4944
J . Phys. Chem. 1984, 88, 4944-4952
Multiphoton Ionization Mass Spectrometric Study of Toluene Clusters in a Pulsed Nozzle Beam Time-of-Flight Apparatus David W. Squiret and Richard B. Bernstein* Department of Chemistry, Columbiq University, New York, New York 10027 (Received: April 2, 1984)
Electron ionization (EI) and multiphoton ionization (MPI) fragmentation studies of van der Waals clusters of toluene are reported. The clusters are formed by expansion of toluene vapor, seeded in He, Ar, or N, carrier gas, through a Gentry-Giese pulsed valve nozzle. The skimmed jet enters the ionizer of the time-of-flight (TOF) mass spectrometer (MS), where the molecules are subjected to E1 (with electrons in the range 20-70 eV) or to MPI (via the 14; transition of the Lbband of toluene at a laser wavelength of 512.4 nm). MPI wavelength scans showed little evidence of the usual jet-cooling effect, yet clusters up to the tetramer were readily observed via both EI-MS and MPI-MS. Bimers (Le,, toluene-He, -Ar, -N2) were also detected. The relative abundances of the individual cluster species varied temporally over the duration of the pulse. Utilizing an iterative deconvolution technique, we have deduced the MPI fragmentation pattern of the toluene dimer and the jet-expanded monomer. Fragmentation of the latter is similar to that of low-pressure toluene, yielding no parent (molecular) ion C7Hsf but a characteristically strong C3H3+peak, resulting from the breakup of the cycloheptatrienyl ion (formed by facile rearrangement of the tolyl ion). The MPI fragmentation pattern of the dimer consisted of a peak at m / z 183 (parent ion minus hydrogen), and no other ions at masses down to the monomer were observed but a substantial peak at m / z 92 corresponding to the toluene molecular ion was observed. The mass spectrum of the dimer at lower m / z resembled that of MPI-excited toluene at very low laser pulse energies, showing an appreciable C4H4+contribution. It thus appears that the dimer ion is quite stable, decaying by elimination of H and by dissociation into a neutral toluene and a toluene ion. In contrast to the C7H8+ions formed by MPI of the monomer (evidently vibronically excited and rapidly fragmenting), those from the dimer dissociation appear to be of low internal energy and relatively stable.
Introduction van der Waals molecules (bimers, dimers, and larger clusters) have been the subject of considerable interest to investigators over the past few They are formed by expanding a vapor seeded in an inert carrier gas through a small orifice forming a cold supersonic jet. Due to the isentropic expansion, the internal temperature of the gas drops and collisional stabilization of van der Waals molecules occurs. The shallow potential well of van der Waals molecules means that low relative velocity collisions and high collision rates will be most effective in stabilizing them. High pressure and large nozzle diameters therefore produce more van der Waals molecules as well as a lower temperature in the uncomplexed beam. The shallow potential well binding the molecules makes for interesting structural and spectroscopic features. Such clusters bridge the gap between isolated gas-phase molecules and condensed media. The principal difficulty in examining such molecules is that a distribution of cluster sizes is produced, regardless of the method used to generate them, and the separation of the behavior of an individual Fluster of a given constitution has been difficult. In addition, when these weakly bound cluster molecules are ionized, they tend to dissociate to clusters of smaller mass, often to the monomer, as a consequence of the shallow potential wells holding the clusters together. Thus, the aspect of these molecules that makes them of interest makes them difficult to study. Laser-induced fluoresence (LIF) spectroscopy has been the method of choice to study these molecules.’-3 Mass spectrometry has been used mainly to identify clusters of a given size. Due to extensive fragmentation, however, the observation of a given n-mer ion cannot be taken as evidence of the same n-mer (neutral) precqrsor, only I n. LIF gives spectral information on a cluster molecule, e.g., an adduct-induced shift in some spectroscopic feature such as wavelength or excited-state lifetimes. Spectroscopic studies of clusters in supersonic beams utilizing LIF or two-photon ionization (2PI) can be divided into pulsed and continuous nozzle experiments. In continuous nozzle beams, LIF has been used to study van der Waals adducts of tetra, ~ benzene clusters* while resonant 2PI ~ i n eN02,6 , ~ t e t r a ~ e n e and Present address: Naval Research Laboratory, Washington, DC 20375. Department of Chemistry and Biochemistry, University of California, Los Angeles, CA 90024.
* Current address:
0022-3654/84/2088-4944$01.50/0
has been employed to examine fl~orobenzene~and benzene clusters.1° A pulsed nozzle source has been used to study fluorobenzene complexes,” fluorene complexes,12ammonia cluster^,'^ and toluene cluster^,'^ the first using LIF while fluorene and the clusters were studied with 2PI. The present study reports a different technique which obtains fragmentation behavior of cluster molecules formed by expansion from a pulsed nozzle, using multiphoton ionization (MPI). The van der Waals adducts of toluene were chosen for study here since there has been considerable related (MPI-MS) work on toluene monomer. There have been some relevant experiments reported during the course of the present study. Smalley observed toluene clusters in a pulsed molecular beam using two-photon ionization (2PI) at 193 rnm.l4 Shinohara has reported observation of ammonia clusters from a pulsed nozzle source via 2PI at 193 mm3.13H e noted that the relative abundance of the various n-mers in the (1) D. H . Levy, Annu. Reu. Phys. Chem., 31, 197 (1980). (2) D. H. Levy, ‘Photoselective Chemistry”, Vol. 1, J. Jortner, R. D. Levine, and S. A. Rice, Eds., Wiley, NY, 1981, Adv. Chem. Phys., 47, pp 323-362. (3) J. A. Beswick and J. Jortner, “Photoselective Chemistry”, Vol. 1, J. Jortner, R. D. Levine, and S. A. Rice, Eds., Wiley, NY, 1981, Adv. Chem. Phys., 47, pp 363-506. (4) R. E. Smalley, D. H. Levy, and L. Wharton, J . Chew. Phys., 64, 3266 (1976). ( 5 ) R. E. Smalley, L. Wharton, D. H. Levy, and D. W. Chandler, J . Chem. Phys., 68, 2487 (1978). (6) R. E. Smalley, L. Wharton, and D. H. Levy, J . Chem. Phys., 66, 2750 (1977). (7) A. Amirav, U. Even, and J. Jortner, Chem. Phys., 51, 31 (1980). (8) P. R. R. Langridge-Smith, D. V. Brumbaugh, C. A. Haynam, and D. H. Levy, J . Phys. Chem., 85, 3742 (1981). (9) K. Rademann, B. Brutschy, and H. Baumgartel, Chem. Phys., 80, 129 (1983). (10) J. B. Hopkins, D. E. Powers, and R. E. Smalley, J . Phys. Chew., 85, 3739 (1981). (1 1) N . Gonohe, H. Abe, N . Mikami, and M. Ito, J . Phys. Chem., 87, 4406 (1983). (12) S. Leutwyler, U. Even, and J. Jortner, Chem. Phys. Lett., 86, 439 (1982). (13) H. Shinohara, J . Chem. Phys., 79, 1732 (1983). (14) R. E. Smalley, Quanta-Ray Advertising Leaflet, PSV-1 Pulsed Valve Source, Quanta-Ray, Mountain View, CA, 1983. (15) D. W. Squire, M. P. Barbalas, and R. B. Bernstein, J . Phys. Chem., 87, 1701 (1983).
0 1984 American Chemical Society
The Journal of Physical Chemistry, Vol. 88, No. 21, 1984 4945
MPI of Toluene Clusters
MULTIPLIER
PERMANENT
ELECTRON GU FILAMENT ASSEMBLY
Figure 1. Schematic of the apparatus showing the electronics configuration. See text for details.
beam varied over the duration of the pulse. Both of these experimenters used comparatively long pulses (ca. 250 ps and ca. 2 ms, respectively) so the time variation of cluster formation is less extreme than in a shorter pulse. Both studies used resonant 2PI as their detection technique. This is an advantage in that 2PI usually fragments molecules to a much lesser extent than resonance-enhanced multiphoton ionization (REMPI) and is therefore a more faithful analytical probe for the clusters. However, 2P1, MPI, and electron ionization (EI) all suffer from the difficulty that (simultaneous or subsequent to ionization) the cluster ions undergo some dissociation, making it impossible to deduce the percentages of the various neutral cluster precursors. Moreover, the strength of a transition or a cross section can be expected to change upon cluster formation. In this study, REMPI-MS is used to study cluster formation and cluster ion fragmentation utilizing a fast pulsed nozzle source. Variation in the mass spectral fragmentation patterns and intensities over the course of the pulse has been used to deconvolute the data and ascertain the MPI mass spectrum of the toluene dimer. The question addressed in this work is the following: how does fragmentation in a dimer (or larger cluster) differ from that of the monomer? Relevant to this is the case of the metal carbonyls, studied extensively by Vaida et a1.16 These molecules generally dissociate before ionization, yielding neutral metal atoms, which then undergo multiphoton ionization. Grant has shown the existence of a range of behavior for different molecules, from ionization fragmentation to dissociation i~nization.’~The same question applies to the fate of cluster molecules undergoing irradiation. Toluene, rather than benzene, is the molecule studied here for the following reasons. Benzene has been reported to form large clusters in a pulsed beam which could unduly complicate the analysis,’* and benzene might show ionization or fragmentation effects that arise from its loss of symmetry upon clustering rather than the cluster formation i t ~ e l f . ’ ~ JToluene, ~ which does not suffer a major symmetry loss on forming clusters, has already been the subject of a detailed MPI-MS study in this 1aborat0ry.l~
-
-
Experimental Section The apparatus used in this experiment has been modified from that previously described (Figure l).lS The ionization chamber of the time-of-flight mass spectrometer (CVC MA-2) has been removed and the ionizer remounted in a new, larger chamber separately pumped by a 1500 L/s turbopump (Sargent-Welch 3 133). The ionizer-electron multiplier distance remains unchanged. To retain electron ionization capability, four 6.0 mm (16) D. A. Lichtin, R. B. Bernstein, and V. Vaida, J . Am. Chem. SOC.,104, 1830 (1982). (17) R. L. Whetten, K.-J. Fu, R. S. Tapper, and E. R. Grant, J. Phys. Chem., 87, 1484 (1983). (18) M. F. Hoffbauer, private communication, 1983. ( I 9) K. H. Fung, H. L. Selzle, and E. W. Schlag, J. Phys. Chem., 87.5 113 (1983).
/ TO TURBO PUMP
0 2 5 m f 1 lens
Figure 2. Schematic of the ionization region of the laser mass spec-
trometer showing the relationships among the various beams. The electron beam extends from the filament to the electron trap. The detector axis is normal to the surface through the center of the ionizer. X 75 mm cylindrical permanent magnets are mounted in a symmetric array around the filament-electron trap axis. This arrangement was necessary to produce minimally acceptable electron trap currents, e.g., 0.004 pA (vs. the normal value for the standard configuration of 0.2 MA). A pulsed molecular nozzle source (Beam Dynamics VCD-1) is mounted at an angle of 26O from the vertical (Figure 2) oriented to pass perpendicularly through the horizontal axis of the timeof-flight drift tube axis at the point where the electron beam crosses it. The nozzle is mounted 0.108 m from the ionization point. A 2.0-mm collimating skimmer (Beam Dynamics) is mounted 13 mm downstream from the nozzle. The pulsed valve is triggered from a Nd:YAG laser (Quanta-Ray DCR-1) pretrigger through a divide-by-two oscilloscope and a variable delay. The delay is set to overlap the ca. 50-ps molecular pulse with either the ca. 5-11s laser pulse output of the Nd:YAG laser pumped dye laser (Quanta-Ray PDL-1) or the ca. 1-ps electron pulse. The laser beam is focused into the ionization zone with a 0.25-m focal length fused silica spherical lens mounted internally to the vacuum system. The output of the mass spectrometer (also operated at 5 Hz) is processed through a wave-form recorder (Biomation 6500) into a microcomputer (DEC MINC-11). Average laser power was measured with a surface absorbing disk calorimeter (Scientech). Typical pulse energies transmitted through the ionizer were 1-6 mJ. Background pressure in the ionization region was always less than 2 X torr while the background in the flight tube was torr. The two regions reached ambient pressures of , I '1, 1
4.0mJ
DIFFERENCE x2
0
z
2
L:
0001.
DIMER SIMULATION x 0.19
0 0 IO
I
pattern differences between the toluene monomer and the lowpressure gas will be discussed first. On the whole, the deduced mass spectrum of the monomer is in good agreement with the result for the gas. Other than the m / z 51 peak in the 4.0-mJ fragmentation pattern, the larger peaks match very well with the spectral differences well within typical day-to-day variations. Minor peaks in the midmass region differ slightly in the two spectra. The C+ ion in the gas spectrum seems virtually absent in the jet-expanded monomer, however. The chief difference in the fragmentation pattern of monomeric toluene from the jet vs. gas shown in Figure 9 is in the lack of one- and seven-carbon mass peaks in the jet mass spectrum. One possible explanation is that the deconvolution attributes virtually all seven-carbon masses to the dimer, so the normal small monomer signal at those peaks could have been swamped. No explanation has been found for the low intensity of the one-carbon peaks except for the possible existence of collisional effects. Perhaps the high local molecular density in the pulsed jet is sufficient for one or two collisions to occur which relax the fragmenting ion sufficiently to preclude the formation of the energetically expensive C+. As seen in Figure 3, a toluene gas pressure of 2.5 X lo4 torr (nominal) is required to match the toluene signal from the jet source; this is significantly higher than normal for MPI fragmentation studies,15J6J9+24-25 and so the possibility of some collisional effects cannot be excluded. Next to be considered is the MPI fragmentation of the toluene dimer. Except for the results at 6.0 mJ, there are negligible yields of bimer (toluene-N2) ions ( m / z 120) and of fragment ions of mass between the toluene monomer ion and the dimer parent "clump". The most intense peak clearly associated with the dimer is m / z 183 (C,H5CH,-.C6HSCH2+ or [M-HI'). In the dimer fragmentation pattern an important contribution is the parent ion ( m / z 92) of toluene. This prominent parent ion intensity is significantly greater than for toluene gas, for which the C7H7+ ion ( m / z 91) predominates. The explanation lies in the appearance energy (AE) of ion [M-H]+ (mass 183), the dimeric equivalent of C7H7+(mass 91) for the toluene van der Waals dimer. If one assumes the AE of m / z 183 to be approximately the same as that of m / z 91, Le., 11.2 eV, the first photon absorbed following ionization at 512.36
MONOMER SIMULATION x 0.24
0.' "
1
"I*
(24) P. Hering, A. G.M. Maaswinkel, and K. L. Kompa, Chem. Phys. Lett., 83, 222 (1981). ( 2 5 ) D. A. Lichtin, R. B. Bernstein, and K. R. Newton, J . Chem. Phys., 75, 5728 (1981).
MPI of Toluene Clusters nm (i.e,, the fifth photon) will excite the dimer sufficiently (0.90 eV above the AE) to induce the formation of the [M-HI+ ion (mlz 183), which then absorbs and dissociates mainly to C7H7+and a toluene neutral. A certain fraction of the internally excited dimeric ions will, however, dissociate to monomer molecular ions before absorbing a fifth photon, using some part of this internal energy to break the van der Waals bond. (If that bond is stronger than 0.90 eV, a two-photon transition would be necessary to reach the AE of mass 91, and m l z 92 terminates the fragmentation path.) The dependence of the branching fractions of the submonomeric mass peaks upon laser pulse energy appears somewhat irregular, but certain results stand out. The peak at m / z 39 (C3H3+)is less intense in the dimer fragmentation pattern than in the monomer, while masses 52 (C4H5+)and 27 (C2H3') are much more prominent in the dimer, with m l z 5 1 nearly as pronounced. m l z 5 1 (C4H3+)is the dominant peak of benzene fragmentation under MPI conditions, which suggests one possible explanation. The C7H7+ion exists in two forms of near equal energy, the sixmembered ring tolyl (methylenebenzene) cation and the sevenmembered ring cycloheptatrienyl cation. Mass 39 is usually considered to originate from the latter, since benzene produces mass 5 1 (C4H3+)as its largest peak and tolyl could be expected to behave much like benzene. If the presence of the toluene adduct in the dimer prevented rearrangement of the m / z 183 ion and the ionic fragmentation and van der Waals dissociation were simultaneous (on a time scale of ca. 20 ps, based on the fragmentation results of Hering, et then benzene-like fragmentation would be expected. Mass 51 is abundant, and mass 64 and other five-carbon ions caused by replacement of -H by -CH3 are present in the dimer spectrum. Mass 52 could arise in the same manner from m l z 39, the second strongest peak in the MPI-induced fragmentation of benzene.25 The strength of the mlz 27 peak is not explained, however. As a whole, the observed low-mass deconvoluted dimer fragmentation pattern resembles that expected from a toluene ion constrained from rearranging into the seven-membered ring form. Another explanation, somewhat simpler, is based on the observation that the mass spectrum of the dimer at lower mlz resembles that of MPI-excited toluene at very low laser pulse energies, showing an appreciable C4H4+contribution. Thus, the submonomer ions may arise from a low-energy toluene ion from the dimer. The dimer ion appears quite stable, decaying by elimination of H and by dissociation into a neutral toluene and a toluene ion. In contrast to the C7H8+ions formed by MPI of the monomer (evidently vibronically excited and rapidly fragmenting), those from the dimer dissociation appear to be of low internal energy and relatively stable. These alternative explanations for the dimer fragmentation pattern may be elucidated by theoretical considerations; the maximal entropy theory of MPI fragmentationz6has already been quite successful in predicting and explaining such data on polyatomic molecules and has now been applied to cluster m0lecules,2~ including the toluene dimer.28 Schlag and co-w~rkers'~ have elucidated the MPI fragmentation mechanism for benzene dimers by a study of their isotope effects. They found that the electronic excitation is effectively localized in one or the other of the benzene rings. This is not inconsistent with the present results for toluene dimers.
Conclusions The E1 results are relatively straightforward. n-meric toluene clusters ( n = 1-4) were detected in the molecular pulse, and the ( 2 6 ) J. Silberstein and R. D. Levine, Chem. Phys. Lett., 74, 6 (1980); J . Chem. Phys., 75, 5735 (1981); Chem. Phys. Lett., 99, 1 (1983). (27) J. Silberstein, N. Ohmichi, and R. D. Levine, following paper in this issue. (28) Several of the more intense peaks in the dimer mass spectrum have +1 and +2 amu peaks associated with them, such as m / z 184 (6 and 5 mJ), 185 (6 and 5 mJ), and 93 (5 mJ), which are stronger than predicted for "C isotopically substituted molecules. These might be attributed in part to the formation of larger ions via hydrogen abstraction from toluene as a consequence of the high local molecular density.
The Journal of Physical Chemistry, Vol. 88, No. 21, 1984 4951 bimer was detected for all three carrier gases at the "leading edge" of the pulse (and at the tail of the pulse for helium and nitrogen). The relative intensities of each of the van der Waals molecules vary strongly with time after the pulsed valve opening. A reasonable explanation for this result is a large cross section for the dimer exchange reaction as follows: bimer
-+
(C,Hg)***X C7Hg C7H,.-C7H, -+
+X
(X = He, N,, Ar) ( 1 )
A deeper potential well binding the toluene dimer is expected relative to that of the bimer leading to a substantial exoergicity for the bimer dimer exchange reaction. This would imply a rapid rate. Under MPI conditions, clusters of up to four toluene molecules have been detected. However, no shift in the two-photon transition frequency was detected within the f5-cm-' resolution of the present experients. No bimer was detected for MPI, but for these time delays the E1 mass spectra also showed virtually no bimer content. The relative percentages of dimer at the different time delays are the same for MPI and EI, but absolute values differ appreciably. The deconvoluted monomer MPI mass spectra resemble the toluene gas spectra. The chief difference is in the lack of onecarbon ions. The dimer MPI mass spectra are composed of two sections. One section, the dimeric equivalent of the molecular peaks in bulk-gas experiments, arises from nascent ions breaking the van der Waals bond before absorbing a fifth photon (the first following ionization). This group of ions gives rise to low internal energy toluene molecular ions and the mass fragmentation pattern arising from their laser photodissociation, a mass fragmentation pattern very similar to that of monomer fragmentation. The second section of the dimer fragmentation pattern is caused by the nascent molecules absorbing one or more photons and losing a hydrogen atom to form mlz 183. Either the C7H7+(excited) portion of this ion is restricted from rearranging into a seven-membered ring or, a more probable explanation, mlz 183 gives, on dissociation, a neutral toluene and a low internal energy C7H7+(as compared to the ion produced when monomer is excited under the same conditions). The C7H7+produces a laser mass fragmentation pattern resembling that of monomer at a lower laser power. The toluene dimer MPI fragmentation mass spectrum is, therefore, made up of a portion resembling toluene monomer results at a longer excitation wavelength than that actually used (giving rise to toluene molecular ion) and a portion resembling toluene monomer results at a lower laser power (leading to larger carbon fragments than monomer does)
-
I
Acknowledgment. D.W.S. would like to acknowledge useful discussions on the design of the apparatus with Drs. K. R. Newton and D. A. Lichtin and aid in construction of the apparatus by Dr. M. P. Barbalas. Valuable advice about the pulsed valve was provided by Prof. W. R. Gentry, Prof. C. F. Giese, and J. Ringer of Beam Dynamics. R.B.B. appreciates helpful suggestions on the interpretation of the MPI mass spectra from Profs. R. D. Levine and M. A. El-Sayed. Support by NSF Grants C H E 78-25187 and C H E 81-16368 is gratefully acknowledged. Appendix The procedure for deconvolution of the MPI mass spectra of toluene clusters was as follows. Taking the data from delays of 157, 171, and 186 p s , pairwise subtracting them ensuring no negative peaks (within experimental error), deleting these small negative peaks, and averaging the resulting mass spectral series lead to an approximate dimer fragmentation series. The dimer mass spectra series was then used in the same manner with the three experimental data series to obtain an approximate set of monomer mass fragmentation patterns. This method led to great latitude in the final spectra so an evaluation procedure of the simulation had to be constructed. The initial requirement for any prospective fit was that the monomer results had to resemble the toluene gas results.
4952
J. Phys. Chem. 1984, 88, 4952-4955 39(exptl) - x[39(dimer)] - (1 - x)[39(monomer)] = 0
A simultaneous equation fit was then used to test the simulation. The 157-ys time delay spectra evidencing the highest dimer concentration were used for this fit. Mass 39 is the largest peak in both the experimental cluster mixture and the monomer spectrum, so it was employed as the fitted datum. The simulation spectra are generated as a series, with the 6.0-mJ total ion current set as 1.O. The lower power spectra are found to decline (in total ion current) in an essentially quadratic dependence, as expected. The 6.0-, 5.0-, and 4.0-mJ values for mass 39 in the monomer, dimer and 157-ys delay experimental spectra were inserted into the equation
(2)
where x is the experimental dimer fraction at that time within the molecular pulse. A perfect fit over all three laser powers with a single x is impossible, but the monomer and dimer were varied to minimize the variation from zero in eq 2. This procedure generated the “dimer simulations” shown in Figure 8. These represent the best present estimates of the MPI mass spectra of the pure dimer species under the specified conditions. Registry No. Toluene, 108-88-3.
The Nonstatistical Multiphoton Ionizatioh-Dissociation of the van der Waals Toluene Dimer J. Silberstein, N. Ohmichi, and R. D. Levine* The Fritz Haber Molecular Dynamics Research Center, The Hebrew University, Jerusalem 91 904, Israel (Received: April 5, 1984)
The persistence of van der Waals bimer ions in the fragmentation pattern of the toluene dimer ion can be accounted for by using a simple, physically motivated constraint. The computations predict that many of these bimers are in predissociative states. Comparison is made with the observed patterns of Squire and Bernstein (preceding paper). Uncertainties in the dissociation energies of the bimer ions preclude however a quantitative check.
Introduction The experimental study’ of the fragmentation, following multiphoton ionization (MPI), of van der Waals dimers provides a critical test of the maximal entropy computationZof the mass spectrum. The considerable frequency mismatch between the vibrations of the monomer and that of the van der Waals bond suggests that the dissociation of such dimers will not be well described by a simple statistical limk3 Indeed we find that the observed’ fragmentation pattern of the toluene dimer is not consistent with the predictions of the extreme statistical limit as previously introduced.2 In particular, there is a qualitative manifestation of this failure. In a given observed fragmentation pattern, where many small (and hence energy expensive) ions are present, one still detects nonnegligible quantities of large ions. This is unexpected in the extreme statistical limit for in that limit the pattern is simply determined by a balance between energetic and entropic considerations. At low levels of excitation, energy wins. One sees mostly the more stable and larger ions. At high levels of excitation, entropy wins and the fragmentation is extensive. The maximum entropy formalism is not limited to computing the fragmentation pattern in the extreme statistical limit. The latter is a special case where the distribution of species (whether ionic or neutral) is constrained only by the universal conservation conditions (energy, elements, charge) and is otherwise of maximal entropy. One can however impose additional constraints. The central result of this paper is that imposing one such constraint whose interpretation is clear-cut suffices to remove the qualitative discrepancy between experiment and the extreme statistical limit. We do not report a quantitative comparison with experimental results since we have too much latitude at the moment for achieving a good fit to experiment. The reason is that input to the computation is the partition function of the dimer ion (and (1) D. W. Squire and R. B. Bernstein, preceding paper in this issue. (2) (a) J. Silberstein and R. D. Levine, Chem. Phys. Lett., 74, 6 (1980); (b) J. Chem. Phys., 75, 5735 (1981). (3) J. Jortner and R. D. Levine, Adu. Chem. Phys., 47, 1 (1981).
0022-3654/84/2088-4952$01.50/0
of any other ionic or neutral potential fragment). Since neither the dissociation energy nor the geometry and certainly not the frequencies of the dimer ion are known, we have too many parameters at our d i ~ p o s a l . ~ A contributing factor to the success of the simple deviance from statistics description proposed below is, of course, that the MPI-fragmentation of toluene monomer is well described by the extreme statistical limit. This is confirmed both by a computation-free test5 and by computational comparisons2b,6with experiment.
The Extreme Statistical Limit The entropy of the distribution of fragments is2 S = -kxX,[ln X, I
+
In x , ~- In x]
(1)
1
k is the gas constant (per molecule). Xj is the number of molecules of species j (whether neutral or charged), and summation is over all potential fragments. X (which is not a priori known) is the total number of fragments
x = cx, J
(2)
xlj is the fraction of molecules of species j in the quantum state i. In the extreme statistical limit the entropy is maximized subject to the following ubiquitous constraints: (a) conservation of probability EXij = 1 i
(b) conservation of energy (4) Any other theory which requires additional input, e.g. the parameters of the transition state for the dissociation of the dimer ion, will have an even wider latitude. ( 5 ) D. A. Lichtin, R. B. Bernstein, and K. R. Newton, J . Chem. Phys., 75, 5728 (1981). (6) N. Ohmichi, J. Silberstein, and R. D. Levine, Isr. J. Chem., in press.
0 1984 American Chemical Society
