J. Phys. Chem. 1985,89, 4181-4188 The position of the P-P bonding curve is the main variable factor here. The maximum in the curve could be thought of as the point of maximum u* filling before the electrons flow from u* to the metal. And this point moves to the right (longer P-P distance) as the metal Fermi level falls. Anyway the “bonding”, “Coulomb”, and ’packing” requirements combine in the known AB2X2structures to give a single minimum, either a made or a broken P-P bond. W e think that for some compounds in this series, for some choice of atoms, there will in fact exist two minima; Le., a possible phase transition is predicted. We would recommend a study of these materials under pressure along the c axis.35 Acknowledgment. We are grateful to the National Science Foundation for its support of this work through Grant C H E 84061 19 and Grant DMR 8217227 A02 to the Materials Science Center a t Cornel1 University. Thanks are due to the members of our research group. C.Z. especially thanks Tim Hughbanks and Sunil Wijeyesekera for sharing their expertise with him. Appreciation is also due to Dr. Jerome Silvestre for providing us with some structural information. We thank Cora Eckenroth for the typing and Jane Jorgensen and Elisabeth Fields for the drawings.
Appendix The extended Huckel method was used in all calculation^.^^
(31) See: Klemm, W. Proc. Chem. Soc. London 1958,329. Schifer, H.; Eisenmann, B.; Miiller, W. Angew. Chem., Int. Ed. Engl. 1973, 12, 694 and references therein.
4181
Table I11 lists the parameters used for Mn and P. The geometry is chosen such that Mn is at the center of an ideal P4tetrahedron, Mn-Mn = 2.8284, Mn-P = 2.45A. A 14K point set is used in the irreducible wedge in the Brillouin zone33to calculate average properties. Registry No. ThCr2Si2, 12018-25-6; Si, 7440-21-3. (32) Hoffmann, R. J . Chem. Phys. 1963,39, 1397. Hoffmann, R.; L i p scomb, W. N. Ibid 1962, 36, 2179, 3489; 1962, 37, 2872. Ammeter, J. H.; Bilrgi, H.-B.; Thibeault, J. C.; Hoffmann, R. J. Am. Chem. Soc. 1978,100, 3686. (33) Pack, J. D.; Monkhorst, H. J. Phys. Rev. B 1977, 16, 1748. (34) In this paper the periodic group notation is in accord with recent
actions by IUPAC and ACS nomenclature committees. A and B notation is eliminated because of wide confusion. Groups IA and IIA become groups 1 and 2. The d-transition elements comprise groups 3 through 12, and the pblock elements comprise groups 13 through 18. (Note that the former Roman number designation is preserved in the last digit of the new numbering: e.g., 111-3 and 13.) (35) A number of high-pressure-induced physical property changes have been observed, many of them attributed to the valence change of A in AB2X2. (a) Eu valence change in EuPd2Si2:Rehler, J.; Krill, G.; Kappler, J. P.; Ravet, M. F.; Wohlleber, D. In ref 3a, p 215. Schmiester, G.; Perscheid, B.; Kaindl, G.; Zukrowsky, J. Ibid. p 219. (b) p(p), linear compressibility and thermal expansion of EuPd2Si2: Batlogg, B.; Jayaraman, A,; Murgai, V.; Grupta, L. C.; Parks, R. D. Ibid. p 229. (c) p(p), T,(p), HC2(p)of CeCu2Si2: Bellarbi, B.; Benoit, A.; Jaccard, D.; Mignot, J. M. Phys. Rev.B 1984,30, 1182. Aliev, F. G.; Brandt, N. B.; Moshchalkov, V. V.; Chudinov, V. V . ; Solid. State Commun. 1983, 45, 215. Aliev, F. G.; Brandt, N. B.; Lutsiv, R. V.; Moshchalkov, V. V.; Chudinov, S. M. JETP Lett. 1982, 35, 539. Aliev, F. G.; Brandt, N. B.; Levin, E. M.; Moshchalkov, V. V.; Chudinov, S. M.; Yasnitskii, R. I. Sou. Phys.-Solid Stare 1982, 24, 164. (d) Mbsbauer studies on NpCo2Si2,hyperfine field B,n(p), isomer shift S(p). NCel temperature TN(p): Potzel, W.; Moser, J.; Kalvius, G. M.; de Novion, C. H.; Spirlet, J. C.; Gal, J. Phys. Reu. B 1981, 24, 6762. Potzel, W. Phys. Scr. 1982, T I , 100.
ARTICLES Time-Dependent Fluorescence Solvent Shifts, Dielectric Friction, and Nonequilibrium Solvation in Polar Solvents G. van der Zwant and James T. Hynes* Department of Chemistry, University of Colorado, Boulder, Colorado 80309 (Received: July 25, 1984; In Final Form: June 13, 1985)
The dynamics of time-dependent fluorescence (TDF) shifts subsequent to electronic absorption by a solute in polar solvents is discussed. It is shown that the TDF shift is directly proportional to the timedependent dielectric friction { ( t ) on the absorbing molecule. This relationship points to the possibility of direct experimental determination of Rt). In addition, several approximate models which go beyond a simple Debye description are discussed. These models include solvent inertia and solvent polarization relaxation via translation and suggest that non-Debye behavior in TDF shifts might be observable. The connection of the TDF shift to related nonequilibrium solvation effects in chemical reactions in polar solvents is briefly described.
I. Introduction The concept of dielectric friction-the measure of the dynamic interaction of a charge or dipole with the surrounding polar solvent-has played a central role over the yeais in discussions of ionic mobility and dipolar orientational relaxation. An admirable review of the concept and its applications is given by Wolynes.’ Our own interest in dielectric friction is in a different and novel arena. In recent years, we have related this concept Present address: Department of Chemistry, University of Oregon, Eugene, OR 94703.
to dynamic polar solvent effects on charge transfer and dipole isomerizations in solution.2-5 Despite the intense interest in dielectric friction, its precise role and quantification have proved elusive1-a melancholy tribute to the difficulty of producing a tractable and accurate theoretical P. G. Wolynes, Ann. Rev.Phys. Chem. 31, 345 (1980). G. van der Zwan and J. T. Hynes, J . Chem. Phys., 76,2993 (1982). G. van der Zwan and J. T. Hynes, Chem. Phys. Lett., 101, 367 (1983). G. van der Zwan and J. T. Hynes, J . Chem. Phys., 78, 4174 (1983); Chem. Phys., 90, 21 (1984). (5) G. van der Zwan and J. T. Hynes, to be submitted. (1) (2) (3) (4)
0022-3654/85/2089-4181$01 .SO10 0 1985 American Chemical Society
4182 The Journal of Physical Chemistry, Vol. 89, No. 20, 1985
description of the dynamics of a strongly interacting charge distribution-polar solvent system. These difficulties suggest that it would be worthwhile to find a direct experimental probe of dielectric friction. This could then be used to examine the phenomenon de novo, to test simple models, and to clarify its molecular ingredients. In this paper, we propose time-dependent fluorescence from excited electronic states as such a probe. We will show that time-dependent fluorescent shifts are directly related to dielectric friction and examine the sort of behavior expected for several simplified models of the solvent relaxation. The theoretical study of time-dependent fluorescence (TDF) from excited electronic states in polar solvents has been pioneered by Bakshiev and M a z ~ r e n k o . ~ - ~A very recent theoretical treatment complementary to our own has been given by Bagchi, Oxtoby, and Fleming.Io Laser techniques have allowed such TDF to be experimentally observed in the nanosecond dornain8Jl8 and more recently in the picosecond time regime.Ilb The models we propose and the conclusions we reach are thus susceptible of experimental scrutiny. Static experimental spectroscopic information12-embodied, for example, in Kosower Z val~es'~-plays an extremely important role in quantifying dipolesolvent and ionsolvent energetics. Such spectroscopically derived correlation parameters reflect molecular level information far more faithfully than do continuum dielectric treatments. These parameters have been usefully exploited for example in the study of the influence of solvation energetics for reactions in polar solvents. In this paper, we extend this idea into the dynamic regime: we make the connection between time-dependent fluorescence shifts, nonequilibrium solvation, and dielectric friction. Let us pause to describe our interest in the T D F problem in connection with chemical reactions. In our recent model studies reaction of dynamic polar solvent effects on chemical is viewed as the passage of a dipole (in an isomerization) or a charge (in a charge transfer) over a parabolic chemical barrier. During the passage, the charge or dipole experiences a frictional interaction with the polar solvent. This friction is often basically dielectric in character and reflects in the simplest cases the lag of solvent dipole orientation compared to the reacting species motion. Alternately stated, the friction describes dynamic nonequilibrium solvation effects on the reaction rate. In particular, the friction leads to deviations from the rate constant predictions of transition state theory, which assumes that equilibrium solvation holds throughout the barrier Since it is well-known that TDF is a probe of nonequilibrium solvation, it is natural to look to the nonreactive TDF problem for insight on nonequilibrium solvation and dielectric friction. A particular goal here would be to discover how fairly drastic continuum solvent either succeed or fail, and if the latter, to discover the essential missing molecular ingredients. The outline of this paper is as follows. In section 11, we discuss the time-dependent fluorescence problem in terms of nonequilibrium free energies, nonequilibrium solvation, and the solvation coordinate. This development distinguishes our treatment from that of Mazurenko and Bakshiev' and M a z ~ r e n k o .In ~ section (6) See,e.&, N. G. Bakshiev, Opt. Spectrosc. (USSR), 16,446(1964),and earlier papers in this series. (7) Y. T. Mazurenko and N. G. Bakshiev, Opt. Spectrosc., 28,490 (1970). (8) Y.T. Mazurenko and V. $. Udaltsov, Opf.Spectrosc. (USSR), 44,417 (1978). (9) Y.T. Mazurenko, Opt. Spectrosc. (USSR), 48, 388 (1980). (IO) B. Bagchi, D. W. Oxtoby, and G. R. Fleming, Chem. Phys., 86,25F (1984). This study generalizes that of ref 6-9 to include the rotation of the solute. (11) (a) W. R.Ware, S. K. Lee, G. J. Brandt, and P. P. Chow. J . Chem. Phys., 54,4729 (1971);R.P.Detoma and L. Brand, Chem. Phys. Lett., 47, 231 (1977);Y.T. Mazurenko and V. S. Udaltsov, Opt. Spectrosc. (USSR), 45,765 (1978). (b) L. A. Hallidy and M. R. Topp, Chem. Phys. Lett., 48, 40 (1977);T.Okamura, M.Sumitani, and K. Yoshihara, ibid., 94,339(1983). (12)See, e.g., C. Reichardt, "Solvent Effects in Organic Chemistry", Verlag Chemie, New York, 1979. (13)E. M.Kosower, "An Introduction to Physical Organic Chemistry", Wiley, New York, 1968.
van der Zwan and Hynes
L
Figure 1. Schematic illustration of nonequilibrium free energies F in the ground and excited electronic states vs. the solvation coordinate. Absorption and prompt fluorescence are illustrated, as is relaxed fluorescence.
111, we make the connection to the time-dependent dielectric friction and probes of the solvation coordinate. In section IV, we present calculated timedependent shifts for several distinct models of the solvent relaxation. A summary and concluding remarks, with a return to reactions, are given in section V. 11. Nonequilibrium Solvation and Time-Dependent
Fluorescence Figure 1 displays a free-energy diagram commonly used for qualitative discussions of time-dependent fluorescence (TDF).I4 Despite their popularity, a certain mystery attaches to such diagrams, since neither coordinate is defined. In this section, we will find the relevant free energies and identify the solvation coordinate z for this diagram. Before delving into details, it is useful to first discuss the phenomenon in qualitative In initial light absorption by an equilibrated ground electronic state g, in which the permanent dipole moment is per the Franck-Condon (FC) principle requires that the nuclear solvent coordinates are "frozenn at the instant of transition to the excited state e with its different dipole moment pea (We neglect any polarizability change.) Thus, while the solvent electronic polarization can instantly adjust, the solvent orientational polarization cannot. Therefore the solvent is out of orientational equilibrium with the e state dipole p,-there is nonequilibrium solvation with respect to pe This leads to an "orientational strain" free energy which contributes to a blue shift in absorption compared to the gas phase. If prompt fluorescence to the ground state now occurs, this same feature is reflected in the emission frequency. In the absence of fluorescence, the solvent orientational polarization will begin to approach equilibrium with pe as solvent molecules reorient. If fluorescence occurs at a certain time t during this transit, then the FC principle dictates that the solvent orientation polarization is that reached by time t in the presence of M,, even though p, has instantly converted to pe at t. Thus after a transition, the solvent is no longer in orientational equilibrium-now there is nonequilibrium solvation with respect to pg. This is reflected in a solvent orientational strain for the ground state and leads to a red shift of fluorescence compared to the prompt fluorescence. Finally, at times sufficiently long for equilibrium solvation to prevail in the excited state, the maximum fluorescence red shift occurs. Equilibrium solvent configurations for the excited state then result, after a FC transition, in nonequilibrium solvation and ~
~~~~~~
(14)N.Mataga in "Molecular Interactions", Vol. 2,H. Tatajczak and W. J. Orville-Thomas, Ed., Wiley, New York, 1981,p 509. Related diagrams can be found elsewhere; examples include E. Lippert, Acc. Chem Res., 3,74 (1970);W. Liptay, Angew. Chem., Int. Edit. Engl., 8, 177 (1969). (15) R. A. Marcus, J. Chem. Phys., 38, 1858 (1963). (16)R. A. Marcus, J . Chem. Phys., 43, 1261 (1965)
Nonequilibrium Solvation in Polar Solvents
The Journal of Physical Chemistry, Vol. 89, No. 20, 1985 4183
an orientational strain in the ground state. A. Nonequilibrium Free Energies. A thwry for the T D F shift problem clearly must treat evolving nonequilibrium solvation dynamics in both the g and e electronic states. In particular, the instantaneous fluorescence frequency w ( t ) requires the calculation of the e and g nonequilibrium free energies F,(t) and Fg(t),shown in Figure 1, in the combination h ( t ) = AU(t) = F,(t) - Fg(f)
whose ingredients we now describe. First
+ SQ
@ - '/(Bop
+ Bor)p2
Fg(t) = q - EPbg
-
(2.8)
Here the equilibrium contribution is
=
%+6
- y2(BOp+ B0,)pg2; 6
q=
q = -3/Rg2 g 4
(2.9)
(2.1)
This relation, due to Marcus,I5 relies on the fact that the entropy is unchanged in an FC transition, so that an energy difference can be replaced by a free-energy difference. The Fs in eq 2.1 are nonequilibrium free energies. To calculate them, we will use a beautiful result of MarcusI5 that a nonequilibrium F can be related to (a) an equilibrium value P and (b) a certain additional contribution related to nonequilibrium orientational polarization in the solvent. The Marcus result applied to the TDF problem for the excited state gives (ignoring vibrational relaxation; cf. section V)
=@
In an analogous fashion, we can write the nonequilibrium ground state free energy as
(2.3)
is the equilibrium free energy for the excited state, including the gas-phase electronic energy @. Here and hereafter we ignore any solvent dispersion contributions to all free energies; these are completely dominated by solvent polarity effects for most polar solvents.I6 The optical and orientational contributions to the c of depend on certain solvent solvent-dependent component 6 response functions Bop.and Bo,. Let us pause to place these in perspective. In a continuum, Onsager cavity description, these are7,15.16
(2.4a)
and the nonequilibrium term is
E?
(2.10)
= -%Bor[pg - d t ) 1 2
The effective dipole moment p(t) in is that which would be in equilibrium with the instantaneous nonequilibrium orientation solvent polarization at instant t. Since the latter does not change e transition by the FC principle, p(t) in in a fluorescence g eq 2.10 is precisely the same as in eq 2.7 for the excited state. B. Dipole Moment p(t) and Solvation Coordinate. To proceed, we need to identify the effective time-dependent dipole moment p ( t ) . To do this, we examine the instantaneous reaction field R(t) experienced by the actual dipole pe in the excited state in the absence of any fluorescence. We show in Appendix A that, subject only to the assumption of linear response, R(t) is given by
-
R(t) = R?
+ A p l '0d s B(s);
Ap
pe - pB
where the equilibrium reaction field for pg is R? =
(Bop
+ Bor)Pg
for a solvent of static dielectric constant c and optical, high-frequency dielectric constant cop, in interaction with a point dipole with point polarizability a. The dipole is here assumed to be encased in a spherical Onsager cavity of radius a with unit dielectric constant. We will use the continuum values eq 2.4 in our discussion for illustration, but it should be realized that eq 2.2 and 2.3 are more general; they rely only on linear response. The solvent term in the equilibrium free energy (2.3) is just 6Q
R? =
= -t/2RFpe (Bop
+ Bor)pe
EYhe - p(t)I
=
-The - p(t)12
(2.13)
The 6 function response Bop6(t)is due to the instantaneous optical electronic solvent response. Its magnitude is Bopin eq 2.3 [in the Onsager description it is given by eq 2.4aI. The orientational response function Bo,(t) has a time dependence and magnitude dependent on the details of the solvent reorientation dynamics (see (2.16)). Independent of these details, one has for the time integral of B ( t ) B
= [ i m d tBop6(t)+ l m d tBO,(t)]5 Bop+ Bo, 0
(2.14)
(2.5) (2.6)
Here R F is the reactionfield-the field on the dipole due to the dipole-induced solvent polarization-when the solvent is equilibrated to the excited state dipole p,. Second, the nonequilibrium contribution to eq 2.2 Bo,
(2.12)
This last is obviously the appropriate field existing just before the initial absorption. The subsequent dynamics are evidently contained in the time integral involving the solvent response function B(t). The detailed form of B(t) depends on the solvent relaxation dynamics to be detailed below. For the moment, we only need to realize that B(t) has two quite different ingredients:
B ( t ) = Bop6(t) + Bor(t) (2.4b)
(2.11)
where Bo, is the function appearing in eq 2.3 [and defined at eq 2.4b in the Onsager description]. This is most easily seen by examining eq 2.1 1 for R(t) for infinite times: R(m) = R?
+ BAp = Bpe= RF
(2.15)
This is the equilibrium reaction field for the excited state dipole as it should be. We pause to illustrate the time dependence of Bo,(t) for an Onsager cavity model coupled with a Debye dielectric relaxation model for the solvent17 (Appendix B) used extensively by Bakshiev and M a z ~ r e n k o : ~ . ~ p,,
(2.7)
is less familiar. It has the form of an equilibrium orientation free energy. [Compare with eq 2.5 and 2.6.1 But the dipole to which it refers is not pe; it is instead p, - p ( t ) . The quantity p(t) is that instantaneous effective dipole which would be in equilibrium with the actual instantaneous orientational polarization in the solvent. Since the latter is not that appropriate for pc except at long times, clearly p(t) is not pc. In fact a t t = 0, the solvent polarization is in equilibrium with pg, both before and immediately after a FC transition. Therefore p(0) equals pg. At sufficiently long times, the solvent is equilibrated to pc and so p(m) = p, We will show how to calculate the apparent dipole moment p(t) at any time t below.
Bo,(t) = BorT;le-'/'~
(2.16)
Here 7, is a certain "longitudinal" solvent relaxation time; it is usually noticeably smaller than the Debye relaxation time 7,,17 extracted from dielectric measurements. We return to this in section IV. (17) See, e.g., H. Frolich, 'Theory of Dielectrics", Clarendon, Oxford, 1958; C. J. F. Bottcher, "Theory of Electrical Polarization", Elsevier, Amsterdam, 1952.
4184
The Journal of Physical Chemistry, Vol. 89, No. 20, 1985
With eq 2.1 1-2.15, we can rewrite the time-dependent reaction field at any time t as
which in turn can be written in the simple form R ( t ) = Bo#,
+ BorIpg[1 - d t ) l + ~ e z ( t ) J I
(2.18)
Rop + RoAt)
Here we have defined what will turn out to be the solvation coordinate
= Bo;lLfds Bo,($)
van der Zwan and Hynes
sfl= hw(0) - ha(-)
+NAt)
(2.20)
Now, the effective dipole moment in the free-energy expressions 2.7 and 2.10 is that which would be in equilibrium with the actual nonequilibrium solvent orientational polarization. It follows without further ado that eq 2.20 gives just that effective dipole moment p( t ) . We can immediately check this central result in the two extreme time limits. First, a t t = 0, the solvent is in orientational equilibrium with p,. Equation 2.19 shows that z(t=O) = 0, and indeed we recover the correct result F(t) = p g . Second, at t = m, the solvent is in orientational equilibrium with pe. Equations 2.19 and 2.20 show that z(t=m) = 1 and that p ( t = - ) = peas required. With eq 2.20 for p ( t ) in hand, we can now examine the timedependent fluorescence shift in detail. Of course the discussion just given should make it clear that z ( t ) = Bo;lJ$ds Bo,(s) is precisely the solvation coordinate in Figure 1. By any time t in the relaxation, the solvent equilibration to pe has proceeded to a certain stage determined by the orientational solvent response Bo,(t). The solvation coordinate measuring this, and defined such that there is full solvent equilibration ultimately, is precisely z . As an example, the simple Onsager-Debye model result (2.16) gives z ( t ) = 1 - e-'/?
(2.21)
, solvation in the When t equals the solvent relaxation time T ~ the excited state is the fraction 1 - e-l complete toward equilibrium.
111. TDF Shifts, Dielectric Friction, and Solvation Coordinate The excited-state and ground-state nonequilibrium free energies from section I1 are easily expressed in terms of the solvation coordinate z ( t ) :
FJt) = @ + 8 e + (B,r/2)(Ap)2[1 - z(t)12 (3.la) Fg(t) =
+ S q + (Bor/2)(Ap)2z2(t)
(3.lb)
(Notice that the "spring constant" is the same for the e and g states.) The T D F frequency then follows from eq 2.1 as
= B,,(A~)~
(s) (
- f2tm i ) ]+ ( A1p ) 2
hw(o)= hwak= hwo+
(sky- sq)+ ( B , , / ~ ) ( A ~ ) ~(3.3)
At t = a,where z = 1, we find the solvent relaxed fluorescence frequency
(3.5)
(3.6)
The solvent coordinate in Figure 1 is precisely defined in terms of the solvent dynamics by eq 2.19. As the solvent equilibrates to the excited-state dipole perz passes from z = 0 to z = 1. The T D F frequency w ( t ) concomitantly diminishes from w(t=O) to w ( t = - ) as this passage is made. Ths most convenient measure of this is the TDF shift compared to the fully relaxed fluorescence red shift: (3.7) With eq 3.2-3.5, this is simply and solely related to the solvent dynamics by A(t) = 1 - z ( t )
= 1 - Bo;IJ'ds
0
Bo,@) = Bo;lJmds
Bor(s)
(3.8)
This shift will concern us in all that follows. A. TDF Shifts and Dielectric Friction. As announced in section I, we can relate the TDF shift A(t) to the time-dependent dielectric friction on a certain rotating dipole. When a reference dipole with moment & = pe - pg rotates in a solvent, it experiences a dynamic torque TmI,due to the finite rate of solvent relaxation. The electrical part of the latter was originally found by Nee and ZwanzigI9 and is293J9920 TSOl"(t)= -s,'ds
{(t
- 4 Q(4
(3.9)
where Q is the angular velocity and { ( f )is the dielectric friction coefficient. Now it can be that { ( t ) is related to the solvent orientational response function Bor(t)by { ( t ) = (AP)2Jmds Bor(s)
(3.10)
Note that only the orientational response appears here; the solvent optical polarization responds instantly and hence cannot cause a torque on Ap. Thus f i t ) is governed by precisely the same solvent dynamics as the TDF shift A(t), eq 3.8. W e can write this key relation in the form A(t) = K t ) / K t = O )
where h w o = @ is the gas-phase frequency. We can recover several important limiting results. At t = 0, where z = 0, the prompt fluorescence frequency is the same as the absorption frequency:
(3.4)
depends only on this orientational strain energy. Equations 3.3-3.5 are not new; they were originally derived by Marcus.16 For the special case of nonpolarizable dipoles (a = 0) and t, = cop, the relaxed fluorescence shift reduces in the Onsager cavity descnption to the well-known result of Ooshika,lSaLippert,lab McCrae,lac and Bakshiev:6 Sfl(a=O)= (2/a3)[
M ( t ) = Pg[1 - 401
- 6 q ) - (B0,/2)(Ap)'
The orientational strain discussed in section I1 is obvious in these two results. Compared to equilibrium conditions, a strain energy (Bm/2)A~* is present for absorption, and a corresponding strain energy occurs for relaxed emission. The relaxed fluorescent red shift compared to absorption
(2.19)
Equations 2.18 and 2.19 are key in identifying p ( t ) . The nonequilibrium orientational reaction field Kr(t)in (2.18) has just the structure of an orientational reaction field in equilibrium with the effective dipole moment p ( t ) :
+ (6Q
h ~ ( m =) hwfl = fiwo
(3.11)
Note that the dipole moment cancels out in the ratio. Equation 3.11 is a major result of this work. It shows that dynamic fluorescence shifts provide a direct experimental route (18) (a) Y.Ooshika, J. Phys. Soc.Jpn., 9,594 (1954); (b) E. Lippert, Z. Narurforsch.A, 10, 541 (1955); (c) E. G. McRae, J . Phys. Chem., 61, 562 (1957). (19) T. W. Nee and R. W. Zwanzig, J. Chem. Phys., 52, 6353 (1970). (20) J. B. Hubbard and P. G . Wolynes, J . Chem. Phys., 69,998 (1978). (21) G. van der Zwan and J. T. Hynes, Physica, lZlA, 227 (1983).
Nonequilibrium Solvation in Polar Solvents
The Journal of Physical Chemistry, Vol. 89, No. 20, 1985 4185
TABLE I: 2 Values for Various Solventsa
solvent H20 CH~OH formamide
2 - Zbcnz: kcal/mol
ethanol 1-propanol 1-butanol
25.6
TABLE II: Solvent Dielectric Properties” and Normalized Initial Friction Values
40.6 29.6 29.3 24.3
CH3CN
23.7 17.3
DMF
14.5
acetone pyridine
11.7 10.0
solvent
€0
HZO CH30H formamide
78.6 32.62 109.5
ethanol
24.4
1-propanol 1-butanol CH3CN
20.33
17.51 35.95 31.4 21.2 13.55
DMF
acetone
“Reference 13. bZbcnz = 54 kcal/mol.
pyridine
to revealing the dielectric friction in a nearly model-free fashion. As shown by Hubbard and Wolynes,20 the dielectric friction coefficient is proportional to the time correlation function of the fluctuating reaction field R . For a spherical cavity of radius a containing a point dipole Apzo { ( t ) = X(AN)z(R.R(t))
(3.12)
t(O)/
im/
e-
tpyr(O)Icon?
1.85
1.44 0.47 0.97 1.15 1.11
tpyr(0)IZ‘ 4.06
5.6 3 2.2 2.24
2.96 2.93 2.56
2.43
2.22 2
1.09
2.37
1.30
1.73
4.5
0.61
1.45
1.9
1.30 1
1.17
2.30
1
“Reference 23. bCalculated from eq 3.22. ‘Calculated from eq 3.21. perpendicular to and much smaller than Marcus eq 3.3 gives, with p,.pg = 0
pg.I3
In this case, the
h(uabs - W ) = -(1/2)Bop(~u,2 - Fg2) + Borp:
(3.18)
Thus A(t) is the normalized correlation function of the fluctuating reaction field due to solvent motion in the presence of the dipole. There are further useful connections of the T D F shift to the friction. The time-integrated shift
Let us now establish an approximate connection to the dielectric friction. In a highly nonpolar solvent such as benzene, Bo, is essentially zero and we can write for any solvent
A = J m d t A ( t ) = L m d t{ ( t ) / { ( t = O ) = { / { ( r = O )
where Bo, refers to the solvent of interest. (Here we have invoked is much smaller than the very mild assumption that Bo, Bar.) But this difference is just the difference of the Z values for a given solvent and of benzene. In combination with eq 3.14, this gives us
0
(3.13)
is proportional to the friction constant {. This important quantity measures the long time integrated effect of nonequilibrium solvation. Further, the initial value (3.14) of the friction coefficient is an important measure of the strength of the friction. It measures the amplitude of the reaction field fluctuations by (3.12). In addition, {(t=O) is required to extract { ( t ) and { from the measured quantities (3.11) and (3.13). Comparison of eq 3.5 and 3.10 shows that this critical initial value {(t=O) is determined by the relaxed fluorescence red shift by {(t=O) = Sfl (3.15) In summary, the dielectric friction coefficient on the absorbing molecule, with apparent dipole moment AM = pe - pg, can be experimentally determined from the T D F shift A ( ? )and the relaxed fluorescence red shift Sfl: T(t) = SflA(t) (3.16) The dielectric friction constant {can be determined experimentally from S f land the time-integrated shift: { = SflA (3.17) These key relations open a new route to experimental investigation of the dielectric friction. We note parenthetically that various aspects of the reaction field correlation friction ( R . R ( t ) ) also follow from eq 3.16 and 3.17 via the connection 3.12. This requires knowledge of the effective dipole moment Ap = p, - pg. The most difficult quantity to obtain here is the excited state moment pe, but this can be obtained from the ratio of absorption and fluorescence solvent shifts.22 B. Z Values and the Initial Time Dielectric Friction. In the Introduction, we stressed the central role played by spectroscopic probes in the quantification of polar solvent energetics. In fact, we can make a connection between the dielectric friction and the Kosower 2 values13 (Table I). These are absorption energies for pyridinium iodide (PI) in various solvents. The highly polar molecule PI, with large pg, undergoes an FC transition to a less polar excited state with the dipole moment p,, thought to be (22) P. Suppan, Chem. Phys. Let?., 94, 272
(1983).
BF
Z
- Z,,, = Borp: = {(t=O)
(3.20)
(This assumes that pe
