9102
J. Phys. Chem. 1994, 98, 9102-9105
The Role of Solute/Solvent Electronic Interaction in the Femtosecond Dynamics of the Bromide/Benzene Cation Contact Ion Pair in Benzene Solution Wlodzimierz Jarzeba,+ Ralph E. Schlief, and Paul F. Barbara' Department of Chemistry, University of Minnesota, Minneapolis, Minnesota 55455 Received: September 27, 1993; In Final Form: June 1 , 1994"
The bromide/benzene cation contact ion pair was prepared in benzene solution by ultrafast excitation of the bromine atom/benzene molecule charge transfer complex. Bromine atoms were prepared in situ by photolysis of a-bromoacetophenone. A simple first-order charge recombination reaction was observed to be the only significant decay process for the ion pair. Despite the apparent simplicity of the charge recombination kinetics, spectroscopic data show that the reaction involves a rapidly formed benzene dimer cation. Various issues of the dynamics involving the dimer cation and the ion pair have been investigated.
Introduction A recent report from this laboratory described the first measurements of the charge recombination kinetics of the bromide/benzenecation contact ionpair (Br/Bz+).l Thiscontact ion pair (CIP) was prepared by ultrafast optical excitation of the charge transfer band of the Br atom/benzene molecule charge transfer complex.
This charge recombination reaction is a promising prototype for studying intermolecular electron transfer at the molecular level. It is relatively simple compared to other recently studied ultrafast intermolecular electron transfer systems," all of which involve donors and acceptors having more electronic, vibrational, and conformational degrees of freedom. Indeed, the Br/Bz+ ion pair is an excellent candidate for detailed theoretical modeling of electron transfer, utilizing such techniques as state-of-the-art nonadiabatic rate calculation^.^ One of the most interesting made on the Br/Bz+ electron transfer reaction is that the recombination rate is much faster in aromatic solvents than in other nonpolar solvents. We have ascribed the acceleration of the rate to a specific interaction between benzene cation (Bz+) solute molecules and the aromatic solvent(Bz), involving the formation of a benzene dimer cation, (BZ)~+. Aromatic dimer cations have been identified in sol~tion3a,~aJ',6,* and in the gas p h a ~ e . ~The J ~ 17.0 kcal/mol gas phase binding energy9 of (Bz)z+ results from "charge resonance" between the mystems of the two benzene molecules.lI Charge resonance interactions are a specific example of donor/acceptor interactions. This paper is concerned with the first ultrafast spectroscopy of the Br/Bz+ electron transfer system in pure benzene solvent. The goals of this study were (i) to further elucidate the role of (Bz)z+ in this electron transfer reaction, (ii) to make dynamic measurements on (Bz)2+formation in solution by directly studying the near-infrared absorption of this intermediate to determine whether alternative solute/solvent complexes such as the delocalized structure Bz6+BrBza+ is involved in the electron transfer, instead of Br(Bz)*+, and finally, (iii) to learn about the spectroscopy and electronic structure of the relaxed contact ion pair (Br/(Bz)z+) that is nominally involved in this reaction. All of these issues are addressed in this paper. t On leave from the Faculty of Chemistry, Jagiellonian University, Ingardena 3, 30-060 Krakow, Poland. Abstract published in Aduance ACS Abstracts, August 15, 1994.
0022-3654/94/2098-9102$04.50/0
Experimental Section Br atoms were prepared by photolysis of a-bromoacetophenone12 (0.05 M ) with 355 nm, 7 pJ, 100 ps pulses in the ultrafast experiments. After an 11 ns delay, sufficiently long to allow formation of an equilibrated mixture of ground state bromine/ benzene (Br/Bz) charge transfer complexes, the charge transfer band was studied by time-resolved pumpprobe laser spectroscopy. Various pump (-1 pJ/pulse) and probe (-0.1 pJ/pulse) wavelengths were employed, using 100 fs pulses (fwhm) focused approximately to a 100 pm spot. Pumpprobe measurements were performed a t the magic angle. The ultrafast laser system and spectrometer have been described in detail elsewhere;IJ3 here we summarize the major points. The output of a synchronously-pumped dye laser (790 nm) was amplified in three dye cells pumped by the second harmonic output of a Nd:YAG regenerative amplifier (500 Hz). Typically, amplified pulses had 8-10 pJ energies and a 100 fs fwhm. These pulses were focused into a quartz cell containing water to generate white light continuum. Pump and probe pulses were independently tunable; they were derived by amplifying 10 nm portions of the continuum. Residual fundamental and second harmonic output from the regenerative amplifier was mixed to generate the third harmonic pulses used to photolyze a-bromoacetophenone. The transient absorption spectra of Br/Bz were recorded with a nanosecond laser spectrometer. It consisted of a Molectron M Y 34-10 Q-switched Nd:YAG laser (10 Hz),a 75 W PTI Xenon lamp system, a lamp pulser, a back-off circuit, a fivestage 1P28 PMT, and an Analogic transient recorder. The lamp pulser and back-off circuit designs were the same as those used by Klaus Schmidt (Argonne). Third harmonic 355 nm pulses typically had 1 mJ energies and a nominal 10 ns width. In both the nanosecond and femtosecond experiments, a 2 mm quartz flow cell was employed. Solutions (300-1000 cm3) were rapidly circulated by a Micropump mechanical pump with a stainless steel pump head. a-Bromoacetophenone (98%, Aldrich) and benzene (ACS Reagent, Fischer) were used as received. Results and Discussion Nanosecond Spectroscopy. Transient absorption studies on radiolytic and photolytic Br atom precursors have shown that Br forms a charge transfer complex with benzene.I2J4 We observe a broad absorption band of the Br/Bz complex centered a t 535 nm, 1 0 0 4 0 0 ns after 355 nm photolysis of a-bromoacetophenone. This spectrum is in good agreement with previous measurements.12 The absorption band appears within our 25 ns instrumental resolution and has a 2 ps lifetime. Previous static spectroscopic studies on charge transfer complexes between strong acceptors and aromatic donors have 0 1994 American Chemical Society
The Journal of Physical Chemistry, Vol. 98, No. 37, 1994 9103
Bromide/Benzene Cation Contact Ion Pair
0. 0
8 0-
0-
0. 0
0
10
Figure 1. Ultrafast pumpprobe transients of the Br/Bz charge transfer band in neat benzene solvent. The pump wavelength was fixed at 550 nm, and the probe wavelength varied from 550 to 700 nm.
shown that both 1:1 and 1:2 donor/acceptor complexes exist in s o l ~ t i o nthe ; ~ ~1:2 complexes dominate at high donor concentrations. Based on these studies, it is likely that many Br atoms in neat benzene are complexed with two or more benzene molecules. There is also kinetic evidence from pumpprobe experiments on the Br/Bz complex in mixed nonpolar solvents that the 1:2 complex is present a t high benzene concentrations, and should dominate in the limit of neat benzene solvent.' To avoid confusion in nomenclature in this paper, the symbol "Br/Bz" will be used to denote the charge transfer complex in general, i.e., Br complexed with any number of benzene molecules. The Br/Bz charge transfer band shape does not change significantly as benzene concentration is varied in such cosolvents as CC14. Unfortunately, this result is inconclusive with regards todetermining therelativeconcentrationof 1:l and 1:2complexes because both should have very similar charge transfer bands, by analogy to other widely studied donor/acceptor pairs.15 Femtosecond Pump-Probe Measurements. Pumpprobe transients of Br/Bz, with the pump excitation fixed a t 550 nm and probe detection at various wavelengths, are shown in Figures 1 and 2. The lifetimes and amplitudes used in fitting this data are summarized in Table 1. Transient probing near the absorption maximum of the charge transfer band (550-600 nm) displays an initial bleach, due to formation of the BrBz+ contact ion pair (CIP). This is followed by rapid bleach recovery within 1.8 ps, assigned tocharge recombination in the CIP. This bleach recovery in pure benzene is more than 40 times faster than that in the limit of infinite dilution in CC14,1 suggesting a specific solvent effect on the CIP in benzene involving the ( B z ) ~ intermediate + or some other charge delocalized intermediate as the "sandwich complex", Bz6+BrBz6+. The kinetics of ground state recovery probed at 550 and 600 nm are reasonably well-fit by a single exponential, consistent with a simple first-order recombination process. The
10
TIME (PS)
TIME (PS)
Figure 2. Ultrafast pump-probe transients in neat benzene solvent, same conditions as in Figure 1, probing absorption in 740-950 nm region.
TABLE 1: Multiexponential Kinetic Parameters for Transient Spectroscopy of the Br/Benzene Complex in Benzene Solvent
1.76 f 0.19 1.54f 0.15d 0.82 f 0.05d 27
550 600 640 700
0.30f 0.07* (-170%).
740 0.20f 0.07' (-35%)' 800 950 0.39 f 0.07c(+230%)1 6000
1.8 f 0.2'
2.0f 0.2' 1.71 f 0.2E 1.4 f 0.2c 1.67 f 0.2d
* 1oC
(4W
signal direction bleach bleach bleach
10 f ' 2 bleach/ absorption ( 149Y absorption absorption absorption bleach
Xpump= 550nm. Absorptionrisetime. Absorptiondecay. Bleach recovery. e Percentage ratio of f / f ' amplitudes. fPercentage ratio of T"'/r" amplitudes. g Xpump = 600 nm.
result at other probe wavelengths, however, indicates that the recombination is more complex, see below. Between 800 and 950 nm, pumpprobe transients show an absorption, rather than a bleach (see Figure 2). This signal can be assigned to absorption by (Bz)26+, an intermediate in the charge recombination process. The rate of decay of this absorption band is equal, within experimental error, to the bleach recovery kinetics a t 550 nm, supporting the idea that (Bz)z+ is the direct precursor for the charge recombination process at high benzene concentrations. The absorption band of (Bz)z+ has been observed previously in a variety of environments.4*.b.6,* At 77 K in a freon matrix, y-irradiated benzene solutions exhibited a broad absorption centered at 900 nm with a 300 nm fwhm, assigned to (Bz)z+.*~ No strong absorption due to Bz+ was observed in this wavelength region a t low benzene concentrations. As the benzene concentration was increased, the (Bz)2+ absorption band shifted only
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The Journal of Physical Chemistry, Vol. 98, No. 37, 1994
slightly? indicating that its spectrum is similar in aromatic and nonaromatic environments. The (Bz)2+ photodissociation spectrum has been recorded in thegas phase, and a broad peakcentered at 930 nm was observed.loa Finally, the absorption due to (Bz)2+ has been observed in benzene solutions with suitable donor, where charge transfer excitation yielded the Bz+ cation which combined with Bz to form (Bz)2+.4a,bt6 Based on our observations that the 900 nm ( B Z ) ~absorption + band appeared within our instrument response function, we can conclude that (Bz)z+ is formed within 200 fs of excitation of the charge transfer band in neat benzene solvent. This strongly suggests that a Bz/Bz complex is already formed in the ground state of the Br/Bz charge transfer complex. Excitation of the charge transfer band directly produces (Bz)z+, albeit in an initial nonequilibrium “Franck-Condon” geometry. Presumably, this initial excited geometry has a larger inter-ring spacing than the ground state Br/(Bz)z complex, since ab initio calculations indicate that parallel ring spacing is greater in (Bz)216 than in (Bz)2+.11 We have no direct information on the geometry of the (Bz)z+ involved in the CIP. A perpendicular or parallel geometry would be consistent with the ultrafast results. Indeed, both geometries are stable isomers of the dimer cation according to theory.” We observed evidence for a blue shift in the (Bz)2+ absorption spectrum caused by vibrational relaxation, occurring immediately after (Bz)z+ formation and taking place on a 200-400 fs time scale. This is evident a t 950 nm, where the pumpprobe transient absorption was fit to a shorter 400 fs component, apparently due to vibrational relaxation, and a longer 1.4 ps component, due to the disappearance of (Bz)z+ by charge recombination. A complementary delay in the bleach a t 700 nm is also probably due to vibrational relaxation (see Figure 1). A donor/acceptor system similar to Br/Bz was investigated. Experiments on the Br/toluene complex exciting at 550 nm and probing at 950 nm showed again prompt formation of the dimer cation and absorption recovery within 1.1 ps, similar to the 1.8 ps for Br/Bz. Compared to pure benzene, mixed solvent solutions with low benzene concentrations exhibited very little absorption a t 950 nm, consistent with the interpretation that the concentration of the (Bz)2+ intermediate is much lower at low benzene concentrations. Solutions with benzene concentrations >2 M had a shortlived absorption signal a t 950 nm. This may have been due to pre-existing ground state 1:2 Br/Bz complexes, see below. Ground State Structure. Little is known about the ground state structure of the Br/solvent complex in pure benzene. As indicated above, there is kinetic evidence that a t least two benzene molecules are sufficiently close to allow extraordinarily prompt formation of (Bz)2+ after optical charge transfer excitation. It is tempting to assume that there are simply two ground state forms, namely a 1:l complex (Br/Bz) and a 1:2 complex (Br/(Bz)2). The actual situation may be more complex. Indeed, Br/(Bz)2 may have two or more stable isomers, including the conventional donor/acceptor structures DAD and ADD, where the ordering signifies proximity within the complex. The ADD structure would be more consistent with the extremely fast formation of the B r / (Bz)z+ CIP after optical excitation, since the two benzene molecules are already in a geometry that is similar to the (BZ)~+ structure. The ADD structure has also been suggested for other 2: 1 donor/acceptor complexes.l5 No information is available on whether the DD geometry involves parallel or perpendicular orientations. Either one would lead to prompt formation of a dimer cation, since both types of isomers are stable structures for the dimer cation. Excited State Structure. The time scale for vibrational relaxation of the Franck-Condon geometry of Br(Bz)2+ CIP (200-400 fs) is comparable to expectations for the vibrational period of Br/Bz and Bz/Bz large amplitudevibrations. Thus the
Jarzeba et al.
\
hVCT
I
Reaction coordinate Figure 3. Schematic energy diagram for the ground and first excited electronic states of benzene-bromine atom complex in neat benzene solution.
relaxation process observed as the blue shifting of the 900 nm absorption band may actually represent “real-time” observation of intermolecular motion between Brand the benzene ring. Related behavior has been observed in the femtosecond spectroscopy of the Br/mesitylene~omplex.~~ Pumpprobe transients of this complex in a variety of solvents exhibit an oscillatory modulation of the bleach a t early times having a -260 fs period. We believe this is due to coherent vibrational motion in the ground state of the complex caused by a combination of vibrational hole-burning (photoselection) and impulsive stimulated Raman scattering. Another donor/acceptor complex, tetracyanoethylene/hexamethylbenzene, has also exhibited ultrafast oscillations, thought again to be due to coherent vibrational motionaZIn contrast, the evolution of the 900 nm Br/Bz absorption band in neat benzene is due to a nonequilibrium distribution in the Br-(Bz)z+ excited state. Figure 3 is a schematic representation of our model for the photophysics of the Br/Bz complex in benzene solution. The initially Franck-Condon geometry involves large enough interring spacings between solute and solvent benzene rings that the electronic character of the Br/Bz+ species is not strongly perturbed by charge delocalization to the solvent. This is consistent with the observation that the Br/Bz charge transfer band shape and position is not strongly affected by varying the mole fraction of benzene in the solvent which in turn varies the relative initial concentration of Br/Bz and Br/(Bz)z. After the 200-400 fs excited state vibrational relaxation period, the energy gap between the ion pair state and ground state has been tremendously reduced due to stabilization by ( B z ) ~ + formation. The reduced energy gap is presumably responsible for the rapid rate of electron transfer between B r and (Bz)z+. We have briefly explored whether photoselection might play some role in the Br/Bz excited state dynamics and recombination kinetics. This was checked by comparing the bleach recovery kinetics monitored a t 600 nm for 550 versus 600 nm pump excitation. No detectable difference was observed, suggesting that selective excitation does little to vary thecharge recombination rate. As stated before, the rate of charge recombination in neat benzene solvent is much faster than in nonaromatic solvent mixtures, due to the (Bz)2+ intermediate, which accelerates recombination. It might be expected that the recombination rate
Bromide/Benzene Cation Contact Ion Pair coefficient increases in time during the 200-400 fs excited state relaxation. This relaxation presumably involves an increase in the “dimer cation” electronic character of the excited state as the two benzene rings attain the proper geometry. We have explored this possibility by analyzing early-time bleach recovery monitored at 550 nm (with 550 nm excitation). Within experimental error, ground state recovery kinetics are first order between 200 and 10 000 fs. We conclude from this result that the rate coefficient of electron transfer does not vary significantly during the vibrational relaxation of the excited state. The apparent lack of a dependence of the instantaneous electron transfer rate on vibrational relaxation of the ( B Z ) ~ may + result from a spread in initial geometries which would mask such an effect. Alternatively, the ( B Z ) ~ +electronic character may be established earlier in the relaxation process than the -100 fs resolution of our experiment. Other Donor/Acceptor Complexes. It is interesting to compare the femtosecond dynamics of the Br/Bz complex in benzene with other previously studied tetracyanobenzene (TCNB) complexes: TCNB/Bz, TCNB/toluene, and TCNB/mesitylene.4a4 All three TCNB complexes exhibited a 10 ps delay in aromatic dimer cation formation, following excitation of thecharge transfer band. The delay has been explained by slow charge separation in the excited state of the charge transfer complex due to energetic and geometric impediments. Why charge separation is so much faster when Br is the acceptor is unknown. Qualitatively, the Br/Bz system may be able tomore rapidly establish the optimal geometry for charge separation than the TCNB complexes because of the spherical shape of Br. As stated above, the major evidence for intermediary of the Br(Bz)z+ structure includes a reasonable match between the wavelength of the observed transient in the Br/Bz experiments with the known absorption band of (Bz)z and the fact that Bz efficiently quenches the BrBz+ intermediate. However, experiments on Br complexes with benzene derivatives suggest that both types of evidence may be a m b i g ~ 0 u s . l ~For example, femtosecond experiments on the Br/mesitylene 1:1 complex in cyclohexane show a 65% component of 2 ps ground state recovery that is accompanied by -2 ps lifetime near infrared absorption transient. This roughly resembles the features we have assigned to the Br(Bz)Z+ absorption. In addition, the Br/benzene derivative complexes involving bulky benzene rings, such as tritert-butylbenzene, diffusion controlled quenching of the ion pair is observed even though these aromatic molecules cannot form dimer cations. These observations suggest that the mechanism of interaction of Br/Bz+ with thesolvent Bzmay bemorecomplex than simply the formation of Br/(B&+. Potentially, structures with more complexdelocalizationof charge, such as Bz*+BrBz*+, may also play a role in this system. Clearly, more research on the Br/Bz and related complexes will be necessary to sort out the complicated donor/acceptor/ solvent interactions that play a role in the charge transfer kinetics of these species in solution.
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Conclusion The charge transfer kinetics of the Br/Bz+ contact ion pair in benzene solution have been studied by time-resolved ultrafast pumpprobe laser spectroscopy. This involves excitation of the
The Journal of Physical Chemistry, Vol. 98, No. 37, 1994 9105 535 nm charge transfer band of the Br/Bz charge transfer complex. The charge recombination reaction in the ion pair is observed to involve the (Bz)z+ intermediate. It is formed concomitantly with excitation of the charge transfer band and subsequently undergoes vibrational relaxation on the 200-400 fs time scale. The instantaneous rate of charge recombination in the contact ion pair does not seem to depend on whether (Bz)z+ is vibrationally relaxed.
Acknowledgment. This research was supported by the Basic Energy Sciences program of the Department of Energy. We would like to thank Klaus Schmidt for assistance with the construction of the nanosecond spectrometer. References and Notes (1) Schlief, R. E.; Jarzeba, W.; Barbara, P. F. J . Mol. Liq., in press. (2) Wynne, K.; Galli, C.; De Rege, P. J. F.; Therien, M. J.; Hochstrasser,
R. M. In Ultrafast Phenomena VIII;Martin, J.-L., Migus, A,, Mourou, G. A., Zewail, A. H., Eds.; Springer: Berlin, 1993; p 71. (3) (a) Gould, I. R.; Farid, S.J . Am. Chem. SOC.1993,115,4814. (b) Gould, I. R.; Farid, S.J . Phys. Chem. 1992, 96, 7635. (c) Gould, I. R.; Young, R. H.; Moody, R. E.; Farid, S.J. Phys. Chem. 1991, 95,2068. (d) Gould, I. R.; Moser, J. E.; Armitage, B.; Farid, S.J . Am. Chem. SOC.1989, 111, 1917. (e) Gould, I. R.; Moody, R.; Farid, S. J. Am. Chem. SOC.1988, 110,7242. (f) Gould, I. R.; Ege, D.; Mattes, S.;Farid, S.J . Am. Chem. SOC. 1987, 109, 3794. (4) (a) Ojima, S.;Miyasaka, H.; Mataga, N. J. Phys. Chem. 1990,94, 5834. (b) Ojima, S.;Miyasaka, H.; Mataga, N. J . Phys. Chem. 1990, 94, 4147. (c) Ojima, S.; Miyasaka, H.; Mataga, N. J . Phys. Chem. 1990, 94, 7534. (d) Miyasaka, H.; Ojima, S.;Mataga, N. J . Phys. Chem. 1989, 93, 3380. (e) Mataga, N.; Kanda, Y.; Okada, T. J . Phys. Chem. 1986,90,3880. (f) Mataga, N.; Shioyama, H.; Kanda, Y. J . Phys. Chem. 1987,91,314. (9) Asahi, T.; Mataga, N. J. Phys. Chem. 1991, 95, 1956. (5) (a) Goodman, J. L.; Peters, K. S . J . Am. Chem.Soc. 1985,107,1441. (b) Goodman, J.L.; Peters, K. S.J . Phys. Chem. 1986,90,5506. (c) Goodman, J. L.; Peters, K. S.J . Am. Chem. SOC.1985,107,6459. (d) Goodman, J. L.; Peters, K. S. J. Am. Chem. SOC.1986, 108, 1700. (6) Bockman, T. M.; Karpinski, Z. J.; Sankararaman, S.; Kochi, J. K. J. Am. Chem. SOC.1992, 114, 1970. (7) (a) Neria, E.;Nitzan,A. J . Chem.Phys. 1993,99,1102. (b) Webster,
F. J.; Schnitker, J.; Friedrichs, M. S.;Friesner, R. A.; Rossky, P. J. Phys. Rev. Lerr. 1991,66,3172. (c) Murphrey,T. H.; Rossky, P. J. J . Chem. Phys. 1993, 99, 515. (d) Sheu, W. S.;Rossky, P. J. Chem. Phys. Left. 1993, 202, 186. (8) (a) Shida, T. Electronic Absorprion Spectra of Radical Ions; Elsevier: Amsterdam, 1988; p 12. (b) Baler, R. E.; Funk, W. J . Phys. Chem. 1975,79,2098. (c) Badger, B.; Brocklehurst, B. Trans. Faraday SOC. 1969,65,2582. (d) Shida, T.; Hamill, W. H. J . Chem. Phys. 1966,444372. (e) Miller, J. H.; Andrews, L.; Lund, P. A.;Schatz, P. N.J. Chem. Phys. 1980, 73, 4932. (0 Badger, B.; Brocklehurst, B. Trans. Faraday Soc. 1969, 65, 2588. (g) Kira, A,; Arai, S.;Imamura, M. J . Phys. Chem. 1972, 76, 1119. (h) Badger, B.; Brocklehurst, B.; Russell, R. D. Chem. Phys. Lett 1967, I, 122. (9) Meot-Ner, M.; Hamlet, P.; Hunter, E. P.; Field, F. H. J . Am. Chem. SOC.1978, 100, 5466. (10) (a)Ohashi,K.;Nishi,N.J.Phys. Chem. 1992,96,2931. (b)Ohashi, K.; Nishi, N. J . Chem. Phys. 1991, 95, 4002. (11) Hiraoka, K.; Fujimaki, S.;Aruga, K.; Yamabe, S.J . Chem. Phys. 1991,95, 8413. (12) McGimpsey, W. G.; Scaiano, J. C. Can. J . Chem. 1988.66, 1474. (13) Johnson, A. E.; Levinger, N. E.; Jarzeba, W.; Schlief, R. E.; Kliner, D. A. V.; Barbara, P. F. Chem. Phys. 1993, 176, 555. (14) (a) Biihler, R. E. Helu. Chim. Acta 1969,51, 1558. (b) Biihler, R. E. J. Phys. Chem. 1972,76, 3220. (c) Raner, K. D.; Lusztyk, J.; Ingold, K. U. J . Phys. Chem. 1989, 93, 564. (d) Yamamoto, N.; Kajikawa, T.; Sato, H.; Tsubomura, H. J. Am. Chem. SOC.1969,91,265. ( e ) Shoute, L. C. T.; Neta, P. J. Phys. Chem. 1990, 94, 2447. (15) Smith, M. L.; McHale, J. L. J . Phys. Chem. 1985, 89, 4002 and
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(17) Jarzeba, W.; Hbmann, A.; Barbara, P. F. To be published.
