12641
J. Phys. Chem. 1994, 98, 12641-12645
Collisional Excitation of CO Molecules by O(lD) Atoms Makoto Abe, Yousuke Inagaki, Larry L. Springsteen,? Yutaka Matsumi, and Masahiro Kawasaki: Institute for Electronic Science and Graduate School for Environmental Science, Hokkaido University, Sapporo 060, Japan
Hiroto Tachikawa Faculty of Engineering, Hokkaido University, Sapporo 060, Japan Received: August 15, 1994@
Rotational distributions of CO molecules excited by collisions with O( 'D) atoms (average relative translational energy 8 kcaumol) were measured by probing the product CO(v = 0-3) with vacuum ultraviolet laserinduced fluorescence. The inelastic collisions are of two types. The adiabatic collisions involve motion on the first excited singlet surface of the COz system, whereas the nonadiabatic collisions begin on this singlet surface but end on the lowest triplet surface before dissociating to O(3P) and CO. The measured rotational distributions for CO(v = 2 and 3) which can only come from the nonadiabatic collisions agree with the results of a trajectory calculation performed on the ab initio lowest triplet potential energy surface of C02. However, the rotational distributions measured for CO(v = 0 and 1) show a significant difference from the trajectory calculation. The reason is that excitation of the v = 0 and 1 levels can be accomplished by either adiabatic (singlet surface) or nonadiabatic (triplet surface) collisions. Vibrational distributions of CO(v = 1-6) excited by collision with thermal O(lD) atoms were also measured and compared with the trajectory calculation.
Introduction
0
Collisional enregy transfer of O(lD) atoms to CO molecules is an important elementary process in gas phase scattering. An O(lD) atom which is generated by UV photodissociation of 0 3 has a large kinetic energy. When this translationally hot O(lD) atom collides with a CO molecule, two energy transfer processes occur: kinetic energy transfer hot O('D)
+ CO
-
O('D)
+ CO(v,J)
(1)
electronic energy transfer
The involvement of a COz quasibound complex in the IBz state is assumed in the quenching of O(lD) atoms by CO The potential curves of Figure 1 show that these processes are associated with the quasibound complexes COz(lB2) and COZ(3B2)for processes I and 11, respectively. To study the electronic quenching of O(lD), Harding et al.5 used laser photolysis of 0 3 , Nz0, or NO2 with a buffer gas as a source of thermal O( 'D) atoms. The nascent population of CO excited to vibrational levels up to v = 4 was determined by time-resolved diode laser absorption spectroscopy. With Cl80 160(lD)as the reactants, the subsequent probing of the fundamental vibrational transition of Cl60 confirms that the O(lD) quenching mechanism involves a long-lived collisional intermediate and forms a Cl60 or Cl80 molecule from the l60=C=l8O intremediate with an equal statistical probability.
+
+ On leave from the Department of Chemistry, Come11 University, Ithaca, NY 14853-1301. Abstract published in Advance ACS Abstracts, November 1, 1994. @
+ co*
TI
OC(v,J)
0
+
O*
+ CO*(v,J)
exchange
nonexchange
(111)
(rv)
Matsumi et a1.6 reported that the electronic energy transfer efficiency of O( 'D) to the internal energy of CO is 3 1 f 7% by measurement of Doppler profiles of the O(3P2)produced in the triplet channel (11). The high efficiency of the electronic energy transfer process is strong circumstantial evidence for the collisional quenching of O(lD) by CO through a COz complex intermediate. In this work, since O(lD) atoms are produced in the 248 nm photodissociation of 03,the C02 quasibound complex is excited above the dissociation threshold by 2638 cm-' in process I and by 18 500 cm-' in process 11. Measurement of rotational and vibrational distributions can reveal the dynamics of these energy transfer processes.
Experimental Section For measurements of rotational distributions of the collisionally produced CO molecules, the experiments were performed with a mixture of O3 (15 mTorr) and CO (300 mTorr) in a 60 x 60 x 60 mm cell. The reactant O(lD) atoms were produced by 248 nm photodissociation of 0 3 with a KrF laser pulse. CO molecules collided with the O(lD) and were detected 0.5 p s later, using vacuum ultraviolet laser-induced fluorescence (VUV LIF). The VUV laser light was generated by four-wave mixing (2wl - w2) with Xe gas (10 Torr), using two dye lasers (Lambda Physik, FL3002) pumped by a XeCl excimer laser (Lambda Physik, LEXTRA-50, 308 nm, 200 mJlp~lse).~ The wavelength of 01 was 249.62 nm, which was two-photon resonant with Xe6p[1/2]o. The wavelength of w2 is 440-640 nm. The laser
0022-365419412098-12641$04.50/0 0 1994 American Chemical Society
Abe et al.
12642 J. Phys. Chem., Vol. 98, No. 48, 1994 (0-0) band
44
x
45
46
I
f
.~. I4
0
c4
0
0.5
1.0
1.5
Roc-0
(A)
2.0
2.5
-
+ 0 COZ system adapted from ref 5 . The horizontal arrow indicates the average energy of the O(’D) atom produced from the photodissociation of 0 3 at 248 nm. Figure 1. Energy diagram for the CO
energies were 0.5 and 5 &/pulse for 01 and m2, respectively. The output of the VUV light passing through the reaction cell was monitored with a V W monochromator and a solar blind photomultiplier. With etalons installed in both dye lasers the spectral width was reduced to 0.25 cm-’, while without them it was 0.4 cm-’. The VUV LIF signal of CO(AIII) was detected by a solar blind photomultiplier (Hamamatsu Photonics, R1259) and a gated integrator (Stanford Research, SR250). When it was necessary to eliminate contributions from roomtemperature CO, the dissociation laser was fired on every other pulse and active base-line subtraction was performed with the gated integrator. For measurments of vibrational distribution of the CO, 2 Torr of Ar gas was added to the mixture of O3 and CO, and the time delay was 5 ps so that the nascent rotational distribution was relaxed to the Boltzmann distribution at room temperature.
I , , ,
,
155.9
The average center-of-mass (c.m.) collision energy Ecoll for O(lD) CO is calculated with the following equatiod
+
where mo and mco refer to masses of an 0 atom and a CO molecule, respectively. EtABis a translational energy in the laboratory frame (LAB) for the O(lD) produced by the photolysis of 03 at 248 nm. Valentini et aL9 reported the internal energy distribution for the 02(lAg) photofragment from the 240 nm photodissociation of ozone in their measurement of CARS spectra of the photofragment 0 2 , 0,
+ hv - O(’D) + 02(’Ag)
The distribution of the internal energy as a fraction of total available energy ranges from 0.1 to 0.5 with an average of 0.25. Assuming that this same fraction holds at 248 nm, the translational energy of O(lD) ranges from 8.0 to 14.5 kcal/mol with an average value of 12.1 kcal/mol, Thus the c.m. collision energy in our experiment is estimated to range from 5.4 to 9.6 kcal/mol with an average value of 8.0 kcal/mol. Since the vibrational frequency w eof CO is 2170 cm-l, vibrational levels of CO attainable with this collision energy are v = 0-1 in channel I and v = 0-9 in channel 11. In Figure 1, the horizontal m o w indicates the average collision energy.
.
,
I
,
156.0 Wavelength (nm)
,
,
,
,
156.1
Figure 2. Vacuum W laser-induced fluorescence spectrum of the high rotational levels of the CO(0,O) and (1,l) Q bands from collision of CO with translationally hot O(’D).
UV photodissociation of ozone at the Hartley band produces O(lD) along with O(3P) with quantum yields of 0.9 and 0.1, respe~tively.~ The second channel is negligible because of its low yield.
Results
CO Rotational Distributions. To determine the rotational distribution of the CO molecules after collision with O(’D), we recorded fluorescence excitation spectra for the (v’, v”) = (O,O), (O,l), (l,l), (0,3), and (0,2) bands of the A’IT-XlC system. Figure 2 shows part of the excitation spectra of the CO A117X 1 c transition. Although the CO AIII-XIX band consists of P, Q, and R branches, there are many perturbed lines due to interactions with triplet states. For the analysis of rotational populations, we simulated spectra of the A’IT-X’Z system. The energy of the rotational levels of the A ’ n state was taken from the tables of Tilford and Simmons’O and Le Floch et al.” The molecular constants for the XIC state were taken from Manz et al.’* The line intensities are calculated using the formula
Z(J”J”) Estimate of the Collision Energy
,
-
= vssJ”NJ~t/g(J”)
where Z(J’ J”) is the fluorescence intensity of a rotational line in the excitation spectrum, Y is the laser frequency, SJ.~.is the Honl-London factor, NJ,, is the population in the J” level, and g(J”) is the degeneracy of the J” level. The collection efficiency of the fluorescence in the detection system is assumed to be constant in a single vibronic band. Since the line width was dominated by the spectral width of the excitation laser and the rotational structures around the band heads were congested, the calculated line spectra were convoluted with the laser line width. The values of NJ,, were adjusted so that the simulated spectra reproduced the observed spectra of each band. For the rotational lines split by the triplet state perturbations, the sum of their intensities was taken into account. For the analysis of rotational population, the spectra were taken at the delay of 0.5 ps after the photodissociation of O3 with 15 mTorr of 0 3 and 300 mTorr of CO. During the delay time of 0.5 ps, only 5-10% of the O(lD) atoms produced by the photodissociation are quenched by CO molecules. To examine the contribution of the collisional relaxation in the rotational distribution of CO after the first collision with O(lD), the excitation spectra were taken at different delay times and analyzed. The results indicated that the collisional relaxation affected the nascent rotational population by less than 10% for a delay of 0.5 ps under our pressure conditions. Figure 3 shows the rotational distributions of the product CO in v = 0, 1, 2, and 3. Rotational distributions for v 2 4 were not obtained, because the vibronic bands with v“ 2 4 were overlapped with the strong sequence bands with v” 5 3 and could not be analyzed. Only the high rotational levels are
Collisional Excitation of CO by O(lD) Atoms
J. Phys. Chem., Vol. 98, No. 48, I994 12643 I
1.0
0.04
I
I
I
I
I
I
+
0.8 c
0.02
.-
3
0.
0
I 0.041
v=l
0.6 -
0
,o”-oo
8
O
1 2 3 4 5 6 Vibrational quantum number, v
Figure 5. Vibrational population of CO produced from collision of CO with thermal O(lD). Populations are normalized to CO(v = 1): (0)this experimental work; (0)this trajectory calculation; (A) Harding et al. (ref 5); (0)Shortridge et al. (ref 5). v=3
0.041
Rotational Quantum Number, J
Figure 3. Rotational populations for v = 0, 1 , 2 and 3 of collisionally produced CO with hot O(’D). Circles give experimental results. Bars give the results of the trajectory calculations for the electronic energy transfer process 11. Shaded bars are for elastic collision (non-atom exchange reaction IV and white bars for inelastic collision (oxygen atom exchange reaction 111).
I
1
I
-0.5 0 0.5 Doppler Shift (cm-’)
Figure 4. High-resolution Doppler profiles of CO(v = 1, J = 9 and 41). Broken curves show the profiles expected if the CO(v,J) were produced via the electronic energy transfer process 11. analyzed for the v” = 0 band because of the strong background signals due to thermal population of the low rotational levels. CO Doppler Profiles. Figure 4 shows high-resolution Doppler spectra of the Q(9) and Q(41) lines in the CO(0,l) vibrational band which were obtained at 0.5 ,us delay. The dashed curve drawn with each Doppler profile shows what is expected for the corresponding rotational feature if the CO(v = 1) is produced from process 11. The simulated Doppler profile was calculated by convoluting a square profile, which corresponds to the shape expected for nascent CO molecules produced from process 11, with a 0.25 cm-l Gaussian. The width of 0.25 cm-’ was determined from both the probe VUV laser width and thermal velocity of a reactant CO molecule.
The observed Doppler profile for Q(41) shows a reasonable agreement with the simulated one. The Q(9) line is much narrower than the simulated one. The J = 9 and 41 levels in the CO(v = 1) do not originate from the same dynamical process. CO Vibrational Distribution. In order to examine the nascent vibrational distribution of CO after collision with O(ID), we recorded fluorescence excitation spectra with a buffer gas (2 Torr of Ar) at a delay time of 5 ,us after the photodissociation of 0 3 . We used these conditions because the vibronic bands with v’ > 3 were hidden by long rotational progressions of strong vibronic bands with v’ 5 3 under the conditions used for the study of the rotational distribution. For this higher pressure and longer delay, the rotational distribution of the product CO was entirely relaxed to a room-temperature distribution by collisions with the buffer gas, while the vibrational distribution was not relaxed but remained nascent. This was confirmed experimentally by changing the delay time. It should be noted that the O(lD) atoms are also translationally relaxed by collisions with the buffer gas within 0.3 ,us, and therefore most of the collisions between O(’D) and CO take place at thermal collision energies. The measured vibronic bands were (v’, V’) = (0,1), ( W , (0,2), (0,3), (1,3), (2,3), (3,4), (4,419 (3,517 ( 4 3 , ( 5 3 , (4,6), and (7,6) in the laser wavelength range between 155 and 167 nm. The spectra for v” > 7 were not detected because the signal-to-noise ratio was not good enough for the analysis. The intensity of the vibronic bands was analyzed using the formula
where Z(V’,V’’)is the fluorescence intensity of the (v’”’’) vibronic band, vV’,,”is the frequency, qjV”is the Franck-Condon factor,13 Nv$pis the population in the v” vibrational state, and is the quantum efficiency of the photomultiplier at the emission frequency for the (v’,v) vibronic bands. The values of were taken from the manufacture-supplied frequency response curve of the photomultiplier. The obtained vibrational distribution for nascent CO is plotted in Figure 5. Qvtv
Trajectory Calculations Figure 6 shows a potential energy surface for the reaction O(3P) CO with a collision angle of 8 = 120”. The calculation is performed at the MP2/6-31G* level of theory.14 Two energy minima which correspond to OC-0 and O-CO complexes are found on PES at the collision region. In order to obtain a more
+
12644 J. Phys. Chem., Vol. 98,No. 48,1994
Abe et al.
i v=o
-3
I
4
1
1 .o
2.0
'1 I
3.0
A
Potential energy the reaction O(3P)+ CO(lCf) -drawnCOfor+ each calculated the MP2/6-31G* level. Contours are kcal/mol. Dots indicate the intermediate complexes Figure 6.
O(3P)
surfaces of at
5
[0-CO] and [OC-01.
-
L -3 a
1
"
0
c: L
-
-1 -2
-3
accurate structure of the complex, geometry optimization of the complex is performed by the energy gradient method. Fully optimized parameters thus obtained are rl = 1.3246 A, 1-2 = 1.1944 A, and 8 = 121.1". The stabilization energy of the complex is estimated to be 18.1 kcaVmo1 relative to the product o ( 3 ~ ) co. The ab initio PES is fitted to the extended LEPS function by means of least squares method. The energy difference from ab initio PES is less than 2 kcaYmol at each point. A classical trajectory calculation for O(lD) CO O(3P) CO was performed on the 3B2potential energy surface. A collision of hot O(lD) with CO produces the COz(lB2) state, which intersects the C O Z ( ~ Bstate. ~ ) Two different collision energies were taken, 8.0 kcaVmol for calculation of rotational populations and 0.9 kcaVmol for calculation of vibrational distribution. Since it was difficult to calculate the whole contour of the lB2 surface, all trajectories start at the seam between the ~ B an z 3B2 surfaces. The seam ranges from 8 = 100" to 130" for the bending angle and r = 1.2-1.5 A for the bond length of the new CO bond. The transition probability was assumed to be constant over this entire seam. The collision parameter b ranged from 0 to 2.5 A. Each initial condition was randomly selected by the Monte Carlo methods. In total, 12 000 trajectories were computed to obtain rotational populations for each vibrational level. The atom exchange reaction (111) and nonexchange one (IV) are distinguished, as shown in Figure 3. At a large b value, no energy transfer may occur. Thus, noncollisional trajectories are also counted for CO(v = 0). In order to estimate this contribution, a series of trajectory calculations were started from a fixed OCO structure, 8 = 110" and r = 1.262 A, which corresponds to a point in the seam between the lB2 and 3B2states. The relative population of v = 0 thus obtained is [v = O]/[v = 11 = 2.3. Contrary to this result, when b ranges from 0 to 2.5 A, the vibrational ratio becomes 4.4.This large ratio includes a contribution of the noncollisional trajectories for the v = 0 level. Then, the vibrational population normalized to v = 1 in channel I1 is estimated to be 2.3/1/0.74/ 0.32 for v = 0, 1, 2, and 3 by this trajectory calculation.
+
+
-
+
Discussion Surprisal Analysis. The measured rotational distributions for CO(v = 0, 1, 2, and 3) were compared to statistical distributions in which available energies are distributed on the basis of the density of product states for O(3P) CO(v,J). Both the prior cal~ulation'~ po(fRf~)and the phase space calculation16 for the CO rational energy give a similar distribution but do
+
-
E
v=3 1
I
0.1
0.2
1
0.3
0.4
0.5
gR
Figure 7. Surprisal plots for CO(v = 0, 1, 2, and 3). Arrows A and B indicate the J = 9 and 41 levels of the CO(v = l), respectively,
where the high-resolution Doppler spectra of Figure 4 were measured. Slopes are the same, 5.3, for all vibrational levels. not reproduce the observed rotational distributions. For the phase space calculation, the potential functions of the interaction were assumed to be van der Waals type V(r) = -C6r-6 for both entrance and exit channels. The c.5constants for O(lD)and 37.8 CO and O(3P)-C0 were estimated as 38.2 x x erg cm6, re~pectively.l~~~* The surprisal for a particular value of gR is defined as
where P(gR) is the observed probability of having the final reduced rotational energy variable gR = f ~ / ( 1 - fv). Plots of I(gR) vs gR for v = 0, 1, 2, and 3 are shown in Figure 7. The surprisal plots for v = 0 and 1 clearly show two regions. The linear part for v = 1 has a slope of 5.3. For the v = 2 data, most of the high J points lie on a line with the same slope of 5.3. The plots for v = 3 at high J are also linear with the same slope of 5.3. The v = 2 and 3 levels of CO cannot be produced by the kinetic energy transfer process I. The surprisal analysis for v = 2 and 3 shows linear surprisal with the same slope, suggesting that most of the CO(v = 2 and 3) are produced from the electronic energy transfer process 11. A part of the low J levels of v = 2 and 3 could be produced in collisional relaxation of product CO(v = 2 and 3) by parent CO(0.3 Torr) because the LIF spectra were measured 0.5 ps after production of O(lD). The CO(v = 0 and 1) can be produced thermodynamically from both processes I and 11. The linear parts of the surprisal analysis in the large gR region for v = 0 and 1 are due to process 11, while most of the nonlinear parts are due to process I. Actually, the widths of the Doppler profiles of Figure 4 show that formation of CO(v = 1, J = 41) requires a large available energy, while formation of CO(v = 1, J = 9) requires less available energy. On the basis of the trajectory calculation results in Figure 3, we deduce that the points in the high J region
Collisional Excitation of CO by O(lD) Atoms of v = 0 and 1 arise from the electronic energy transfer process II. This is supported by the observed same slope in the surprisal analysis. The positive values of f3R which charaterize the linear rotational surprisals indicate that the rotational energy is much less than expected statistically. The dynamical constraint against rotational excitation is strong in process I1 because potential hopping is required at the seam. Comparison with Trajectory Calculations. We now compare the rotational and vibrational distributions measured experimentally with the results of classical trajectory calculations for process I1 performed on the ab initio first excited triplet potential energy surface of COz. Since only relative populations were experimentally determined, it was necessary to scale the results for comparison with the trajectory calculations. Because the high J levels of CO are collisionally produced in process 11, results of the trajectory calculations are scaled at the “linear surprisal” regions of each v level to provide good overlap for the J > 40 region of v = 0, J > 30 for v = 1, J > 20 for v = 2, and J > 10 for v = 3. The calculated rotational populations are shown overlaid on the experimental results in Figure 3 with white and shaded histograms. The shaded histograms represent population of the CO(v,J) without the 0 atom exchange (IV), while white ones refer to those with the 0 atom exchange (111). Process IV is associated with the high J levels and large collision parameters b, while process I11 with the low J level and small b values. Since the CO(v 2 2) are produced only in process 11, the experimental and calculated rotational distributions of CO(v = 2 and 3) show good agreement. The differences between measured and calculated rotational distributions for CO(v = O and 1) are due to the contribution of the CO produced in process I. This population difference between the trajectory calculation and experimental results in CO(v = 1) starts at J = 0 and ends at J = 25, peaking at around J = 11. It is interestinbg to compare this rotational distribution due to process I with that of the CO(v = 1) photofragment from the 157 nm photodissociation of C02, because the energetics are quite similar and also the dissociation occurs on the same ~ B surface, z
+ 157 nm -CO + O(’D) + 10.1 kcaVmol hot O(’D) + CO - CO + O(’D); (Ecoll)= 8.0 kcaVmol CO,
Miller et al.19 reported the CO(v = 1) rotational distribution by a pump-probe technique using VUV LIF. The distribution peaks at around J = 15 and is highly excited, terminating abruptly at the energetic limit (J = 28). This is indicative of dissociation occurring from a bent intermediate state COz( lB2). Consistent with this is a measurement of an angular anisotropy parameter of B = 0 for the photofragment.20 Since the agreement between the photodissociation and our collision results for CO(v = 1) due to process I is fairly good for the rotational distributions, we deduce that this bent state COz(lB2) plays an important role as an intermediate state in the translational energy process of O(lD) CO. Vibrational Distribution. Harding et aL5 and Shortridge et al.4 measured the collisionally produced CO(v) from thermal O(‘D) CO by infrared absorption. Figure 5 shows our results along with their results. The vibrational distribution obtained
+
+
J. Phys. Chem., Vol. 98, No. 48, 1994 12645 from our trajectory calculations is also shown in the same figure. The agreement is satisfactory. Experimentally, the v” = 0 population was difficult to measure because of the strong background signals from parent CO. Because the trajectory calculations predict the relative population (v = O)/(v = 1) = 2.3, the vibrational distribution in process I1 is estimated by combining the results of the trajectory calculation with the experimental results. The rotational distributions for process I1 may be estimated from the linear-suprisal analysis. Then, the electronic energy transfer efficiency from O(‘D) to internal degrees of freedom in CO is calculated to be 29%. In our previous measurement with Doppler spectral analysis of the product O(3P2) for process 11, Matsumi et a1.6 reported an efficiency of 31 f 7%. Slanger and Black21reported collision of CO with O(lD) generated by 0 2 photodissociation at 147 nm and showed an E-V transfer efficiency of 40 f 10%.
Acknowledgment. This work is partly defrayed by Grantsin-Aid in Priority Fields of “Free Radical Science” and “Photoreaction Dynamics”, and an International Cooperative Program from the Ministry of Education, Science and Culture, Japan. H.T. is indebted to the Computer Center at the Institute for Molecular Science for the use of the computing facilities. Y.I. thanks the Japan Society for Promotion of Science for a Fellowship to Japanese Junior Scientists. References and Notes (1) Tully, J. C. J . Chem. Phys. 1974, 61, 61; J. Chem. Phys. 1975, 62, 1893. (2) Kinnersley, S. R. Mol. Phys. 1979, 38, 1067. (3) Zahr, G. E.; Preston, R. K.; Miller, W. H. J . Chem. Phys. 1975, 62, 1127. . (4) Shortridge, R. G.; Lin, M. C. J . Chem. Phys. 1976, 64, 4076. 151 Harding. D. R.: Weston. R. E.. Jr.: Flvnn. G. W. J . Chem. Phvs. 1988,88, 35901 (61 Matsumi. Y.: Inaaaki. Y.: Morlev. G. P.: Kawasaki. M. J . Chem. Phys. 1994, 100, 315. (7) Hilber, G.; Lago, A,; Wallenstein, R. J. Opt. Soc. Am. 1987, B4, 1753. (8) Marinero, E.; Rettner, C. T.; Zare, R. N. J . Chem. Phys. 1984,80, 4142. (9) Valentini, J. J.; Gerrity, D. P.; Phillips, D. L.; Nieh, J.-C.; Tabor, K. D. J. Chem. Phys. 1987, 86, 6745. (10) Tilford, S. G.; Simmons, J. D. J. Phys. Chem. Ref. Data 1972, 147, 1972. (1 1) Le Floch, A. W.; Laumay, F.; Postas, J.; Field, R. W.; Brown, C. M.; Yoshino, K. J . Mol. Spectrosc. 1987, 121, 337. (12) Manz, A. W.; Maillard, J.-P.; Roh, W. B.; Rao, K. N. J. Mol. Spectrosc. 1975, 57, 155. (13) Walter, I. M.; Hepbum, J. W. J . Chem. Phys. 1988, 88, 6658. (14) (a) Hariharan, P. C.; Pople, J. A. Theor. Chim. Acta. 1973, 82, 213. (b) Francl, M. M.; Pietro, W. J.; Hehre, W. J.; Binkley, J. S.;Gordon, M. S.; DeFrees, D. J.; Pople, J. A. J . Chem. Phys. 1982, 77, 3654. (c) Frish, M. J.; Binkley, J. S.;Schlegel, H. B.; Raghavachari, K.; Melius, C. F.; Martin, R. L.; Stewart, J. J. P Bobrowicz, F. W.; Rohlfing, C. M.; Kahn, L. R.; DeFrees, D. J.; Seeger, R.; Whiteside, R. A,; Fox, D. J.; Fleuder, E. M.; Topiol, S.; Pople, J. A. Ab-initio molecular orbital calculation program GAUSSIAN86 Camegie-Mellon Quantum Chemistry Publishing Unit: Pittsburgh, PA, 1986; p 18. (15) Levine, R. D.; Bemstein, R. B. Ace. Chem. Res. 1974, 7 , 393. (16) Pechukas, P.; Light, J. C.; Rankin, C. J . Chem. Phys. 1966, 44, 794.. (17) Slater, J. C.; Kirkwood, J. G. Phys. Rev. 1931, 37, 682. (18) Kramer, H. L.; Herschbach, D. R. J. Chem. Phys. 1970,53,2972. (19) Miller, R. L.; Kable, S. H.; Houston, P. L.; Burak, I. J . Chem. Phys. 1991, 96, 332. (20) Stolow, A.; Lee, Y. T. J . Chem. Phys. 1993, 98, 2066. (21) Slanger, T. G.; Black, G. J . Chem. Phys. 1974, 60, 468. I
