo-' CP E,," kcal mol-' IO4AQb Ea!c . kcal mol-'
500
550
600
650
700
Wavelength (nm) Figure 2. Steady-stateabsorption (a, top) and emission (b, bottom) spectra of merocyanine 540 in four solvents at 25 OC.
from the absorption and fluorescence spectra. It is given by
k, = 2.88
X 1 0 - g n 2 ( ~ ~ 3 ) A ~ 1 ( g , /dgIn u )Y~ t , .(4a)
where t, (in units of M-1 cm-I) is the decadic molar extinction coefficient a t frequency Y (in units of cm-I) and gl and g, are the electronic degeneracies of the lower and upper electronic states. The term in the angular brackets is the weighted frequency function
where I ( v ) is the fluorescence intensity. The integrals in eqs 4a and 4b involve integration over the whole electronic band. A comparison of calculated and experimental values of k , for aromatic dyes shows that the SB equation is accurate to within 20-25%.40*43 3. Results
A. PhotoisomerizationRates and FluorescenceQuantum Yields. The typical spectra used in calculating k, by the SB equation are shown in Figure 2. The calculated values of k,are listed in Table I. Previous studies have shown that the measured value of k, increases linearly with n2,as predicted by the SB e q ~ a t i o n . In 3~~ a homologous series, n increases with the size of the alkyl group: in the series methanol to 1-octanol, n increases from 1.3288 to 1.4249, and in the series acetonitrile to nonanenitrile, n increases from 1.344 to 1.426. Hence, k, should increase by 15% in going from methanol to 1-octanol and by 13%in going from acetonitrile to nonanenitrile. The 20-25% accuracy of t h e SB equation,
methanol ethanol 1-propanol 1-butanol 1-pentanol 1-hexanol I-heptanol 1-octanol acetonitrile propionitrile butanenitrile pentanenitrile hexanenitrile heptanenitrile octanenitrile nonanenitrile
7.7 3.6 1.4 1.1 0.53 0.64 0.41 0.25 19.7 18.0 10.6 9.95 10.8 7.23 4.74 3.40
2.53 f 0.10 3.37 f 0.10 4.29 f 0.10 4.60 f 0.10 5.22 f 0.24 5.25 f 0.12 5.66 f 0.41 6.09 f 0.20 1.70 f 0.16 1.86 f 0.10 2.33 f 0.20 2.50 f 0.10 2.63 f 0.20 3.02 f 0.20 3.43 f 0.32 3.76 f 0.44
1.5 1.5 2.5 2.3 1.6 1.5 1.5 3.0 1.02 1.08 1.07 1.5 2.4 3.5 8.3 8.0
4.57 f 0.94 4.89 f 0.22 5.27 f 0.24 5.40 f 0.29 5.24 f 0.80 5.33 f 0.38 5.53 f 0.49 6.02 f 0.60 4.70 f 0.20 4.89 f 0.26 5.03 f 0.32 5.35 f 0.39 5.80 f 0.83 6.18 f 0.74 6.81 f 0.97 6.82 f 0.95
Theviscosity prefactor,qo, andviscosity activation energy,& obtained from refs 12b,c or by fitting viscosities at different temperatures to eq 14. Arrhenius parameters obtained by linear least-squares fit of semilogarithmic plots of (1 /@f) - 1 over temperature range 0-80 OC. Error is calculated as described in the Appendix.
however, masks the variation in k , due to n2. This would explain why the values of k, listed in Table I do not vary systematically with n2as one would predict from the SB equation. The average value of k, calculated from the SB equation is 0.47 f 0.04 ns-1 for the alcohols and 0.49 f 0.03 ns-I for the nitriles. The SB values of k, are comparable to the value in methanol (0.42 ns-I) calculated by Davila et al.30 and the value in ethanol (0.4 ns-1) calculated by Aramendia et al.32a Fluorescencequantum yields a t 25 "Care listed for comparison in Table I. Within experimental error, our valueof Or for methanol agrees with the one reported by Davila et al.30 for methanol (Of = 0.13 f 0.01) but differs from that (Of = 0.26) reported by Hoebeke et al.33cThe value of Or for ethanol agrees with the one reported by Aramendia et al.32a for ethanol (Of= 0.15 f 0.02) but differs from that (Of= 0.39) reported by Hoebeke et al.338 Despite these differences, the fluorescence lifetime, T I , calculated from the equation, T f = @f/k,, using our measured values of Of and k,, is 0.27 ns in methanol and 0.44 ns in ethanol. These values coincide with the literature values of 0.2330band 0.43 ns,32 respectively. The nonradiative rate measured at 25 OC,which was calculated by using eq 3, depends strongly on the solvent. It decreases in a homologous seried as the length of the alkyl chain
The Journal of Physical Chemistry, Vol. 97, No. 10, 1993 2341
Effects on the Cis-Trans Isomerization
3 7 -
ik I
0
Alcohols Nitriles
\
J
2
4 6 Number of CH2 groups
a
Figure 3. Correlation of the nonradiative rate at 25 O C with the number of CH2 groups in the alkyl chain of solvent.
3.0
3.2
3.4
3.6
1OOO/T (K-') Figure 4. Typical Arrhenius plots of (1 /@r) - 1 for merocyanine 540 in four solvents. Lines are linear least-squares fits of data. Arrhenius
increases, and it is faster in an alcohol solvent than in a nitrile parameters are listed in Table 11. solvent with the same alkyl chain (Figure 3). Ideally, we would like to obtain ki, from knr. This can be linear regression of ln[(l/@r) - 11 vs l / T a r e listed in Table 11. difficult becauseof the presenceof theother nonradiativechannels In keeping with the terminology of Waldeck and co-workers,12 (ki, and kilt). Fortunately, photoisomerization is the dominant we will call E, the single-solvent activation energy. The value pathway for nonradiative decay in MC 540. This can be seen as of E. changes by =1.5 kcal mol-' in going from methanol to follows. Because the triplet quantum yield of MC 540 is typically and by =2.1 kcal mol-l in going from acetonitrile to less than 0.05 in the temperature range of our m e a s ~ r e m e n t s , ~ ~ ~ -1-octanol ~~ nonanenitrile. The values of E, in methanol and ethanol are ki, can be neglected relative to ki, + ki,. It has been assumed consistent, respectively, with the values of 4.8 and 5.5 kcal mol-l in previous studies that internal conversion does not occur in the which were previously reported.3hJ2 Note that the linearity of N ~tate.3~33~ Hoebeke et have recently measured @randthe the plots of In[(l/@f) - 11 vs 1 / T is also consistent with k,, = photoisomerization quantum yield, @isom, of MC 540 in glycerol/ ki,. The rate of internal conversion depends weakly on temethanol mixtures. The values of @isomin these mixtures were perature.45 Therefore, if internal conversion is a significant determined by directly detecting P by flash photolysis. By varying pathway to nonradiative decay, lowering the temperature should the relative amounts of glycerol and ethanol in the mixtures, they cause k,, to approach a constant value correspondingto ki,. If ki, were able to vary the viscosity of the mixtures. As the viscosity were not negligible compared to kiw, the plots of In[( l/%) - 11 of the mixtures is increased, @r increases, while @isom decreases. vs 1/T would exhibit non-Arrhenius behavior, deviating from For all values of the viscosity, @r aiwm = 1. Furthermore, the linearity at low temperatures. viscosity dependences of @r and @isomare the same, within experimentalerror. This is consistentwith observations made by 4. Discussion Aramendia et al.32who showed that the photoisomer directly arises from the first excited singlet state. These results imply A. Separation of Solvent Effects. As others have previously that internal conversion does not occur or is not a major pathway the decrease in the rate of photoisomerizationof for nonradiative decay in MC 540. Therefore, we can essentially MC 540 can be correlated to the solvent shear viscosity (Table equate the nonradiative rate to the rate of photoisomerization, I). The rate also depends on solvent polarity, as demonstrated i.e., kn, = ki,. by the fact that the rate is faster in alcohols than in nitriles. This B. Single-Solvent Activation Energies. Although the intensity is readily apparent when one compares the rate in 1-propanol to of the fluorescence decreased with increasing temperature, the that in nonanenitrile at 25 O C . The solvent shear viscosity for shape of the fluorescencespectrum did not. Davila et al.30b have these solvents is the same and equal to 1.95 cP, but the rate is also found that the absorption spectrum of MC 540 does not 5 times faster in 1-propanol than in nonanenitrile. To further change with temperature. Therefore, on the basis of the SB understand the nature of this dependence on the solvent, solute/ equation, the temperature dependence of k, will be determined solvent friction and dielectric interactions must be separated. primarily by the temperature dependence of n2. As discussed in This separabilityis achieved by using the activated barrier crossing section 2, n2 does not vary much with temperature. Over the formula modest temperature ranges of our measurements, k,is relatively independent of temperature, varying by at most 396, well within the accuracy of the SB equation. According to eq 3, the where EOis the intrinsic barrier height, R is the gas constant, and temperature dependence of kn,can thus be determined from the T is the absolute tem~orature,~2.42.*The prefactor, F(J3, is a temperature dependence of (1/@PI) - 1. Figure 4 shows that plots dynamical quantity that depends on the solute/solvent friction, of ln[(l/@r) - 11 vs 1 / T are linear (correlation 2 0.98). The 1: In this form, frictional effects, and hence the viscosity linearity of the plots in Figure 4 can be rationalized if eq 3 is dependence of the rate, can be compared to models for F(0. combined with an Arrhenius equation for k,, Solvation effects are manifested by solvent-induced variations in EO.^^*^^,^^-^^ In order to address the dynamical aspects of knr An, W - E a l R T ) (5) solvation in the photoisomerization of MC 540, treatments that to give only emphasize the static aspects of solvation will be avoided. Specifically, we will not assume EO to be the sum of a solvent(6) ( l / @ J - 1 = A, exp(-E,/RT) independent barrier height and a term that varies linearly with some empirical solvent polarity parameter.' Although the where A$ = A,,/k,. The Arrhenius equation is an appropriate effect of solvent polarity on the photophysics for a few systems form for the nonradiative rate because of photoisomerization. has been explained with some success by this approach,ll*47one Equation 6 has been previously used to obtain the activation must be cautious in interpreting the results. The use of a single energies for MC 540 and for other systems which undergo parameter to characterize complex solute/solvent interactions is photoisomerizati~n.~~*~ The Arrhenius parameters obtained by
+
Onganer et al.
2348 The Journal of Physical Chemistry, Vol. 97, No. 10, 1993 an oversimplication. These parameters are derived from the solvent-induced spectral shifts of the dyes. These polarity parameters have physical relevance only if the solvation around the dye used to define a particular solvent polarity parameter is similar to the solvation around the solute that is being studied. The ET(30)scale and the Kamlet-Taft ?y* scale are two commonly used polarity scales.* The E ~ ( 3 0scale ) is based on the effect of solvent on the lowest energy electronictransition of the pyridiumN-phenoxide betaine dye, whereas the Kamlet-Taft T* scale is based on the effect of solvent on T r* transitions of a variety of nitroaromatics. The E ~ ( 3 0 scale ) is sensitive to hydrogenbonding effects, whereas the **scale is not. Thus, the use of the ?F* scale instead of the E ~ ( 3 0scale ) to characterize the effect of polarity on a photophysical property may lead to differences in the interpretation of the solvent effect. Another drawback in using these polarity parameters is that they refer to the effect of solvation on Franck-Condon transitions. This solvation, which is described by the Onsager reaction field on the excited state,49may not correspond to the solvation in regions of the potential energy surface that are relevant to the chemical reaction. Ephardt and F r ~ m h e r zhave ~ ~ argued that the effect of solvation on EOcan be better estimated by the Born energy. The Born energy corresponds to the energy required to charge a sphere of radius r in a medium with dielectric constant e. The solvation energy, which results from a difference in the Born radii of the fluorescent state and the transition state, is proportional to l/c. This approach is also rather restrictive because it assumes equilibrium solvation and neglects the effect of specific solute/solvent interactions. B. Kr~mersTheory.Thesimplest way to treat frictional effects is through the Kramers theory.50 In the Kramers theory, the reaction is modeled by the motion of a particle in a one-dimensional potential well, with the motion modified by frictional drag and a random fluctuating force. In the 'intermediate"-friction regime, the dynamical factor F({) is given by
2.5
Y
1.o
-
where oris the reactant well frequency,O b is the barrier frequency, and Zr is the reduced moment of inertia of the isomerizing moieties. The Kramers equation is an approximate expression for the flux of population across a barrier. It assumesthat the time-dependent distribution of isomerization coordinates and velocities obeys a Fokker-Planckequation. Equation 8 governs the behavior when solvent collisions or friction induce barrier recrossing. In this regime, F( {) decreases as { is increased. The parameter wdr/{measures thestrength of frictional forces inhibiting passageover the barrier comparedto conservative forces which drive the solute down the barrier. If wd,/{
