J. Phys. Chem. 1991, 95, 5003-5007
5003
Time-Resolved Transient Resonance Coherent Anti-Stokes Raman Scattering Spectra of Dlphenylhexatrlene Ion Radicals in Polar Solvents and Exciplex-Formlng Systems Toshio Kamisuki and Chiaki Hirose* Research Laboratory of Resources Utilization, Tokyo Institute of Technology, Nagatsuta, Midori-ku, Yokohama 227, Japan (Received: September 19, 1990; I n Final Form: December 5, 1990) The transient resonance CARS (coherent anti-Stokes Raman scattering) spectra of all-trans-l,6-diphenyl-l,3,5-hexatriene (DPH) cation and anion radicals were measured in exciplex systems with a strong electron acceptor (l,Cdicyanobenzene, DCNB) or with a strong electron donor (Nfldimethylaniline, DMA) in 1,4-dioxane,as well as in polar solvents with different polarity. Close correspondence between the signals of the DPH/DCNB system and those of DPH in polar solvents such as acetone and ethanol confirms that the species produced in the solvents is the cation radical. Timeresohred CARS measurements revealed that the effective lifetime of the cation radical is strongly dependent on solvent polarity giving values of 1.5 ps, 4.0 ps, and 72 ns in acetone, ethanol, and 2-methyltetrahydrofuran,respectively. The ion radicals in DPH/DCNB and DPH/DMA systems had respective lifetimes of 110 and 61 ns, which are much more longer than the lifetime of exciplex fluorescence (equal to or less than 10 ns), indicating that geminated ion pairs or solvated free ions were produced from the charge-transfer (CT) exciplexes. Furthermore, biphotonic production of the cation radical in polar solvents was found to set in at higher UV power, competing with solvent-assisted monophotonic ionization. Introduction The solvent effects on all-trans-l,6-diphenyl-1,3,5-hexatriene (DPH) in the lowest electronic excited (SI, S2,and TI) states and the dynamical behavior of DPH have been investigated by several authors.Id In polar solvents, the transient absorption spectra of DPH'+ and DPH'- were observed by pulse radiolysis and laser-photolysis techniques by Almgren and Thomas;' the pulse radiolysis of DPH in 1,2-dichloroethanegave rise to both DPH'+ and DPH'- in the region of 600-650 nm, while DPH'+ was produced through two-photon ionization by the pulse photolysis of DPH in such solvents as ethanol and 1,2-dichloroethane and in various micellar media. Although the Raman frequencies of DPH in the SIand T I states were obtained in a nonpolar solvent by using the transient resonance CARS (coherent anti-Stokes Raman scattering) technique! little information has been available on the vibrational spectra of the DPH ion radials. Very recently, we reported the transient CARS spectra of DPH in acetone (irradiation at 337 nm) and suggested that the long-lived (about 2 ps) species was the cation radical DPH'+.9 The population of the species was found to be linear with UV power, indicating a rather unusual occurrence of monophotonic ionization, and the occurrence of solvent-assisted ionization from the SIstate was suggested. Vibrational information about DPH exciplexes may be of great significance in order to identify the photoproduced ionic species. The present paper describes the transient resonance CARS spectra of the DPH exciplex systems containing a strong electron donor or a strong electron acceptor as well as those of the DPH solutions in polar solvents with different polarity. Generation of DPH'+ in the UV-irradiated solutions was confirmed. The results of time-resolved CARS measurements and the UV power dependence of CARS signals of DPH'+ are also reported and discussed in connection with solvent-assisted photoionization. A brief comparison is made among the vibrational features of DPH in the S, SI, and TI states, and those of the cation and anion radicals. Experimental Section DPH (Aldrich) and DCNB (Aldrich) wre purified by vacuum sublimation following recrystallization from ethanol. N,N-Dimethylaniline (DMA; Wako Junyaku) was used without further purification. The solvents (acetone, ethanol, 1,Cdioxane, 2methyltetrahydrofuran (MTHF), and n-heptane) of spectrograde (Wako Junyaku) were used without further purification. All sample solutions of 1-3 mM concentration were degassed by repeated freeze-pumpthaw method under vacuum. Author to whom correspondence should be addressed.
The experimental apparatus used to obtain transient resonance CARS spectra has been described elsewhere.'O Briefly, two dye lasers, which were pumped by the UV beam from a nitrogen laser (Molectron, UV-24, 9 mJ/pulse, 10 ns pulse width), were used for the CARS experiment. Part of the UV beam was used for the photoexcitation in the case when no delay time was imposed between sample excitation and CARS generation. In the timeresolved CARS measurement, the third harmonic (355 nm) of a Nd:YAG laser (Molectron, MY-32, 15 ns pulse width) output was used for photoexcitation and the nitrogen laser which pumped the two dye lasers was operated by taking an appropriate delay time (between 30 ns and 2.5 ps) from the 355-nm pulse. As for the dye lasers, the w1 beam (frequency-fixed pump beam) was set in resonance with the transient absorption of the investigated species in the region of 580-650 nm. The CARS signal was normalized against the intensity of the probe beam w2 which was scanned over the region where the difference w1 - o2goes across the Raman frequency. The dependence of transient resonance CARS signals on UV (337 nm) laser power was measured by inserting an appropriate number of glass plates, each with a transmittance of 0.70 at 337 nm. The UV power was 0.3 mJ/pulse when no attenuation was made. A quartz lens having focal length of 250 mm was used to focus the UV beam inside sample solutions with the estimated radius at beam waist of about 0.5 mm which is much larger than the beam waists of the w1 and w2 beams (less than 0.1 mm). All the measurements were performed at room temperature. Results and Discussion A. Resonance CARS Spectra of DPH Exciplex Systems. The vibrational spectrum of exciplex systems provides the vibrational frequencies of the involved ionic species and the derived values are often decisive in the identification of the species produced in different environment such as in photoirradiated solutions, especially when the visible spectrum is overlapped by that of other species. The spectroscopic study of exciplex systems containing DPH, however, has not been reported before. Our initial exKohler, B. E.; Itoh, T. 1.Phys. Chcm. 1988, 92, 5120. Itoh, T.; Kohler, B. E. J . Phys. Chcm. 1988, 92, 1807. Kohler, E. E.; Spiglanin, T. A. J . Chcm. Phys. 1984, 80, 5465. Alford, P. C.; Palmer, T. F. Chem. Phys. Leii. 1986, 127, 19. Jones, G. R.; Cundall, R. B. Chem. Phys. k i t . 1986, 126, 129. Itoh, T.; Kohler, B. E. 1. Phys. Chcm. 1987, 91, 1760. Almgren, M.; Thomas, J. K. Phoiochcm. Phoiobiol. 1980, 31, 329. (8) Kasama, A.; Taya, M.; Kamisuki, T.; Adachi, Y.; Maeda, S. In Time-resolved vibraiional speciroscopy; Atkinson, G . H. Ed.;Gordon: New York. 1987: D. 304. (9) Wakiabe, M.; Kamisuki, T.; Akamatsu. N.; Hirose, C. Chem. Phys. k i t . 1990, 170, 451. (10) Maeda, S.; Kamisuki, T.; Kataoka, H.; Adachi, Y. Appl. Spectrosc. Rev. 1985, 21, 211.
0022-3654/91/2095-5003302.50/00 1991 American Chemical Society
5004 The Journal of Physical Chemistry, Vol. 95, No. 13, 1991
Kamisuki and Hirose
TABLE I: obscncd Ramn Frequencies of the S, SI,and TIStates and Tbose of the Cation and Anion R a d W of DPH CARS So solid
Raman'
SIin n-heptane CARSb ( 1 575) 1545 1493 1328 1280? 1235 1186 I164 1148 1135?
1608 1596 1589 1255 1 I44
998
974 898?
TI in
n-heptane CARS" 1570 1550 1270 1200 1 I85 1 I64
anion radical
cation radical C d 1588 1588 1532 1534 1444? 1442?
e
f
P
1570 1521 1440?
1569 1518
1587 1545
1302 1248 1185
1272 1215 1 I76
1269 1213 1177
1292 1267
1302 1244 1183
dianion
1 I65 1145 1128 1050 lo00 852 603? 538
1 I30
987
998
997
988
988
547
549
550
'Reference 8. bReference 9. The 1575-cm-I band is not a SIsignal. cIn acetone solution, this work. dDPHt/DCNB- exciplex in pdioxane solution, this work. c(Produced by Na reduction) in THF solution, this work. fDPH-/DMAt exciplex in p-dioxane solution, this work. r(Produced by Na reduction) in THF solution, this work. periments focused on 1,4-dioxane solutions of DPH with a strong electron acceptor (1,4-dicyanobenzene, DCNB) or a strong donor (N,N-dimethylaniline,DMA). The transient absorption band of DPH in the SIstate has been reported to peak at about 640 nm (15600 cm-l; in cyclohexane)."*'* Both the cation and anion radicals of DPH have absorptions peaking at 600-650 nm (16 700-1 5 400 cm-I), which depend on the solvent to some extent, and the absorption peak of the cation radical occurs at slightly shorter wavelengths than the peak of the anion radical The broad absorption bands of the SIstate, the cation, and anion radicals overlap so that it is virtually impossible to distinguish the three species from the visible spectrum alone. For a solution of about 1 mM DPH and 0.1 M DMA the transient CARS spectrum was observed when w1 was set at 15 950 cm-l resonant with the absorption band of the anion radical," as shown in Figure la. A broad fluorescence spectrum with its maximum at about 19 300 cm-' was observed in the solutions, instead of the DPH SI So fluorescence band peaking at 21 900 cm-' (in n-heptane). Figure 1, b and c, shows the resonance CARS spectra (wI= 15 950 cm-l) of DPH'- and DPHZ-, respectively, which were produced by chemical reduction by Na metal in a tetrahydrofuran (THF) solution. A remarkable similarity of Figure la, b proves that a CT type exciplex (DPH'--DMA*+)* was actually formed, although a close inspection indicates that the relative intensity and spectral widths differ between the two spectra, indicating a slight difference in the electronic absorption spectra. The dianion DPH" shows absorption peaked around 630 nm in T H F solution with the width broader than that of the anion radical. I I It has been reportedg that, on irradiation with 337-nm pulsed light, DPH in acetone or ethanol gave rise to the transient CARS signals which were distinct from the bands reported for DPH in the SIand T I states,* or for the anion radical. No such signals showed up when DPH was dissolved in nonpolar 1,4-dioxaneor n-heptane, and the Raman (or CARS) bands were ascribed to the cation radical DPH'+ from several prices of circumstantial evidence. The transient resonance CARS signals which were observed for DPH ( 1 mM)/DCNB (0.1 M) in 1,4-dioxaneand for DPH (1 mM) in acetone are shown in Figure 2, a and b, respectively. The observation was made in the resonance condition with the absorption band of the cation radical by setting wlat 16 300 cm-'. When combined with the fact that a broad fluorescence spectrum with the peak at about 21 000-22000 cm-' showed up for the
-
(1 1) Couldbeck, R.A.; Twarowski, A. J.: Russell. E. L.: Rice. J. K.: Birsc.
R. R.: Switkes.E.: Kliner. D. S. J . Chen
I
,
I
,
1600
1
1
1200
1400
1600
n
I
1400
,
,
,
izbo
1000
1
'
1
1000
'
14b0
izbo
1000
(Raman shift/cm-')
Figure 1. Resonance CARS spectra of DPH/DMA (DPH 1 m M DMA 0.1 M) in 1,Cdioxaneon UV irradiation at 337 nm (a), the DPH anion radical (b), and the dianion (c) (both produced by chemical reduction by Na metal) in THF, at wI = 15 950 cm-I. The bands marked by s are due to the solvent.
DPH/DCNB system we can conclude that CT exciplex with an electron being transferred from DPH to DCNB, (DPH'+DCNB'-)*, has been formed. The virtually exact coincidence of the transient CARS signal in question (Figure 2b) with that of the cation radical in the DPH/DCNB exciplex system (Figure 2a) establishes our previous assignment. As described in the following section, however, the lifetime of the CARS signals of DPH/DCNB system was much longer than that of the fluorescence, indicating that the CARS signals in Figure 2a are due to the so-called geminated ion-pair state or solvated free The widths of CARS signals of the DPH/DCNB system are about 1.5 times broader than those of DPH'+ alone in acetone. The derived values of Raman frequencies are summarized in the last three columns of Table I. One notes that the overall features of Figure la, b are similar to those of Figure 2a, b despite the fact that the two sets were obtained under different resonance conditions. (14) Mataga, N. Pure Appl. Chcm. 1984.56, 1225. (15) Mataga, N. J. Mol. Struct. 1986. 135, 279.
The Journal of Physical Chemistry, Vol. 95, NO. 13, 1991 5005
Transient Resonance CARS Spectra of DPH
,I?.J L 30119
30118
1
do0
1
1
1100
,
1
do0
'
1doo
II
L J :29011s
0011s
1
1600
.
1
'
1460
,
1
1
( Raman shift
1
1200
/ cm-l
1
1
1
lobo
J44
12UllS
)
Figure 2. Transient resonance CARS spectra of DPH/DCNB (DPH 1 mM; DCNB 0.1 M) in 1,4-dioxane(a) and DPH (1 mM) in acetone (b) by setting at w, = 16 300 cm-I on UV irradiation at 337 nm.
do0 1doo (Raman sliift/cm-')
Id00
I 4-
c.
n
-- II -
I "-._-.-
50
100
11s
t
50
do0
l0OllS
Figure 4. Time-resolved resonance CARS spectra of DPH/DCNB in 1,4-dioxane at wI = 16350 cm-l under UV irradiation at 355 nm. Semilogarithmic plots of the density population (n, arbitrary unit) of the
-1
I
cation radical vs the delay time are shown on the bottom.
I+
I
fa00
1500
?
do0
n-
1600
(Raman shift/cm-')
i
z'p
Figure 3. Time-resolved resonance CARS spcctra of DPH in acetone (a), ethanol (b), and MTHF (c), respectively, by using the delay time f d between 20 ns and 2.5 ps, at wI = 16350 cm-I, under UV irradiation at 355 nm. The plots at the bottom are the semilogarithmic plots of the population density (n, arbitrary unit) of the cation radical vs the delay time.
B. Transient TimeResolved Resonance CARS Spectra of DPH in Polar Solvents and of the Exciplexes with DCNB and DMA. We have already reported a preliminary value of 2 ps as the lifetime of the DPH'+ in acetone solution9that was estimated from the temporal change of the signal intensity of the time-resolved CARS spectra. The results of subsequent experiments are shown in Figures 3 and 4. The temporal behavior of the UV-produced DPH'+ in acetone, ethanol, and MTHF are shown in Figure 3a, b, and c, respectively. The insets a t the bottom of the spectral curves are the semilogarithmic plots of the number density (n, in arbitrary units) of the cation radical against delay time t . The density was estimated from the simulation of the CARS signal at 1588 cm-I, which depends on the Raman resonant (thus on number density of the cation radical) and background (Raman nonresonant; due to solvent molecules) signals. The plots indicate
that the cation radicals undergo a simple exponential decay process and the lifetimes of 1.5 ps, 4.0 ps, and 72 ns are derived for the species in acetone, ethanol, and MTHF solutions, respectively. The values correlate with the dielectric constant t which are 20.70 for acetone, 24.55 for ethanol, and about 7.6 for MTHF (assuming the same value as that of THF) at 25 OC, and thus the stabilization of the cation radical and hence the lifetime seem to strongly depend on solvent polarity. It is quite likely that retardation of the recombination process of geminate ion pairs or solvated free ions is effective in the highly polar solvents due to the stabilization of polarization. It should be mentioned here that the DPH solution in l,r)-dioxane (t = 1.88) or n-heptane gave the CARS signals of only the S,state with the lifetime less than 10 ns. Figure 4 shows the time-resolved transient resonance CARS spectra of the DPH/DCNB (a) and DPH/DMA (b) systems in 1,Cdioxane; the lifetimes were estimated to be 60 and 110 ns, respectively. A much shorter lifetime of 110 ns was observed for exciplex fluorescence of both systems. These findings strongly indicate that the long-lived species giving rise to the CARS signals are not the exciplexes themselves but rather nonfluorescent geminated ion-pair species or solvated free ions generated from the exciplexes. C. UV Power Depenaenee of DPW+CARS Sign& A previous paperg reported that the dependence of both the transient absorption and the CARS signal at 1588 cm-' on the excitation UV power was linear in the acetone solution for lower power conditions, and a monophotonic process for the formation of the cation radical was suggested. The monophotonic mechanism of DPH'+ photogeneration was reexamined for the exciplex DPH/DCNB system in 1,4-dioxane together with the remeasurement of acetone solution in the present work. Biphotonic contribution was also studied in acetone and MTHF solutions for higher UV power conditions of excitation. Figure 5 plots the population densities of the cation radical, which were estimated from the dependence of the CARS signal a t 1588 cm-' on UV power in the DPH/DCNB system in 1,4-
5006 The Journal of Physical Chemistry, Vol. 95, No. 13, 19191
Kamisuki and Hirose
n I
A0
l4
12
10 plates
0.5
1 .o
(a)
0.3 mJ
-
, -
950 Id00
1000
0.07
0.15
950
1000
950
(Raman shift/cm-')
sc, 0.15
I
I,, ) 0.5 1.0 Figure 5. UV (337 nm) power dependences of the population density of the DPH cation radical for DPH/DCNB in 1,4-dioxane(a) and DPH in acetone (b) under the condition that the population of the SI state is not saturated (less than 0.01 and 0.03 mJ for (a) and (b), respectively). Iuvdenotes the relative intensity of the UV pulse reduced by using glass plates, each with a transmittanceof 0.7. The power of exciting UV beam is indicated by the number of glass plates that were placed in front of the sample cell. The relative population density was derived by simulating the CARS signal at 1588 cm-I against Raman-nonresonant background signal. w , was set at 16 300 cm-I. (
dioxane (a) and those of DPH in acetone (b). UV power, Iuv, of up to 0.01 and 0.03 mJ was employed to obtain the results shown in Figure 5 , a and b, respectively, which is the region where no serious saturation of the SI-,So fluorescence took place. The 337-nm UV beam of about 0.3 mJ/pulse partly separated from the N2 laser output was attenuated by glass plates having a transmittance of 0.7 per plate, and the irradiating power Iuvis indicated on the abscissa by the number of plates. Figure Sa shows that the population in DPH'+ in the UV irradiated DPH/DCNB system grows linearly with the UV power as expected from the exciplex formed via the SI state of DPH. In the case of DPH in acetone (Figure 5b) too, the reexamined result is the same as for the exciplex case, although the apparent UV power a t which the saturation starts differs due to the formation efficiency of DPH'+ and to the difference in concentration. We expect that the population of DPH'+ in DPH/DCNB would be proportional to that of the photoproduced SI species; the similarity of the two plots is a strong indication that DPH'+ in acetone also results from the SI state without further absorption of the UV beam. Subsequent measurement a t higher UV power (from 0.07 to 0.3 mJ instead of less than 0.03 mJ in the above-described experiments) proved, however, that the apparent one-photon production of DPH*+ was only a part of the whole story. It should be noted that the 1588-cm-' band of DPW+ is extremely intense and can be observed a t very low UV excitation power while the second strongest band at 998 cm-I is accompanied by the SIsignal at 974 cm-I which serves as an excellent reference for our purpose. Figure 6 shows the power-dependent feature of the CARS signals a t 998 cm-l (the second intense band) of DPH'+ in acetone (a) and in MTHF (b), and those of the signal at 1588 cm-I of DPH'+ in M T H F (c). The CARS signal at 974 cm-l in Figure 6a, b originates from DPH in the SIstate and the intensity of the 998-cm-l (DPH") band is seen to get much stronger than that of the 974 cm-I (SI) band as the UV power doubled stepwise from right to left (from 0.07 to 0.3 mJ). A simulation of the signal profile showed that the population of the SIstate changed by not
,
1000 1500 (Raman sliift/ein-')
Figure 6. Dependence of the resonance CARS signal at 998 cm-l of DPH in acetone (a) and in MTHF (b), and the signal at 1588 cm-' in MTHF (c), on the UV (337 nm) power Iuvin the high-power region (up to 0.3 mJ) where the SI (fluorescent) state as examined by the Raman signal at 975 cm-' was practically saturated. Iuvwas attenuated by 1/2 on each step.
more than 150% (indicating strong saturation of the SIpopulation) for a 4 times increase of the UV power while that of DPH'+ changed in proportion to the UV power, suggesting the participation of another photon for the production of the cation radical from DPH in the SI state. The above findings indicate that the DPH cation radical in polar solvents is produced either by so-called direct photoionization (electron ejection; but a solvent-assisted contribution is important) in one or two steps, M 2hv M'+ e- when enough UV power is available, or by solvent-assisted monophotonic ionization, viz., M* (M+-.solv-)* Msolv.*+or M+-.(solv),,- ion M hv pair. The biphotonic process is dominant In the MTHF solution, while appreciable monophotonic process takes place in the acetone solution. It seems probable that the interaction between the photoexcited DPH (SI) and solvent molecules surrounding the DPH plays an important role in the monophotonic ionization process. D. Vibrational Frequencies of the DPH Ion Radicals and DPH in the S, SI,and TIStates. Table I presents the observed Raman frequencies for the ion radicals and different electronic states of DPH. The frequencies for the two sets each of the cation and anion radicals, namely, one for the solution in polar solvent and the other for the species in exciplex forming environment, are in good correspondence to each other within experimental error. As stated before, the spectral patterns for the both ion radicals are similar, suggesting that the electronic natures of both the anion and cation radicals in both the ground and the excited states, which are responsible for the resonance CARS enhancements of specific vibrations, are similar. On the other hand, the resonant feature of DPH in the SIstate is quite different from those of the ion radicals or the TI state. The most intense band near 1600 cm-', which seems to be the characteristic band of the TI state and ion radical species and thus serves as an important key band for the vibrational assignment and characterization of the electronic states related to resonance, is absent in the spectrum of the SIstate. Many bands with medium intensity in the 1100-1300-~m-~region were observed,
+
- -
+
-
+
-
J. Phys. Chem. 1991,95, 5007-501 1 suggesting a complicated nature of the potential surface in the SI(possibly 2’ A,) state. The apparent similarity in the spectral features of the ion radicals and the TI signals may be explained by a simple MO scheme of HOMO and LUMO in the DPH (neutral) ground state. The assignment of the observed vibrational modes of the DPH ion radicals is in progress at the moment.
Conclusions The long-lived species observed in such polar solvents as acetone, ethanol, and M T H F was confirmed to be the cation radical by comparing the transient CARS spectrum with that of the 1,4dioxane solution of DPH containing DCNB as a strong electron acceptor. The formation of the charge-transfer exciplex in the 1,4-dioxane solution of DMA/DCNB was confirmed by comparing the CARS spectrum with that of the chemically produced DPH*-.
5007
The lifetime of the cation radicals produced in acetone, ethanol, and MTHF was derived as 1.5 ps, 4.0 ps, and 7 2 ns, respectively, indicating the role of polarity-dependent solvation in the stabilization of the species. The lifetime of the cation radical in the DPH/DCNB system and that of the anion radical in the DPH/DMA system were 60 and 110 ns, respectively, when the concentration of the electron donor or acceptor was 0.1 M in 1,Cdioxane. These lifetimes are longer by a t least 1 order of magnitude than the fluorescence lifetime of the solutions, indicating the formation of the geminated ion pairs or solvated free ions. Some evidence of both monophotonic and biphotonic processes of the DPH cation radical in the polar solvents such as acetone and methyltetrahydrofuran was deduced, suggesting the presence of competition between the ionization by electron ejection from a twephoton-excited state and the solvent-assisted photoionization from the S I state in acetone.
Collisional Energy Transfer at Hlgh Temperatures from the Biased Walk Model Robert G . Gilbert* School of Chemistry, Sydney University, NS W 2006, Australia
and I. Oref* Department of Chemistry, Technion-Israel Institute of Technology, Haifa 3200, Israel (Received: October 19, 1990)
A new expression for the probability, P(E,E’), of collisional energy transfer for large polyatomics at high temperatures is deduced by using the assumptions of the biased random walk (BRW) model [Lim, K. F.; Gilbert, R. G. J . Chem. Phys. 1986,84,6124] and of a Gaussian equilibrium population distribution. The latter approximation is shown to be very a m r a t e at sufficiently high temperatures. Simple analytic expressions are obtained for P(E,E’), which show the correct limiting weak- and strong-collision forms as the duration of a collision approaches zero and infinity, respectively. Expressions arc found for the average energy-transfer quantities (A&), (A&), and (a2), as functions of initial energy E’. The expression for (AE,,,)explicitly shows the change of sign as E’is varied over a wide range of E’. The new BRW expression for P(E,E’) can be accurately approximated as Gaussian. The derived expressions will be of use in modeling high-temperature processes such as combustion.
Introduction An understanding of the rates of collisional energy transfer between a highly excited molecule and a bath gas is necessary for the interpretation of unimolecular and recombination rate coefficients and finds particular application in modeling studies, such as for combustion and atmospheric chemistry.’s2 The fundamental means of quantifying this energy-transfer process is as the rate coefficient R(E,E’) for collisional energy transfer from internal energy E’to E.’ Methods are now available for classical trajectory calculations of R(E,E’)and of its moments (only the first or second moment being required for computing quantities such as falloff curves). However, such trajectory calculations require computer codes and computational resources that are not readily available at present; nor do they provide physical insight per se. Tractable approximate models are therefore essential tools for practical applications. For the purposes of providing approximate models for collisional energy transfer, it is useful to reexpress R(E,E’) in terms of the probability per collision P(E,E’) P(E,E3 R(E,E?[Ml/ w (1) where [MI is the concentration of bath gas and o is an arbitrary (but phYsica1ly reference frequency’ A
* Author for correspondence. 0022-3654/91/2095-5007$02.50/0
number of models for P(EB’), applicable to various temperatures, bath gases, and excited molecules, have been given in the literature. However, no model presented hitherto has been properly applicable to large molecules and/or elevated temperatures, for reasons that are discussed in detail below. We present here the first model which is able to overcome these defects. We commence by briefly reviewing some well-known results for P(E,E’). Experimental variables determined by P(E,E’) or R(EB’) include falloff curves (the dependence of a rate coefficient on pressure) and the time evolution of the average energy. It is often found that these observables depend strongly on one moment of R(E,E’) or P(E,E’) and are less sensitive to other details of the functional form of these distributions. One such moment is the overall energy-transfer rate coefficient RE,,’,or equivalently, the mean energy transferred per collision, (PEaI1)= R F . ~ [ M ] / W RE,,’= L m ( E- E’)R(E,E’)d E
(2)
(3) ( I ) For example: Gilbert, R. G.; Smith, S. C. Theory o/Unimolemkrr and RecombiMrion Reoc?ionF;Blackwell Scientific: Oxford, U.K.,1990. Gilbert, R. G . In?. Rev. Phys. Chem., in press. (2) For example, Oref, I.; Tardy, D. C. Chem. Reo. 1990, 90, 1407.
0 1991 American Chemical Society
