J. Phys. Chem. 1987, 91.4714-4723
4714
B
A
Figure 4. Fluorescence line shapes for the 4 3 ' : transitions. The excited modes are (a) A, (b) K, (c) L,and (d) S. Slit resolutions are (a) 10, (b) 10, (c) 25, and (d) 3 0 cm-I. The traces have been justified in frequency and the intensities made arbitrarily equal. Sum of four to nine scans.
excitation merges below 1300 cm-l as for coronene-h12below 2300 cm-I. This is of course a consequence of the increased densityof-states in coronene-d12. Thus, the 4i3: emission (Figure 4D) is a broad featureless line, not unlike the k' profile shown in Figure 6d of I. Presumably, a similar, large number of dark levels are
effectively coupled to the e2, mode in coronene-d,, at 1294 cm-' (level S), as in coronene-h,, a t 2356 cm-I (level k' in ref I). This behavior of the v i mode is in stark contrast to that of the h,, equivalent, where a broad doublet with a 45-cm-I separation was observed. We believe this comparison strengthens our previous suggestion for the origin of the splitting: a small number of dark states does not constitute an adequate statistical basis for a normal distribution in the line profile. Going from coronene-h,, to coronene-d12,the same mode u i will couple anharmonically to the same set (symmetrywise) of dark combination states if available. On increasing the density-of-states, however, the number of dark states contributing to the red-shift in the dI2species has increased to the extent that the splitting is conspicuously absent. As a further corroboration, consider the Li3: fluorescence (Figure 4C) from level L, 974 cm-' above the origin. With a slight stretch of the imagination a weaker component to the blue of the main peak may be identified as lining up with Figure 4A,B. While the signal-to-noise ratio is poor, the same feature appears in each separate scan making up the final picture, so the peak is real. This behavior mimics the splitting of the 4i3: line (at 1348 cm-') in coronene-h12,except that the vibrational energy of the gateway state is now 974 cm-'. We believe that the effects of vibrational relaxation to a limited subset of dark states are being observed in both cases. Again, the energy domain for the appearance of this intermediate region is consistent with the increase in the density-of-states in the perdeuterio species. The lifetimes measured for coronene-d,, in a jet were about 200 ns longer than those for the perhydro species. This is consistent with a decrease in the intersystem crossing rate in perdeuteriated coronene. The same general trend of near-constant lifetime, as vibrational energy increases, also prevails in the perdeuteriated species. Such lifetime data are included in Table I.
Conclusion This work complements and completes a jet spectroscopic investigation of coronene started in I. Most of the ez8vibrations (1 1 out of 12) are now identified. Two tentative interpretations of interesting spectral line profiles in I are now established beyond reasonable doubt. Registry No. Coronene-dr2,16083-32-2.
Solvent Reorganization in Optical and Thermal. Electron-Transfer Processes: Solvatochromism and Intramolecular Electron-Transfer Barriers in Spheroidal Molecules Bruce S. Brunscbwig,* Stanton Ehrenson,* and Norman Sutin* Department of Chemistry, Brookhaven National Laboratory, Upton, New York (Received: February 25, 1987; In Final Form: May 4, 1987)
1I973
Expressions for the shift of the absorption and emission band maxima of a solute with changes of the solvent's dielectric properties are presented for solute molecules with shapes and polarizabilitiesthat can be approximated by those of a spherical dielectric-continuum cavity. The derivations use a nonequilibrium thermodynamic approach developed by Marcus. A series of accurate approximations for the solvent shifts are presented. The new expressions are shown to reduce to equations previously derived by McRae, Ooshika, Mataga, and others when point-dipole approximations are made and particular values of the solute polarizability are assigned. The polarizability of the solute can be related to the internal dielectric constant of the cavity, and general expressions for the band shift in the point-dipole limit containing the dielectric constant of the cavity as a parameter are derived.
The interaction of a solute with surrounding solvent is sensitive to the charge distribution and polarizability of the solvent (and solute) molecules. If the solvent interacts with the solute in a specific manner (e.g., by hydrogen bonding or other donor-acceptor interactions), large changes in the energy of the solvated 0022-3654/87/2091-4714$01.50/0
molecule can occur with changes in solvent. Even if the solutesolvent interactions are nonspecific, changes in the energy of the solvated molecule will still occur with changes in solvent, especially for polar solvent molecules or solutes that have high charges or dipole moments. The various solute-solvent interactions lead to 0 1987 American Chemical Society
Solvent Reorganization in Electron-Transfer Processes
Figure 1. Plot of the energy of the reactants (precursor complex) and products (successor complex) as a function of nuclear configuration. X is the vertical reorganization energy, Eabsis the energy of the absorption band maximum, and AGO is the free energy difference between the ground and thermally relaxed excited states.
a solvent dependence of charge-transfer spectra'-7 and of the rates of electron-transfer Both of these effects arise from the differences in the solute-solvent interactions in the initial and final states of the charge-transfer process. Photon and/or electron capture or loss can also affect the structure of the molecule directly (i.e., change its equilibrium bond distances and angles), and these structural changes also need to be considered. The absorption of a photon by a solute is governed by the Franck-Condon p r i n ~ i p l e ~which * ~ , ' ~requires that the nuclear configurations and momenta of the solute and solvent molecules be the same before and immediately after the optical transition. Subsequent to the optical transition, the intramolecular nuclear configurations and orientations of the surrounding (polar) solvent molecules relax to the configurations appropriate to the thermally equilibrated electronically excited state. These nuclear configuAs shown in this ration changes are illustrated in Figure figure, there are two contributions to Eabs,the energy of the absorption maximum: these are AGabo, the free energy difference between the thermally equilibrated ground and excited states (Ge0 - Ggo), and A, the distortion or reorganization energy of the electronically excited state (Le., the energy difference between the Franck-Condon and the thermally equilibrated excited state^).^,^^^^ Eabsis related to these quantities by l.439
The reorganization energy of the excited state has two contributions; one (but) arises from the difference in solvent polarization between the Franck-Condon and the thermally equilibrated excited state and the other (A,,) from the corresponding changes in the internal structure of the solute m o l e ~ u l e . ~Provided ~'~~~~ that hydrogen-bonding and other specific interactions between the solute and the solvent are absent or can be neglected, the (1) Ooshika, Y . J. Phys. SOC.Jpn. 1954, 9, 594. (2) Lippert, E. Z. Naturforsch., A 1955, IOA, 541. Lippert, E. Ber. Bunsenges. Phys. Chem. 1957, 61, 562. Lippert, E. Z . Elektrochem. 1957, 61, 962. (3) Bayliss, N. S.; McRae, E. G. J. Chem. Phys. 1954,58, 1002. McRae, E. G. J. Phys. Chem. 1957,61,562. McRae, E. G. Spectrochim. Acta 1958, 12, 192. (4) Marcus, R. A. J. Chem. Phys. 1965, 43, 1261. (5) Liptay, W. Angew. Chem., Int. Ed. Engl. 1969, 8, 177. Liptay, W.; Walz, G. Z . Naturforsch., A 1971, 26A, 2007. Liptay, W. In Excited States; Lim, E. C . , Ed.; Academic: New York, 1974; p 129. (6) Mataga, N.; Kaifu, Y.; Masao, K. Bull. Chem. SOC.Jpn. 1955, 28,690. Mataga, N.; Kaifu, Y.; Masao, K. Bull. Chem. SOC.Jpn. 1956, 29, 465. Mataga, N. Bull Chem. SOC.Jpn. 1963, 36, 654. Mataga, N.; Kubota, T. Molecular Interactions and Electronic Spectra; Marcel Decker: New York, 1970; pp 371-410. (7) Hush, N. S. Prog. Inorg. Chem. 1967, 8, 391. (8) Marcus, R. A. J. Chem. Phys. 1963.38, 1335. Marcus, R. A. J. Chem. Phys. 1963, 39, 460. Marcus, R. A. J. Chem. Phys. 1963, 39, 1734. (9) Sutin, N. Annu. Reu. Nucl. Sci. 1962, 12, 285. (10) Sutin, N. Prog. Inorg. Chem. 1983, 30, 441. (11) Marcus, R. A.; Sutin, N. Comments Inorg. Chem. 1986, 5 , 119. (12) Marcus, R. A.; Sutin, N. Biochim. Biophys. Acta 1985, 881, 265.
The Journal of Physical Chemistry, Vol. 91, No. 18, 1987 4715 solvent shift of the absorption band arises from changes in AGabo and A,,. If the excited state decays to the ground state partly by photon emission (for which the energy change is defined to be positive), then the energy of the emission maximum is similarly determined by AG,,' = Gg0- Ge0 = -AGabso and A (eq 1b) E,, = -(Ace,' + A) = AGabso- h (1b) where X is now the reorganization energy of the ground state. Use of the same value of X in eq l a and l b implies that the free energy surfaces of the ground and excited states are (displaced) mirror images. The assumption that the surfaces are harmonic with identical force constants fulfills this condition and is made throughout this paper. From eq l a and l b it follows that the sum and difference of the energies of the absorption and emission maxima are equal to 2AGabSoand 2X, respectively. These are important results since, provided that the excited state is the same for both absorption and emission, the free energy changes and reorganization energies can be directly obtained from the energies of the absorption and emission maxima. In addition to harmonic free energy surfaces, use of eq l a and l b implies that the absorption and emission spectra have no vibrational structure and are Gaussian in shape. This requires that the frequencies of all vibrational and orientational modes contributing to X satisfy the condition hv, 200 cm-l, at room temperature the solvent modes, but not the intramolecular modes, will be in the high-temperature limit. However, eq l a and l b may still be valid under certain conditions even when a particular mode is not in the high-temperature limit. For example, if hv,, >> 2kT (Le., the intramolecular mode is of very high frequency), eq l a and l b are still good approximations provided that X,,/hv,, 0 or >5.13 These limits relax as hv,, approaches 2kT. An entirely analogous situation obtains in thermally induced electron-transfer reactions.8-'0,12 In such reactions the charge distribution and polarizability of a molecule change as a result of a thermally activated electron transfer from one redox site in the molecule to another. Thermal electron-transfer reactions are also governed by the Franck-Condon principle. Classically, this requires that the reactants reorganize prior to the electron t r a n ~ f e r * ~so~that ~ ' ~ energy , ~ ~ is conserved in the actual electron-transfer step. In effect, the electron transfer occurs at the nuclear configurations appropriate to the intersection of the two energy curves in Figure 1. The reorganization energy, AG*, constitutes the barrier to the thermal electron transfer and is related to AGO, the free energy difference between the thermally equilibrated reactant and product states ( G P O - G , O ) , and to A, by eq 2 when the energy surfaces are harmonic with identical force AG* = (AGO X)2/4h (2)
+
constants. As noted for eq 1, eq 2 is only valid in the high-temperature limit.8,10.12*14 Within this limit the same two parameters (AGO and A) determine both the energy of the charge-transfer transition and the barrier to thermal electron transfer. In this paper the relation between the solvent's dielectric properties, the energy of the optical charge-transfer transition, (1 3) Because optical transitions do not originate from the minimum of the potential energy well but from the zero-point vibrational level, the classical expressions for X in terms of the energies of the optical absorption and emission bands are in error by this zero-point energy. The relative magnitude of this error ([X,,(classical) - X,,(quantum)] /X,,(quantum)) decreases with increasing X,,/hu,, and is 5. If a correction for the zero-point energy is introduced, then Eabs= AGabro+ X - hu,,/2 and E,, = AGabloX + hu,,/2. These expressions are good approximations when Xln/huln> 1 provided A,, is sufficiently large so that the absorption and emission bands are structureless. When X,n/hu,n< 1 the relative error becomes large; however, due to the small magnitude of XI, the absolute error in AEaboor AE,, is not appreciable. (14) Brunschwig, B. S.; Logan, J.; Newton, M. D.; Sutin, N J. Am. Chem. Soc. 1980, 102, 5798.
4716
The Journal of Physical Chemistry, Vol. 91, No. 18. 1987
Brunschwig et al.
and the barrier to thermal electron transfer is first derived folPolar Solvent Vacuum lowing an approach developed by Marcus."s8 The model is then made specific for solute molecules with shapes and polarizabilities that can be approximated by those of a spherical low-dielectric cavity. Closed-form expressions are given for the case where the charge is localized on two sites. A number of a ~ t h o r s ' ~ have J~-~~ considered the problem of nonspecific solute-solvent interactions and have derived equations for the solvent shifts of absorption and emission band maxima. The relation between these equations is examined, and it is shown that several different theoretical approaches yield the same expressions when similar assumptions are , made. M The Net Free Energy Change. Consider an uncharged sphere of internal dielectric constant Di, immersed in a solvent of dielectric M constant D. The reversible work (Le., the free energy change) Figure 2. Illustration of the thermodynamic cycle used to relate the to charge the sphere to an arbitrary charge distribution is given energy of the absorption band maximum in a solvent, Eab,(D),to the by26 energy of the same transition in a vacuum, Eabs(v). M, M*, and M**
where a is the radius of the sphere, rj and rk are the distances from the center of the sphere to the charges ej and f?k, respectively, rjk is the distance between charges ej and ek,Ojk is the angle between rj and rk, P,(z) are the ordinary Legendre functions, and the e in W,(e,D) and in g,,(e) represents the charge distribution (e,,e2, ...I within the sphere. The first term on the right-hand side of eq 3a is the internal work of charging the sphere; the second is the contribution to the work due to the changing polarization of the surrounding medium (relative to a solvent with D = Din). The work required to transfer the charged sphere from a solvent of dielectric constant D'to another of dielectric constant D is given by w(e,D',D) = Wc(e,D)- W,(e,D/) (4) In order to discuss the shifts of the absorption and emission maxima, it is necessary to define a reference medium. For the present purpose we choose a vacuum as the reference medium2' and denote the work required to transfer the charged sphere from a vacuum (D'= 1) to a medium of dielectric constant D by w(e,D) w(e,l,D). Substitution for W, from eq 3, and noting that the first term in eq 3a cancels, yields the following expression for w(e,D)28
are the ground state, thermally relaxed excited state, and Franck-Condon excited state, respectively; X(D) and X(v) are the reorganization energies in the polar solvent and vacuum, respectively; AGo(D,) and AGo(v) are the free energy differences between the ground and thermally relaxed excited states in the polar solvent and vacuum, respectively; w(e*,D,) and w(e,D,) are the reversible work required to transfer the excited and thermally relaxed ground states, respectively; from a vacuum to the polar solvent; and wFc(e*,D)is the work required to transfer the FranckCondon excited state from a vacuum to the polar solvent. The dielectric constant D is used to indicate both the static (D,) and optical (D0J dielectric constants of the medium.
- - - - - -C -C -0 0.0
I
I
I
I
0.2
0.4
0.6
0.8
I
re,d/o
w(e,D) = (q2/2a)(1/D - 1) (1/2a)Itl/Din- 1 / ~ 1 g g n ( e ) / (+l [ n / ( n + n=l
[l/Din - 11Zgn(e)/(l n= I
1 ) 1 ~ i n / ~ )
+ [n/(n + 111~1n11
(5)
~~
(15) Dodsworth, E. S . ; Lever, A. B. P. Chem. Phys. Lerr. 1984,112, 567. (16) Bagchi, B.; Oxtoby, D. W.; Fleming, G. R. Chem. Phys 1984, 86, 257. (17) Beens, H.; Knibbe, H.; Weller, A. J . Chem. Phys. 1967, 47, 1183. (18) Koutek, B. Collect. Czech. Chem. Commun. 1978, 43, 2368. (19) Bakhshiev, N. G.;Knyazhanskii, M. I.: Osipov, 0. A,: Minkin, V. I.; Saidov, G. V. Usp. Khim. 1969, 38, 1644. (20) Kober, E. M.;Sullivan, B. P.; Meyer, T. J. Inorg. Chem. 1984, 23, 2098. (21) Creutz, C. Prog. Inorg. Chem. 1983, 30, 1. (22) Saito, H.; Fujita, J.; Saito, K. Bull. Chem. SOC.Jpn. 1968, 41, 863. (23) Brunschwig, B. S.; Ehrenson, S.; Sutin, N. J . Phys. Chem. 1986, 90, 3657. (24) Brady, J. E.; Carr, P. W. J . Phys. Chem. 1985, 89, 5759. Brady, J. E.;Carr, P. W. J . Phys. Chem. 1982, 86, 3053. (25) Kamlet, M. J.; Abboud, J. L. M.; Taft, R. W. Prog. Phys. Org. Chem. 1980, 13, 485 and references cited therein. (26) Kirkwood, J. G. Chem. Phys. 1934, 2, 351. Kirkwood, J. G.; Westheimer, F. H . J . Chem. Phys. 1938, 6, 506. (27) The choice of the reference medium is arbitrary, and the analogous equations for a nonpolar reference medium are readily derived (see Appendix A).
Figure 3. Plot of the product of the cavity radius and the reversible work required to transfer a spherical cavity with an embedded dipole from a vs. nonpolar solvent as reference medium to a polar solvent, uwnP(e,Ds), the position of the dipole. Din = 2 for the cavity while D,= 80 and Do, = 1.8 for the polar solvent. The dipole has a length of 0.2~1, unit charges, and is oriented along a cavity radius. The abscissa is rendlawhere rend is the distance from the center of the cavity to the end of the dipole located farthest from the cavity center. The solid line is calculated from the full expressions (eq 5 and 7b): the results obtained from the approximation presented in Appendix A (eq A4a) are indistinguishable from the solid line. The short dashes are calculated from the dipole approximation (eq 8a-8c).
where q is the sum of the charges within the sphere ( q = Ce,) and the summation is now from n = 1 to n = m with the n = 0 term being shown explicitly. This first term is a Born charging term that is zero if the sphere has no net charge. (In eq 3 and 5 and certain subsequent equations the symbol D is used to denote the static and/or optical dielectric constant; if only the static dielectric constant is appropriate, then the symbol D,is used.) The relation between the net free energy change for an optical or thermal charge-transfer process in a polar solvent, AGo(D,), and the free energy change for the same process in a vacuum, (28) Note that the internal dielectric constant Dinof the sphere need not equal unity when it is surrounded by a vacuum.
The Journal of Physical Chemistry, Vol. 91, No. 18, 1987 4717
Solvent Reorganization in Electron-Transfer Processes AGo(v), is shown in Figure 2 where M, M*, and M** represent the molecule in its thermally relaxed ground state, thermally relaxed excited state, and Franck-Condon excited state, respectively. The net free energy change for the charge transfer is given by AGo(D,) = ACo(v) + Aw(D,) (6a) W D , ) = N e @ , ) - w(eiJ,)
(6b)
where w(ei,D,) and w(ef,D,) are the work required to transfer a sphere of internal dielectric constant Din containing a charge distribution appropriate to the initial (ei) and final (ef) states of the molecule, respectively, from a vacuum to a polar solvent of static dielectric constant D,.A simple closed-form approximation to w and Aw can be derived when Din2 for metalloproteins to the extent that the polar polypeptide chains can reorient in response to a change in internal charge distribution. (In addition, solvent molecules may penetrate the interior of the protein.) In the limit such configurational changes can be incorporated into A, and D,,then refers to the dielectric response of the "reorganized" solute. Despite these complications, for many simple inorganic complexes and organic molecules the assumption that D,, u 2 appears to be appropriate. (37) Abe, T. Bull. Chem. Soc. Jpn. 1965,38,1314. Abe, T.; Amako, Y.; Nishioka, T.; Azumi, H. Bull. Chem. SOC.Jpn. 1966,39, 845.
The Journal of Physical Chemistry, Vol. 91, No. 18, 1987 4721
Solvent Reorganization in Electron-Transfer Processes
tively. An alternative expression that is widely used3s6makes results33 and are therefore considered together. differing assumptions for a depending on the context. Thus, a The solute-solvent interaction is generally modeled as a which point-dipole interaction operator in the reaction-field m e t h ~ d . ~ , ~ . is ~ ~approximated by a3/2 in eq 24a to calculate R, and flop are in turn used (eq 24b) to calculate Ror. The expression for flop Second-order perturbation theory is used to find the energy change used in eq 24c and 24d, however, is obtained by setting a = 0 since in the solvent-solute system, and the terms in the perturbation the polarizability of the solute is expected to contribute only a theory results are identified with the classical electrostatic reaction small correction to Substitution of these expressions for the fields developed by O n ~ a g e in r ~ which ~ the solute molecule is reaction fields into eq 24c and 24d and taking the difference in reduced to a point dipole at the center of a single spherical cavity shifts between the two states give eq 25a with the coefficients F, contained in a continuous dielectric solvent. The reaction field and F2 given by eq 22a and 20b, respectively. is the electric field felt by the solute due to the orientation and/or electronic polarization of the solvent by the solute dipole. The Solvent Dependence of Absorption and Emission Bandwidths reaction field due to the polarization of the solvent is given by33 Information about X can also be obtained from the widths of R = (2/a3)(p + a!?)[@ - 1)/(2D l ) ] (24a) the absorption and emission bands7 Within the semiclassical Franck-Condon approximation the full width at half-maximum where p is the dipole moment measured for the solute molecule of the (Gaussian-shaped) bands is given by eq 26a. in a vacuum and a is the polarizability of the solute. The reaction field due to the total polarization, R,, or the electronic polarization, r = ( A ~ i , 2 ) ~ / [In8 (2)] = Xhjhvj coth (huj/2kT) (26a) Rop, is obtained by substituting for a into eq 24a, solving for the ( A ~ l / 2 ) ~ / [In8 (2)] = C2hjkT (26b) reaction field, and then using the appropriate (D,or Dop)dielectric constant. The field due to the orientational polarization, Q,,, can ( A ~ 1 / 2 ) ~ / [In8 (2)] = 2X0,,kT + Xinhuin (26c) then be obtained by taking the difference between the field due to the total polarization and that due to the electronic polarization. Equation 26a assumes that the individual vibronic transitions due
+
nor
= 9s - Oop
(24b)
The shifts in the energies of the ground and Franck-Condon excited states of the solute relative to their energies in a vacuum are given by eq 24c and 24d336233 w(e,D) = -r(e)Qor(e) wFc(e*,D) = -p(e*)R,,(e)
- (1 /2)p(e)QOp(e) + ... ( 2 4 ~ )
- (1 /2)p(e*)QOp(e*)+ ...
wFde*,D) = Ne*$,)
+ A,,,
(24d) (24e)
where e* represents the charge distribution of the excited state. The reaction field in the Franck-Condon excited state due to the solvent's orientation polarization is unchanged, while that due to the solvent's electronic polarization has changed to reflect the dipole moment of the excited solute molecule. From Figure 2, the shift in the energy of the Franck-Condon excited state is also given by eq 24e. If the Clausius-Mossotti expression (eq 21) is used for a, eq 24a solved for the reaction field, and norcalculated from eq 24b, the shift of the absorption band (AEab,(D,v) = wFc(e*,D) - w(e$,)) is given by Mabs(DTv) = (l/a3)[2Pg0(Pg0 - kU,O)F, +
beo2 - Pg02)F*(V)l + c
= pi(& + 2)/3
(25b)
to the high-frequency intramolecular modes are sufficiently broadened by the low-frequency (solvent and intramolecular) modes so that the resulting bands are structureless. Equation 26a reduces to eq 26b in the high-temperature limit and to eq 26c for a continuum solvent and a solute with a single high-frequency intramolecular mode (for which hi,/hvi, 0 or >> 1). In this latter case the solvent acts to broaden the individual vibronic absorption lines, with each line having a Gaussian width of Av,/* = 4[X,,,kT In (2)l1l2. The overall absorption profile will begin to reveal vibronic structure when X,, < (l.5hul,)2/[ 16kT In (2)]. At room temperature this corresponds to A,, < (hvin)2/1000 for energies in wavenumbers. From this latter relationship it is inferred that at room temperature vibronic structure will generally only be observed in very nonpolar solvents. 5. The bandshape will not be symmetric when 0 < Xi,/hu,, Under these conditions the measured bandwidth (taken as the sum of the low-energy and high-energy half-widths) is still given approximately by eq 26c; however the band maxima are no longer given by eq l a unless hi, N 0. The semiclassical bandwidth expressions evidently remain valid for a broader range of conditions than do the band-maximum expre~sions.'~ In the high-temperature limit the shift in the bandwidth function, I?, with solvent relative to its value in a vacuum is given by Al?" = 2kTX,,,(D) (27)
A vacuum reference medium is an idealization since individual where F, and F2(v) are given by eq 1l b and 18b, respectively, vibronic lines will be observed in a vacuum because of the absence and C includes dispersion terms and other small contributions. of the low-frequency modes responsible for the broadening in The first two terms in eq 25a are identical with those in eq 18a = 0). However, eq 27 can still be used to predict solution (X, while the last term is generally ignored. It has been s h o ~ n ~ * , ~the ~ change in the bandwidth with changes in solvent. If the that when a point dipole ( p o ) is embedded in a dielectric sphere point-dipole approximation is used, the AI'" for both emission and (D,,), the dipole moment measured for the system in a vacuum absorption becomes ( p ) is related to po by eq 25b. Alternatively, po can be interpreted as the dipole moment that would be measured if the solute molecule (actually the molecular cavity) had a = 0 (Le., D,, = where F , is given by eq 1 lb. The width of the absorption or 1). It should be noted that eq 25a is often written in terms of emission band will therefore normally increase with increasing p (rather than p " ) with the [(D,, 2)/312 included in Fl and F2. static dielectric constant of the solvent regardless of whether the A number of frequently used expressions'" for the shift in the ground or excited state is the more polar. energy of the absorption band are obtained as special cases of eq Conclusions. Two approaches to the calculation of the shift 25 upon substituting the various values for D,,discussed earlier of absorption and emission band maxima with changes in solvent (Table I). The value D,,= 1 corresponds to setting CY = 0 in all properties have been discussed. The more general approach, the expressions for R; under these conditions po = p and F1and developed by R. A. Marcus, is based upon the use of a thermoF2(v) are given by eq 20a and 20b,5333respectively. Another dynamic cycle to calculate the energy of a charged sphere in a approach is the approximation a = a3/23,6,33(Le., D,,= [(D,, + solvent with a nonequilibrium polarization. This approach yields 2)/312 = 4) with F1and F2(v) given by eq 22a and 22b, respecexpressions for the band shifts that are applicable to an arbitrary distribution of charge within a single spherical cavity and is not restricted to the point-dipole case. A series of very accurate ( 3 8 ) Frohlich, H.Theory ofDielectrics; Oxford,: New York, 1958; p 166. analytic approximations to the full reversible-work expressions (39) Orttung, W. H.J . A m . Chem. SOC.1978, 100, 4369.
+
4722
Brunschwig et ai.
The Journal of Physical Chemistry, Vol. 91, No. 18, 1987
are presented. The second approach uses Onsager's reaction-field formalism to calculate the energy of a point dipole embedded within a spherical cavity. Expressions for the shift of the band maxima are derived from the Clausius-Mossotti relationship for Din. These expressions are general for a point dipole and allow the assignment of a value for Oh appropriate to the specific system considered. Within the point-dipole approximation, both the reversible-work approach and the reaction-field method (and a method introduced by Abe) yield identical expressions for the shift in the absorption and emission band maxima with changes in solvent properties. The reaction-field equations most often used are based upon assumptions about the magnitude of Dinthat are difficult to justify. Two common choices of Dh are 1 and 4. The former choice, eq 20, should only be used when the solute molecule is known to have a uery low polarizability, and Din = 4 seems too large in the absence of large intramolecular configurational changes. Arguments are presented that a value of Din= 2 represents a better compromise for most small molecules and expressions for this case are derived. For most applications the more general expressions for F, and F2 (eq I l b and 18b) are to be preferred since the assignment of Dincan then be made explicitly. Finally, the use of any of the equations based upon a point-dipole approximation are suspect when the charge separation is about the same magnitude as the radius of the cavity or when the dipole is not centrally located within the cavity. In either of these cases use of eq 17a and 17b with work terms given by eq 5 is more appropriate. Acknowledgment. This work was performed at Brookhaven National Laboratory under Contract DE-AC02-76CH00016 with the U S . Department of Energy and supported by its Division of Chemical Sciences, Office of Basic Energy Sciences.
Appendix A. Closed-Form Approximation for the Net Free Energy Cbange Equations for the absorption or emission energies and the shifts of these energies with solvent analogous to those derived for a vacuum as the reference medium can also be derived for the case where a nonpolar solvent is chosen as the reference medium. This reference medium is defined as having optical and static dielectric constants equal to Dinof the solute; w,,(e,D) is then given by wnp(e,D)= W e J ) - Wc(e,Din) = (q2/2a)( 1 /D - 1)
+
w(e,D) = w,,(e,D) - wnp(e71)
('45)
The work required to move a charged sphere from the nonpolar reference medium to a vacuum, wnp(e,l), can be approximated in a similar manner to that used to obtain eq A4. If D = D,[( 1/2) 61, where 6 is small, is substituted into eq A1 and the expansion is carried to first order in 6, the following expression results
+
wnp(e,l) = (1/2aDin)(I/[1/2
+ 61 - l)Celep(xij, cos O i j ) ij Ma)
s(xij, cos
cl)J
=
C(n + l)xij"Pn(cos a i j ) / [ ( n+ 1) n
+ nw] (A6b)
w = D i n / D = 1 /( 1 / 2
+ 6)
Following Kirkwood and Westheimer26we obtain (to first order in 6) wnp(e,l)= (1 /6a)(So/2 S k
+ S,/3
- 6(S0/3
+ 4S,/9))
= xe1f?fk(xjj, cos 8jj) ij
(A7a) (A7b)
+ x2)1/2
(A7c)
~ / ~cos ( l cl) sI(x, cos 6) = ( l / ~ l / ~ ) { ~ ~ d t / [-f (2t)
+ t2)1/2])
so(x,cos 0 ) = 1/(1 - (2x) cos
cl)
(A74
( 1 / 2 a ) [ l / ~ - l/~in15gn(e)/(1+ [ n / ( n + 1 ) 1 ~ i n / ~ ) n= I
(All where wnp(e,D) = w(e,enp,D). Similarly, the outer-sphere reorganization energy is given by xou,(D) =
When one of the charges is located at the center of the sphere, rj = 0, Sjk, = 1, and [In (1 - xii)]/xii = -1. For the case where two of the charges are on the same radius, cos ojk = 1, S j k = -[In ( - xjk) 1/xjk* Equation A4a cannot be used to calculate the shift of the band maximum relative to its position in a vacuum since the approximation breaks down for D,= 1; however the shift relative to a nonpolar solvent as reference medium (0, = Do, = Din)can be calculated from eq A4a. Numerical calculations of w,,(e,D,) show that for D,> 10 eq A4a gives results that are within 5% of those calculated by using the full expressions. The error is almost independent of both D,and r,rk/a2. Even for values of D, = 5 the values of w,,(e,D,) calculated from eq A4a are in error by less than 20%. Values of the work required to move a charged sphere from a vacuum to a polar solvent can be calculated from w,,(eJ). From eq 5 and AI, w(e,D) is given by
wnp(AeJ'op) - wnp(Ae9s)
= w(Ae,D,,) - w(Ae,D,)
(A2)
The following closed-form approximations for the work required to move a charged sphere from a nonpolar to a polar solvent, w,,(e,D,), and for the net free energy change, Aw,,(D,), can be derived when Din 0.99 and 19 0 do the values obtained by quadrature differ by as much as 1% from the summation values.
-
Appendix B. Closed-Form Approximation for the ReorganizationEnergy L, can be calculated from &,,(D)= wnp(Ae,D,) - wnp(Ae,Ds), eq A2, and the approximations given above (eq A4a). However, because Do, N 2, the condition used to derive the approximate expressions (Dq>> Oh) is no longer satisfied and the contribution from the optical dielectric constant (wnp(e,Dop))will not to but be calculated correctly. Fortunately, the numerical value of wn,(e,Dop) is small since the prefactor (1 /Do,- 1/Din) in the expression for w,,(e,D,,) (see eq A4a) is usually not large and the resulting error in &,,is therefore not significant. The values of A,, calculated in this manner are compared with the values calculated from the full expressions in Figure 7. For most cases the error in A,, is less than 5%. If the second term in eq 9b can be neglected (Le., 0,. = Do, and wnp(eJop) = 0), then &, = -w?,(eJs). Under these conditions and provided the charge transfer is between only two sites (and, as above, Din
