J. Phys. Chem. 1990, 94, 11-84
77
P(E,E’), although for a single-channel reaction the results are not sensitive to this functional form. For these illustrative calculations, the functional form chosen for P(E,E’) was the biased random walk model:32
2. This sensitivity analysis suggests that slight variations in the choice of the assumed LJ parameters will not significantly change the energy-transfer values obtained from VLPP experiments. Strictly speaking, analysis of VLPP data should use a total elastic collision number Z,,, to determine the diffusion coefficient P(E,E’) = !/2s-17r-I/2exp[-(s2z + E - E’)2/4s2] (A.l) D and a single measure of the energy-transfer rate, such as RE,,?. In practice, the latter is written as the product of an inelastic where reference collision number Z and ( A E 2 ) . In the absence of exz = -8 In [ p ( E ) exp(-E/kBT)]/8E perimentally determined diffusion coefficients, one often has to (-4.2) use the same set of LJ parameters to calculate both D and Z . In p ( E ) is the density of states, and s is a parameter with the disuch a situation, the only sure method of comparing gas/gas mensions of energy which can be estimated ~emiempirically,~~ energy-transfer quantities from experimental VLPP data obtained obtained accurately from trajectory c a l c ~ l a t i o n s or , ~ (as ~ ~ in ~ ~the by using different Z values is to reanalyze the data by using a present case) obtained by fitting eq 28 to the experimental data. consistent set of gas/wall and gas/gas collision numbers for the Note from eq 2 and A.l that two systems being compared. The R(E,E’) obtained from such manipulation of the appropriate master equation (eq 28) involves ( A E 2 ) = 2sz s4z2 (A.3) an implicit dependence on Z through the definition of the diffusion coefficient D which appears on the left-hand side of eq 29. Hence, This functional form for P(E,E’) comes from the physically plausible assumption that the internal energy of the excited a proper comparison of theoretical results with pressuredependent VLPP data requires using the to compute a molecule varies randomly during a collision, subject to the contheoretical pressure dependence by solution of eq 28, and comstraint of microscopic r e v e r ~ i b i l i t y . ~In~ the , ~ ~calculations, the usual p r a ~ t i c e ~of ~ -taking ~ * D = D ( Z ) was used. The products paring experimental and theoretical values of dkuni/dp. This comparison if desired can be through a “trajectory” value of Z ( A E ) , Z ( AEdown) and Z ( A E 2 ) are listed in Table I. ( AE2)1/2,etc., obtained by taking the dkUni/dpof the VLPP kUni It can be seen that the products Z ( A E ) = RE,,I. Z ( AEdown), and Z ( A E 2 ) = RE,,2are indeed functions of D as indicated by computed from the trajectory value for RE,2and seeing what value of (AE2)IIZ,etc., gives the same dkuni/dpwith the Z used to and (AEdown) have a eq 29. It can also be seen that (AE2)1/2 relatively weak dependence on D: variation is only of the order interpret the original experimental data. Once again, this last comparison is convenient but relies on an arbitrary Z . We have although Z (from of l0-15% in (AEbm)and ca. 30% in ( AE2)1/2, reported such comparisons elsewhere.I3 which D has been obtained) has been varied by over a factor of
+
Trajectory Simulations of Collisional Energy Transfer of Highly Vibrationally Excited Azulene Kieran F. Limt and Robert G. Gilbert* Department of Theoretical Chemistry, Sydney University, NS W 2006, Australia (Received: January 10, 1989; In Final Form: April 27, 1989)
Trajectory calculations are reported on highly vibrationally excited azulene colliding with monatomic bath gases. An accurate valence force field was used for the intramolecular potential, and atom-atom Lennard-Jones and exponential repulsive forms for the intermolecular potential. Energy-transfer parameters were obtained using the method of Lim and Gilbert, which computes the mean-square rate of energy transfer. This computation converges with ca. lo2 trajectories. The results show that the energy transfer is through multiple interactions dominated by the repulsive part of the potential. The results are compared with extensive experimental data for this system. The computed energy-transfervalues are similar to those obtained experimentally, and indicate that, for all except the lightest bath gases, the method may be used to calculate average energy-transfer values a priori with sufficient accuracy to predict falloff curves. However, the trajectory results are in poor accord with experiment for light bath gases (He and Ne), because the repulsive part of the assumed potential functions is too hard.
Introduction Being able to predict the rate of energy transfer in collisions between a highly excited molecule and a bath gas is essential for the a priori calculation of the rate coefficients of unimolecular and recombination reactions; moreover, it is also required to extrapolate data, obtained over a limited pressure and temperature range, to any pressure and temperature. The theory for this process can be approached from two different directions: trajectory calculations (which purport to be essentially exact simulations of the physical system) and approximate models. These approaches are complementary: the former are laborious and hard * Author for correspondence.
’ Present address:
Department of Chemistry, Stanford University, Stan-
ford, CA 94305.
0022-3654/90/2094-0077$02.50/0
to interpret phenomenologically but invoke minimal approximations, while the latter should be easier to evaluate and give physical insight. Most importantly, the correctness of approximations made in a particular model can be tested against the benchmark of “exact” trajectory simulations (Le., this will test the approximations made by a model in the dynamical aspect of the problem, given that the underlying potential is the same in both approaches). The trajectories should clearly employ a realistic potential, and the proper test of this is that the results are in accord with reliable experiments. Some of the most extensive and reliable experimental data on this energy transfer have been obtained for highly excited azulene. In the past few years, the deactivation of azulene molecules by various bath gases has been studied by Hippler, Troe, and cow o r k e r ~ ’ -(herein ~ called HT) and by Barker and co-workers
0 1990 American Chemical Society
Lim and Gilbert
78 The Journal of Physical Chemistry, Vol. 94, No. I , 1990 (herein called BCI).~-~The (AE) values that were first reported by BG6 and HT1,2are very similar in magnitude (at least for an initial excitation energy E’ = 30 644 cm-I), but the observed qualitative trends were quite different. The data of BG have been interpreted to suggest that at high energies (E’in excess of IO4 cm-I), (AE) has a strong (almost linear) dependence on energy while the data of H T have been interpreted to suggest that (AE) has only a weak dependence on energy. In this paper, the theory developed in the preceding article9 (denoted paper 1 ) for the economic and essentially exact calculation of average energy-transfer properties by trajectory simulation is applied to azulene/monatomic bath gas collisions. The objectives are ( I ) to make overall comparison with experiments (which is essentially a test of the potential functions used), (2) to gain qualitative understanding of the experimental findings and (3) to identify possible sources of (systematic and other) discrepancies between theoretical calculations and experimental determinations of energy-transfer quantities. The method developed in paper 1 is summarized as follows. It relies on computing the mean-square energy-transfer rate, I f R(E’,E) is the (second-order) rate coefficient for collisional energy transfer from initial energy E’ to final energy E (with dimension concentration-l X energy X time-’), then
RE,,2is related to the mean-square energy transferred per collision, ( AEZ),through RF32= Z ( A E 2 )
(2)
where Z is a collision number; other quantities such as (AE) are then found through an appropriate functional form for R(E,E’). is The classical mechanical expression for
where b is the impact parameter, kB Boltzmann’s constant, I.L the reduced mass of the colliding species, E , the relative translational energy, and B(E,E’,E,b) the probability that the substrate molecule will undergo the collisional transition from E’to E. In terms of a finite number of trajectories, is evaluated by using the expression
ad2 N
RF,2= hm lim ( 8 k B T / ~ p ) 1 / 2 - c ( A E i ) 2 d-m N-m N j=l
(4)
Here d is the maximum impact parameter (Le., the actual upper bound of the integration over b in eq 3), and AEi is the energy change in the ith trajectory, which starts with initial conditions b, E,, and E’. N is the number of trajectories. Details of the method of evaluation, particularly the means of choosing the initial conditions from appropriate distribution functions, are given in paper I . One important advantage of the present method is that all contributions to the summation in eq 4 are positive, which usually ensures more rapid conversion than if one were computing the average energy transferred, ( AE),rather than the mean-square quantity ( A E 2 ) . ( I ) Hippler, H.; Lindemann, L.; Troe, J. J . Cfiem.Pfiys. 1985,83, 3906. (2) Hippler, H. Ber. Bunsen-Ges. Pfiys. Cfiem. 1985, 89, 303. (3) Hippler, H. Private communications. (4) Hippler, H.; Otto, B.; Troe,J. Ber. Bunsen-Ges.Phys. Cfiem. 1989, 93, 423. (5) Damfn, M.; Hippler, H.; Troe, J. J . Chem. Pfiys. 1988, 88, 3564. (6) Rossi, M. J.; Pladziewicz, J. R.; Barker, J. R. J . Cfiem. Pfiys. 1983, 78. 6695; Barker, J. R.; Golden, R. E. J . Pfiys. Cfiem. 1984, 88, 1012. (7) Shi, J.; Barker, J. R. J . Cfiem. Pfiys. 1988, 88, 6219. (8) Barker, J. R. J. Pfiys. Cfiem. 1984, 88, 1 1 . (9) Lim, K. F.; Gilbert, R. G. J. Pfiys. Cfiem.,preceding paper in this issue.
The Deactivation of Azulene Azulene is particularly suited to energy-transfer investigations using laser excitation/relaxation techniques for reasons discussed by Barker.s In the studies by BG, azulene was excited either to initial energy E’ N 17 500 cm-I or to E’ E 30 600 cm-I. Collisional relaxation was monitored by time-resolved infrared emission (IRE) from CH stretching modes.lO-” These experiments were interpreted to suggest that (AE) is strongly dependent on the average energy ( E ) . This qualitative behavior is consistent with statistical theories since the more energy a molecule possesses, the more energy it can transfer to a collider in a single collision.12.13 This qualitative behavior has also been observed in theoretical calculations for smaller systems (e.g., CH4 ArI4), but to date, there have not been any theoretical studies of systems of comparable size to azulene (CloHs). The method of initial laser excitation used by H T was similar to that employed by BG. Collisional relaxation was monitored by time-resolved ultraviolet absorption (UVA).I5 The UVA experiments were interpreted to suggest that ( h E ) is proportional to ( E ) at low energies and independent of (or only weakly dependent on) ( E ) ,over a large range of higher ( E ) values; this is consistent with other UVA experiments with different reactant molecules in a large number of bath gases.2*1b1s The discrepancy between the results of the two groups seems to hinge primarily on the calibration of their intensity versus ( E ) or absorption coefficient versus ( E ) curves, as well as on possible assumptions implicit in their analyses. The discrepancy does not seem to be an artifact of the experimental method per se, since UVA experiments on benzeneI9 and perfluorobenzene20 also suggest that (AE)is dependent on ( E ) . Another possibility (which is suggested by inter alia the results of this work) is that ( A E ) is strongly dependent on ( E ) at low ( E ) but essentially independent of ( E ) at higher ( E ) . As well as its experimental convenience, azulene is also well suited to theoretical investigation by classical trajectory simulations for the following reasons: (i) its properties are well characterized, which is essential for modelling; (ii) it is a planar molecule, so that its geometry and vibrational motions can be described by a potential consisting only of stretching, bending and out-of-plane deformation terms; (iii) at the energies of interest, azulene has a high density of states, in excess of 1OI8 states per cm-I, so that the discrete energy ladder is well approximated by an energy continuum; and (iv) any potential terms associated with interactions between azulene and a collider molecule will act as a perturbation of the azulene eigenstates and, together with (ii) will result in a mixing of states so that quantized behavior will probably not be observed.
+
Specification of Solution of Classical Equations of Motion The present study is based upon the validity of applying classical mechanics to the collisional energy-transfer process in question. Now, Sceats2‘ has suggested the low excitation per mode invalidates the use of classical mechanics. However, notwithstanding the low excitation of each individual vibrational mode, the high density of states, the rapid internal redistribution of energy (on a time scale faster than the average time between collisions),1’ (IO) Forst, W.; Barker, J. R. J . Cfiem.Phys. 1985, 83, 124. ( I I ) Shi, J.; Bernfeld, D.; Barker, J. R. J. Chem. Phys. 1988, 88, 621 1. (12) E.g., Schranz, H. W.; Nordholm, S. Inr. J . Cfiem.Kinet. 1981, 13, 1051. (13) Oref, I. J. Cfiem. Pfiys. 1982, 77, 5146 and references therein. (14) Hu, X.; Hase, W. L. J. Phys. Cfiem. 1988, 92, 4042. (15) Brouwer, L.; Hippler, H.; Lindemann, L.; Troe, J. J . Pfiys. Cfiem. .. 1985, 89, 4608. (16) Hippler, H.; Troe, J.; Wendelkin, H. J. J . Cfiem. Pfiys. 1983, 78, 671-19
(17) Hippler, H.; Troe, J.; Wendelkin, H. J. J . Cfiem. Pfiys. 1983, 78,
6718. (18) Heymann, M.; Hippler, H.;Troe, J. J . Cfiem.Pfiys. 1984, 80, 1853. (19) Nakashima, N.; Yoshihara, K. J . Cfiem. Pfiys. 1983, 79, 2727. (20) Ichimura, T.; Mori, Y.; Nakashima, N.; Yoshihara, K. J . Cfiem. Pfiys. 1985, 83, 117. (21) Sceats, M. G . Private communication.
Collisional Energy Transfer of Excited Azulene
The Journal of Physical Chemistry, Vol. 94, No. 1, I990 79 TABLE I: Observed and Predicted Vibrational Frequencies (cm-I) and Assignments of Azulene
valence force field
e~perimental~~
I-
\
/
C H stretches
Figure 1. Bond orders and assumed charge distribution of azulene used to specify potential functions.
and mixing of eigenstates all strongly suggest that azulene/collider systems should be well described by classical mechanics. The solution of Hamilton’s equations requires specification of the azulene intramolecular potential and of the azulene/bath gas intermolecular potential. Intramolecular Potential. A valence force field was used to describe the azulene molecule with the force constants chosen according to Lindner’s rules.22 This potential consisted of contributions of harmonic stretches, harmonic bends, torsions about CC (partial) double bonds, and out-of-plane harmonic wags, with interactions between atoms not directly bonded to each other assumed to be zero: V= Vharmonic stretch + vharmonic bend + Vtorsion + 2 vharmonic wag (5) Since azulene has a large number (48) of vibrational modes, even at high excitation energies of ca. 30000 cm-’, each of azulene’s vibrational modes will be, on average, only in the v = 1 or v = 0 vibrational states, where anharmonicities in the potential are not important. This validates the use of harmonic terms in the potential. Note that this is a weak assumption since the local modes are specified by harmonic terms: the mixing of local modes to give normal modes will give anharmonicities in the intramolecular potential. Further, it has been demonstratedl4sZ3that the use of a harmonic potential instead of an anharmonic potential (or vice versa) does not significantly alter the calculated energy-transfer quantities. The potential terms for the in-plane vibrational motions were taken to beZ2 vhaharmonic stretch
= f/zf,(r - re)2
f, (mdyn A-l) = 5.000, C-H = -31.514/rO2 + 130.0171r~4- 7O.269/ro6, C-C Vharmonicknd
= j/2(0.572 mdyn
A rad-2)(tl -
(6) (7) (8)
The equilibrium bond lengths re, and bond angles e,, for the carbon skeleton were taken from the observed geometry of azulene?2 The value of the ro (which varies from one C C bond to another) was that calculated.22 For the C H stretches, re was taken to be 1.084 A and Be for the C C H angles were taken to be symmetric about each C H bond. Bond orders p (required for the torsion given below) are shown for the various bonds in Figure 1; the corresponding values of re and ro, as (p, re, ro),are (1.500, 1.404, l.4), (1.125, 1.404, 1.4l), (1.750, 1.49, 1.476), (1.125, 1.380, 1.398), (1.625, 1.391, 1.398), and (1.375, 1.385, 1.403). The potential terms for the out-of-plane vibrational modes were taken to be22 Vtorsion= (0.075 mdyn A ) p sin2 T
(9)
(10) = h(O.45 mdyn A p is the bond order of the CC bond, the assumed bond orders being shown in Figure 1. T is the torsional angle defined so that the planar configuration gives T = 0. cp is the angle between the C H vharmonic wag
(22) Lindner, H. J. Tetrahedron 1974, 30, 1127. (23) Bruehl, M.; Schatz, G. C. J . Phys. Chem. 1988, 92, 7223.
C C stretches
C-C stretch C C H bends
CCC bends
AI 3098 3072 3037 3037 2968 1579 1457 1448 1396 1210 1268 1160 971 900
825 680 406
Bl 3077 3042 3018 1536 1480 1443 1378 1049 1300 1216 1117 712 1012 987 486 323
BI 3039 3034 3033
AI 3043 3034 3033 3033 3030 1992 1787 1715 1569 975 1357 1193 1149 807
2037 1922 1807 1635 1223 1480 1328 1284 1198 1073 667 404 327
754 600 373
ex~erimental~~ valence force field B2 1058 965 952 795 762
A2 C C H deformations 941 91 1 813
CCC deformations 542 331 189
zero-point energy
304 240 30 892
A2
8 2
1252 1072 982
1323 1178 1116 914 873 597 46 1 336 175
739 406 175 33 268
bond and the plane defined by the carbon skeleton. The normal-mode frequencies predicted by the assumed intramolecular potential are compared to the experimental fundamental f r e q ~ e n c i e sin~ Table ~ I. Azulene/Collider Interaction Potential. There are neither theoretical calculations nor experimental benchmarks for the azulene/monatomic interaction potentials, nor are there any data on van der Waals molecules for these partners. Since the classic work of Stace and Murre11,25-26generalized Lennard-Jones (12-6-4) potentials have been commonly used in classical trajectory studies (e.g., ref 14, 27, 28). Recently, exponential repulsive terms (and/or Morse interactions) have also been used (e.g., ref 23, 29, 30). The main type of interaction potential used in this work was that of the sum of pairwise generalized Lennard-Jones (LJ) terms. The azulene dipole was mimicked by placing point charges on each of the carbon atoms, giving rise to charge-induced dipole interaction terms in the intermolecular potential. The relative charges on the carbon atoms were taken from Wheland and M a n r ~ and ,~~ their magnitudes were scaled to give agreement with the experC m). Hence the hydroimental dipole of 1.0 D (3.34 X gen-collider interactions were given by V = 4eH[(uH/r)l2- ( ~ ~ / r ) ~ ]
(1 1)
and the carbon-collider interactions were given by V = 4tc[(cc/r)12 - ( ~ ~ / r q) 2~~ (] 8 ~ o ~ # ) - ’ (12) r is the atom-atom center-of-mass separation, q is the charge on (24) Chao, R. S.;Khanna, R. K. Spectrochim. Acta, 1977, 3 3 4 39. (25) Stace, A. J.; Murrell, J . N. J . Chem. Phys. 1978, 68, 3028. (26) Stace, A. J. Mol. Phys. 1978, 36, 81. (27) Hippler, H.; Schranz, H. W.; Troe, J. J . Phys. Chem. 1986,90,6158. (28) Schranz, H. W.; Troe, J. J . Phys. Chem. 1986, 90, 6168. (29) Grinchak, M. B.; Levitsky, A. A,; Polak, L. S . ; Umanskii. S. Ya. Chem. Phys. 1984, 88, 365. (30) Bruehl, M.; Schatz, G. C. J . Chem. Phys. 1988, 89, 770. (31) Wheland, G. W.; Mann, D. E. J. Chem. Phys. 1949, 17, 264.
80 The Journal of Physical Chemistry, Vol. 94, No. I , 1990 TABLE 11: Lennard-Jones Potential Parameters for Azulene/Monatomic Path Gas Collisions (Eq 11-13) bath gas He Ne Ar Kr
Molecular 2.55
IO 0.20
TABLE 111: Potential Parameters (Eq 18-20) and Energy-Transfer Results (Normalized to the Reference Collision Frequency of HT') for Azulene/Monatomic Bath Gas (He and Kr) Collisions with an Exponential Repulsive Potential" helium krypton Kr/He ratio
Xe
parameter^',^
2.82 32 0.39
3.47 114 1.63
3.66 178 2.46
4.05 230 4.00
3.5007 3.3407 57.8545 32.3416 1.185 0.588
3.7208 3.5585 61.5361 34.3997 1.203 0.514
BH,
3.3847 3.2268 48.1379 26.9098 1.170 0.635
each carbon atom, shown in Figure 1, a is the polarizability of the collider atom, and to is the permittivity of free space. The parameters uH,tH, uc, and tc, for the interactions between a reactant atom (H or C) and the monatomic bath gas M, were taken to be given by
The a are the LJ radii and t the corresponding LJ well depths.I6 The parameters XI and X2 are fitting parameters which were determined as follows. A local rectilinear coordinate system was defined relative to the azulene center of mass. The potential (given by the sum of the individual pairwise LJ terms) projected onto any one of the six rectilinear axes ( f x , f y , f z ) had a zero ( V = 0) at some distance rc and a minimum value E,. An "overall molecular radius" a was defined to be the value of rc averaged over the six axes and an "overall molecular well depth" t was similarly defined to be the average value of E,,,. For consistency with experiment, XI and X2 were chosen so that the overall molecular radius u and well depth t were the same as the LJ radii uLJ and well depths t L j used by HT. The a, uH, UC, tH, and tc values used in the trajectory calculations are listed in Table 11. A "softer" exponential repulsion was used in some trajectory simulations instead of a r-12 repulsion. In such calculations, the intermolecular potentials eq 11 and 12 were replaced by V = AH exp{-BHr) - 4tH(aH/r)6,
hydrogen collider
(14)
V= Ac exp(-Bcr) - 4 ~ ~ ( a ~-/q2a(8to~r4)-1, r ) ~ carbon collider (15) where A H , Ac, BH, and Bc are obtained from uH, uc, tH, and tC by requiring that Ai exp{-Bir} = 4ti(q/r)I2, i = H or C
at r =
ai
Exponential Repulsion
A-'
AH, K Ac, K N (E2)Il2cm-l - ( A E ) , cm-I
4.194 3.972 3.305 X lo6 5.913 X IO6 120 554 320
N (AE2)1/2, cm-I -(A/?), cm-I
180 690 425
(AE2)1/2, cm-I - ( A E ) , cm-I
220 75
( A E 2 ) ' l 2 cm-I , - ( A E ) , cm-]
247 44
Bc, A-1
Atom-Atom Parameters 2.8501 3.0127 2.6993 2.8593 17.5718 29.3327 9.8229 16.3966 1.117 1.137 0.965 0.840
Lim and Gilbert
(16)
and r = 2'/6ui. This gives the relationships
Bi = 0 , - ' ( 2 ' / ~ - I)-I In (4), Ai = 1.02(4ti exp{uiBi}), i =
H or C (17) where, in addition to criteria given by eq 16, the Ai's have been scaled by an empirical 1.02 factor to improve agreement between the overall molecular radii and well depths and the LJ radii and well depths. ac, uH, tc, and tH are the same LJ parameters and have the same values as those in eq 1 1 and 12, and in Table 11. The parameters AH, Ac. BH, and Bc are listed in Table 111. Selection of Initial Conditions For the azulene/monatomic bath gas systems, it can be assumed that, after laser excitation, internal conversion to the highly excited vibrational manifold of the electronic ground state produces vibrationally hot molecules. Further, since total momentum and angular momentum are conserved during the laser excitation/ internal conversion sequence (which occurs faster than any collisional process*), it can be assumed that the excited azulene
3.389 3.234 19.09 X lo6 21.20 X lo6 120 270 107
r-12 Repulsion 120 329 148
0.49 0.33
0.48 0.35
Experimental, HT'l3 339 154
1.54 2.05
Experimental, BG6v7 399 144
1.62 3.27
'Simulations for T = 300 K, E' = 17 500 cm-I. N = number of trajectories in calculation. Results for r-I2 repulsion duplicated from Tables V and VI for comparison. molecules are translationally and rotationally cold. The laser photons are not absorbed by the bath gas, so that the collider molecules are translationally cold. Further, since only monatomic bath gases are considered in this study, the collider molecules have no internal (vibrational and rotational) energy. The initial vibrational phases were chosen from a microcanonical ensemble at energy E' E' = ( E ) = (average thermal energy at 300 K) hv
+
= 919 cm-'
+ hv
(18)
where hv is the laser excitation energy. The azulene angular momentum was selected from a thermal ensemble at 300 K. The initial orientation of the azulene molecule relative to the monatomic collider was chosen by randomly rotating the azulene molecule about its center of mass through Euler's angles. The initial impact energy E, was selected from a Boltzmann distribution, eq I O of paper I . The initial impact parameter b was chosen from the range bmi, 8 b < b,,,, according to where {is a random number (with uniform distribution) between 0 and 1. This procedure selects b according to the impact-parameter binning algorithm of paper 1, and with relative statistical weighting given by eq 11 therein. Equation 4 gives the algorithm for obtaining the rate of energy transfer from trajectories, by accumulation of the mean-square energy transfer from trajectories which can be "binned" according to their impact parameter. For azulene/helium trajectories, three bins were used to test the convergence of eq 22 of paper 1, Le., the convergence of the integration over impact parameter in eq 3: 1st bin 0 Q b* < 1.0 2nd bin 1.0 8 b* < 1.5Il2 3rd bin
1.5II2 Q b*
< 1.5
where b* is the reduced impact parameter (b*)2 = b2[(uLJ)2a(2,2)*]-1 (20) The above bins were used to calculate all final trajectory results using eq 22 of paper I . Convergence of RE,2with increasing b can be seen more clearly if the trajectories can be resorted into even smaller (sub)bins and eq 43 of paper 1 evaluated as a function of increasing impact parameter (or as a function of the "counter" j ) . Table IV and
Collisional Energy Transfer of Excited Azulene
The Journal of Physical Chemistry, Vol. 94, No. 1. 1990 81
TABLE IV: Convergence of the Mean-Square Rate of Energy Transfer, RE,,*,with Increasing Impact Parameter b for Azulene/Helium Traiectories at T = 300 K O
TABLE V Results of Trajectory Simulations (Using Generalized Lennard-Jones Intermolecular Potentials) for the Collisional Energy Transfer between Highly Vibrationally Excited Azulene Molecules and Monatomic Bath Gases at 300 K' He Ne Ar Kr Xe Zb = 812 Z = 474 Z = 523 Z = 469 Z = 478 E' = 17500 cm-' 50.8 44.8 101 64.4 RE,,z, m3 387
0-0.5 0.5-1.0 1.0-1.5 1.5-2.0 2.0-2.5 2.5-3.0 3.0-3.5 3.5-4.0 4.0-4.5 4.5-5.0 5.0-5.5 5.5-6.0 6.O-6.5 6.5-7.0
0-0.1 1 0.1 1-0.22 0.22-0.33 0.33-0.45 0.45-0.56 0.56-0.67 0.67-0.78 0.78-0.89 0.89-1.00 1.00-1.11 1.11-1.22 1.22-1.34 1.34-1.45 1.45-1.56
2
0-0.5 0.5-1.0 1 .0-1.5 1.5-2.0 2.0-2.5 2.5-3.0 3.0-3.5 3.5-4.0 4 .O-4.5 4.5-5.0 5.0-5.5 5.5-6.0 6.0-6.5 6.5-7.0
E' = 30644 cm-l 0-0.1 1 2 0.1 1-0.22 5 0.22-0.33 3 0.33-0.45 7 0.4 5-0.56 9 0.56-0.67 3 0.67-0.78 IO 0.78-0.89 6 0.89-1.00 17 1.00-1.11 33 1.11-1.22 25 1.22-1.34 28 1.34-1.45 17 1.45-1.56 15
5 3 7 9 3
IO 6 17 33 25 28 17 15
0.2 23.5 44.1 45.8 74.5 92.0 121.4 469.9 520.1 531.1 562.0 563.8 563.9 563.9 0.2 8.6 26.4 27.8 35.9 77.9 147.6 527.6 622.1 639.0 756.6 761.7 762.1 762.1
N, = number of trajectories having b in the indicated range. 1
(32) Hase, W. L. QCPE 1983, 3, 14
s-l (cm-1)2 N ( A E 2 ) 1 / 2cm-I ,
180 690
RE,,2r
m3 SKI(cm-1)2
509
N ( A E 2 ) 1 / 2cm-l ,
180 792
120 351
120 329
120 306
E' = 30644 cm-l 159 87.5
54.2
49.9
120 409
120 340
120 323
120 46 1
120 579
"Collision number Z (used for converting RE,,2to the more convenient energy-transfer quantity ( A E 2 ) 1 / 2taken ) to be that of ref I . N = m3 s-l. number of trajectories in calculation. b Z in
achieve convergence to within the accuracy of typical experimental data. The uncertainty in our final energy-transfer results can be estimated from data such as that in Figure 2 to be ca. i20%.
Calculations As stated, the method involves calculating the mean-square rate of energy transfer, which is related to the mean-square energy transfer per collision, ( A E 2 ) ,by the collision number Z ( M 2 ) = R E , , 2 / Z . Interconversion between ( A E 2 ) and ( A E ) is then achieved through assuming a physically reasonable functional form for the collisional energy-transfer probability distribution function, P(E,E') = R ( E , E ' ) / Z . For convenience, the results of the trajectory simulations will be discussed in terms of the nth moment of the probability of energy transferred per collision, (AE") (normalized to the collision numbers used by HT), rather than measures of the rate RE,,,. However, it is essential to emphasize that it is the latter which is both computed from trajectories and extracted from experimental data, and the factorization into a collision number and a mean energy transfer per collision is through a collision number whose definition is nonunique. The convergence of the first and second moments of AE from the calculations of this work are plotted in Figure 2 for a set of azulene/He trajectories. The impact parameter has been varied between 0 and 7.4 A, where the rotational and translational temperature is 300 K, and the azulene has initial internal energy E' = 17 500 cm-'. Note that eq 4 effectively consists of a double summation over both impact parameter b and the number of trajectories N . in the jth (impact parameter) bin. For the purposes of Figure 2, the double summation has been contracted to a single summation over the number of trajectories by weighting the trajectories in each bin accordingly. It can be seen that, for a large highly excited molecule such as azulene, ( A E 2 ) ' / 2has converged after ca. 50 trajectories to within 20% of the final value; for this particular case, the first moment ( AE)also converges with about the same number of trajectories. However, Grinchak et al.29and other workers have demonstrated that, for molecules that are smaller than azulene and/or at lower excitation energies, ( AE2)1/2 converges more quickly with the number of trajectories than ( AE). This check of proper convergence in both number of trajectories and maximum impact parameter is most important and has caused difficulties in the past. Our results show, for example, that it is inaccurate to assume that one can use as the maximum impact parameter (Le., the actual upper limit of the integration over b in eq 3) simply the quantity C T ~ [ Q ( ~ , ~ ) *A] ~number / ~ . of previous calculations (including some of our 0wn33934)are incorrect as a result of this invalid assumption. Instead, it would appear that a value at least 20% higher should always be used to obtain proper convergence. (33) Gilbert, R. G.J . Chem. Phys. 1984, 80,5501. (34) Lim, K . F.: Gilbert, R. G. J . Chem. Phys. 1986,84, 6129.
82
Lim and Gilbert
The Journal of Physical Chemistry, Vol. 94, No, I , 1990
TABLE VI: - ( A E ) (cm-’) for Azulene/Monatomic Bath Cas Collisions at 300 K‘ (A) and (AE2)’12(em-’) for Azulene/Monatomic Bath Gas Collisions at 300 Kb (B) Xe Ar Kr He Ne exptl, HT3.4 exptl, BG6,’ IECTi2 this work (Lennard-Jones) potential)
A
- 5Oi -tE
Z c = 8 1 2 Z = 4 7 4 Z = 5 2 3 Z = 4 6 9 Z = 478 E ’ = 17500 cm-’ 15 120 160 I54 160 1I O 44 92 134 I44 476 476 476 476 476 425 244 163 148 133
exptl, HT’ exptl, BG6*7 IECT’* this work (Lennard-Jones potential)
79 98 849 519
E‘ = 30644 cm” I79 205 176 236 849 849 35 1 216
225 210 849 164
225 222 849 154
exptl, HT3.4 exptl, BG637 this work (Lennard-Jones potential)
220 167 690
E ‘ = 17500 cm-l 288 347 246 309 46 I 35 1
339 324 329
347 273 306
exptl, HT’ exptl, BG6.7 this work (Lennard-Jones potential)
221 247 792
E’ = 30644 cm‘’ 360 391 356 434 519 409
420 399 340
420 417 323
A
2
,
509 --
“All values normalized to Z cited by Hippler et a1.I Numbers in parentheses are ( A E ) values obtained directly from trajectory simulations ( n = 1 in eq 33 of ref 9) rather than through RE,,2.*All values normalized to Z cited by Hippler et al.’ c Z in
r
A
C
m3 s-I.
The results of the trajectory calculations using LJ-type potentials (eq 1 1-1 3) are tabulated in Table V. The ( AE2)values reported therein for convenience are obtained from the calculated values for each bath gas by using the Lennard-Jones collision number ZLJas the reference collision number Z , where the LJ parameters were taken to be the same as those used by HT. I f the shape of P(E,E”) = R(E,E’)/Z is known, then any moment or measure of P(E,E’) is sufficient to generate the full P(E,E’)distribution and hence yield all the other moments. The interconversion between these quantities here is made assuming the biased random walk (BRW) mode1;33*34 this model is based upon the constraints of energy conservation and microscopic reversibility, together with the weak assumption (for which there is ample ju~tification)~) that energy transfer during a collision is pseudorandom. The values of ( A E ) , etc., are not particularly sensitive to the functional form of P(E,E’): e.g., if the interconversion is made by using the exponential down instead of the BRW model, the results change by much less than the experimental and computational uncertainty in the ( A E ) values. (AE) values can be obtained from ( A E 2 ) 1 /values 2 by fitting to eq 11 and 27 of ref 34 or (more simply) to eq A.3 of ref 9 and eq 25 and 26 of ref 33. The ( A E ) and ( A E 2 ) ’ / 2values thus obtained (for LJ-type potentials, eq 11-1 3) are compared with those derived from experiment in Table VI. (Note that all the ( A E ) and ( , 1 E 2 ) 1 / values 2 are normalized to the reference collision frequencies used by HT.) Trajectory simulations using “soft” exponential repulsive potentials (eq 13-16) were performed for the azulene/helium and azulene/krypton systems at E’ = 17 500 cm-’ and T = 300 K. Those results are compared to the calculations using the “hard” LJ potentials (eq 11-1 3) in Table 111. Comparison with Experiment
It can be seen that the results of this work (Tables 111-VI1 and Figures 3 and 4) are a real improvement over those of a statistical theory such as the impulsive ergodic collision theory ( IECT)I2 (indeed, the latter gives very poor accord with experiment). The general trends in the results of the exact simulations of this work
He Ne A r Kr Xe Figure 4. ( AE2)II2for azulene/monatomic bath gas collisions at E’ = 30644 cm-I and T = 300 K. (W) Experimental data of Shi and Barker;7 ( 0 )HT
experimental data;l (A)present trajectory calculations, using
hard (Lennard-Jones) intermolecular potential. All values normalized9 to reference collision numbers used by HT.’
TABLE VII: (AE)(E’=30644cm-’)/(AE)(E’=17500cm-I) Azulene/Monatomic Bath Gas Collisions at 300 K He Ne Ar Kr Xe 1.05 1.49 1.28 1.46 1.41 exptl, HT133.4 1.91 1.76 1.46 2.02 exptl, BG6l7 2.23 IECTl2 this work
1.78 1.22
1.78 1.44
1.78 1.33
1.78 1.11
1.78 1.16
for
av 1.34 1.88 1.78 1.25
(Lennard-Jones potential) are (i) ( A E ) and (AE2)’12have the correct order of magnitude: there is sufficient accuracy, at least for the heavier bath gases, for use in the calculation of falloff rate coefficients (Le., to within ca. 30% accuracy); (ii) ( A E ) and ( AE2)’I2show an increasing trend with increasing excitation energy E‘ as observed experimentally; and (iii) ( A E ) and ( A E 2 ) ’ I 2have the wrong trend (compared to that of experiment) with increasing bath gas mass (from helium to xenon). It must be emphasized here that this work has used exact trajectory simulations, with only three assumptions: (i) the assumed potential energy surface(s); (ii) the assumed functional form for P(E,E’);and (iii) the applicability of classical mechanics and the numerical accuracy of the calculation. Assumption (ii) is a weak assumption and only affects the ( AE) values reported in Tables 111, VI, and VII; its effect on ( A E ) is
Collisional Energy Transfer of Excited Azulene
The Journal of Physical Chemistry, Vol. 94, No. 1. 1990 83
less than the experimental uncertainty, and it does not affect any of the general conclusions stated above. Hence discrepancy with experiment must be due to the breakdown of assumptions (i) and/or (iii). Assumption (i) is discussed in the next section. This leaves only the question of the applicability of classical mechanics, which is almost a philosophical point. As this question has been discussed e l s e ~ h e r eit, ~will ~ not be discussed further here. The numerical accuracy of the calculation was checked by demanding that the total energy of the bimolecular system being simulated be conserved to within 1 cm-'. Irrespective of agreement with experiment, the results reported here can be contrasted with, and can serve as a future benchmark for testing, models (such as the IECT and the BRW models) which make dynamical assumptions about energy transfer in addition to the above assumptions (i)-(iii).
ambiguity. The trajectory calculations for each bath gas can be considered a theoretical test of the energy dependence of ( A E ) for some model system. It is important to note that the microscopic reversibility requirement demands that (AE)must be zero at equilibrium, regardless of the actual broadness of the P(E,E') or R(E,E') distributions; Le., at thermal equilibrium ( ( E ) = ( ( AE) is equal to zero, although (A,??)and the average downward energy transfer (AEdown)may be quite large. For azulene at 300 K, (E)thermal== 1000 cm-'. At high energies where ( E ) is greatly are all suitable in excess of (E)theml,( M ) (,A E 2 ) ,and (&!?down) measures of the P(E,E') and R(E,E') distribution^.^ In the following discussion about reactant molecules at high energies, ( A E ) and ( AEdown)will both be used. Apart from azulene which remains contentious, energy-dependent ( A E ) values have been observed inter alia for CS2,36 S02,37benzene,I9 and perfluorobenzene.20 However, data from ethylcycloheptatriene, isopropylcycloheptatriene, and toluene systems have been interpreted to suggest that ( AE) is at most only weakly dependent on the excitation energy E', for E'? lo4
Dependence of Energy-Transfer Results on Potential The calculations in this work used generalized LJ potentials ("hard potentials"), and potentials consisting of exponential repulsions with r4 and r"' attractions ("soft potentials") with the same spherically averaged potential parameters. It can be seen from Tables 111, V, and VI that the energy transferred per collision, as measured by either ( A E ) or ( AE2), decreases in magnitude along the series from helium to xenon bath gas. This trend was observed for both the initial internal energies, E'= 17 500 cm-I and E' = 30 644 cm-l, for azulene. This is the opposite trend to what has been observed experimentally by both HT and BG. Since the present calculations represent essentially exact classical evaluations of the appropriate quantities (and quantum effects are unlikely to be important), the most likely origin of this discrepancy is the assumed form for the intermolecular interaction potential. The recent trajectory study by Bruehl and S ~ h a t which z ~ ~ investigated the variation in energy transfer with systematic variation of bath gas also observed this decreasing trend for the systems CS2/He, CS2/Ne, and CS2/Ar. In the case of CS2 systems, the e ~ p e r i m e n t a (AE) l ~ ~ values also decreased in magnitude through the series from helium to argon. It can be seen from Table 111 that the soft potentials decrease the amount of energy transferred per collision for both helium and krypton bath gases for an initial azulene internal energy of E' = 17 500 cm-l. The same attractive terms were used for both the hard and soft intermolecular potentials which were chosen so that both hard and soft potentials for a given bath gas had the same overall well depth and molecular radius. In each case, ( AE2)'I2decreases by ca. 20% in changing from the hard to the soft potential. This result is in excellent agreement with the work of Bruehl and S c h a t who ~ ~ ~systematically varied the depth and steepness of the intermolecular potential in their study of energy transfer in the CS2/He system. The decrease in (AE) on softening the repulsive wall brings the trajectory results closer to experiment. From this, we conclude that a softer form for the repulsive part of the monatomic/substrate interaction could certainly be found for the lighter bath gases that would give accord with experiment. Table 111 shows that a softer form for the heavier bath gases brings little change to the quite acceptable accord between calculated and observed energy-transfer parameters. From this we conclude that the true intermolecular interaction is much softer than the LJ repulsion which has been commonly assumed in trajectory calculations on energy transfer. The resolution of the precise form of this repulsive potential must await data from quantum chemical calculations and/or van der Waals spectroscopy. Energy Dependence of ( A E ) Notwithstanding any uncertainties in the potential, which makes obtaining quantitative agreement with experiment difficult and also confuses trends when progressing along a series of bath gases, the qualitative variation in ( AE2) (or ( A E ) ) is free of any such (35) For example: Porter, R. N.; Raff, L. M. In Dynamics of Molecular Collisions; Miller, W. H., Ed., Plenum: New York, 1976. ( 3 6 ) Dove, J. E.; Hippler, H.; Troe, J. J . Chem. Phys. 1985, 82, 1907.
cm-l.
16,I 7
A simplistic representation of the energy dependence of ( A E ) is to take the ratios of the ( AE) values at E' = 30 644 and 17 500 cm-l. This is shown in Table VII. Note that (30644)/(17 500) = 1.75. Now, IECTI2 predicts ( A E )(IECT, 30644)/ ( A E )(IECT, 17500) = .78
BG's average ( AE) ratio of 1.88 is in almost perfect accord with statistical trends; however, the values of ( A E ) obtained by the statistical theory are in gross error. HT's average ( A E ) ratio of 1.34 shows a much weaker dependence of ( A E ) on the average excitation energy ( E ) . The discrepancy between the two groups is a problem since their "different" methods (IRE and UVA) are intrinsically the same experiment. The resolution of the disagreement must lie in refinement of either their spectroscopic observable versus internal energy calibration or the analyses of their data, or both. The calculations of this work lend credence to HT's conclusion that ( AE) is only weakly dependent on initial energy E'at sufficiently high E'. This is consistent with some preliminary results from HT5which suggest that azulene at even higher excitation energies (E'up to ca. 60000 cm-l) has ( A E ) values that are very similar to those at E ' = 30000 cm-l. Mechanism of Energy Transfer The time dependence of the internal energy during a collision, as shown in the E i ( t ) curves of ref 34, show that fluctuations in E i ( t )during a collision occur when the separation, re,, between the monatomic collider and the closest atom in the azulene molecule is less than ca. one atomic diameter, e.g., in the case of azulene/argon collisions, when re, is less than ca. 3.7 A. (uC and uH for the individual argon atom LJ interactions are ca. 3.3 A.) This implies that changes in the reactant internal energy Ei are governed by the repulsive parts of the individual atom-atom contributions to the intermolecular potential. The dependence of energy transfer on the (short-range) repulsive walls of intermolecular potentials has been noted elsewhere: (i) Landau and Teller3* noted that only the short-range (Le., repulsive) forces are of importance in energy transfer; (ii) the semiclassical models of energy transfer proposed by inter alia Schwartz et al.,39 Miklavc and Fischer,40and McDonald and Rice4I either explicitly or implicitly assume that the energy transfer is determined directly by the short-range repulsive forces; (iii) Bruehl and SchatzZ3noted that the ( AE) values calculated from their classical trajectory simulations depend on the steepness of the repulsive part of the assumed intermolecular potential, not (37) Hippler, H. Ber. Bunsen-Ges. Phys. Chem. 1985, 89, 303. (38) Landau, L.; Teller, E. Phys. Z . Sowietunion 1936, 10, 34. (39) Schwartz, R. N.; Slawsky, Z. I.; Herzfeld, K. F. J . Chem. Phys. 1952, 20, 1591; Schwartz, R. N.; Herzfeld, K. F. J . Chem. Phys. 1954, 22, 767. (40) Miklavc, A,; Fisher, S. F. J. Chem. Phys. 1978, 69, 281. Miklavc, A. J . Chem. Phys. 1980, 72, 3805. (41) McDonald, D. B.; Rice, S. A. J . Chem. Phys. 1981, 74, 4918.
The Journal of Physical Chemistry, Vol. 94, No. 1 , 1990
84
on the well depth or the attractive terms. However, all of the above studies have been quantitative and have not actually applied the physical ideas inherent in an atom-atom interaction model to further qualitative understanding of collisional energy transfer. The foregoing discussion shows that the energy transfer is governed by multiple atom-atom interactions and therefore dependent on the repulsive part of the interaction potential. A full qualitative picture of the energy-transfer mechanism involves concepts associated with the BRW model and will be discussed in a later paper by the present authors (indeed, it will inter alia be pointed out therein that our earlier work applying this model to trajectory simulation^^^ can in fact be reinterpreted as an approximate means of evaluating eq 3). Without invoking those (BRW model) concepts, the atom-atom interaction "model" explains (i) the experimental observation that ( A E ) and (AEdown) values are of similar magnitude for a wide variety of systems with different reactants but all with the same bath gas;42and (ii) the o b ~ e r v a t i o nthat ~ ~ .calculated ~~ ( AE)and (A&,,) values strongly depend on the repulsive (but not the attractive) part(s) of the intermolecular potential. For example, consider a homologous series of alkane/M systems, where M is some (eg., monatomic) bath gas; all are governed by the same basic carbon/M or hydrogen/M repulsive interactions, which will be similar for different (organic) reactants. Hence all will have similar energy-transfer quantities. These trends are indeed seen e ~ p e r i m e n t a l l y .Finer ~ ~ details will be determined by the full Hamiltonian for the particular system. In a trajectory simulation study, changes to the attractive part of the intermolecular potential will not affect the multiple short-range atom-atom interaction nature of the collision and hence will not affect the amount of energy transferred per collision. On the other hand, simulations using hard repulsive potentials will, for the lightest bath gases, predict energy-transfer values that are very different from those using soft repulsive potentials. (Other workers23have noted these qualitative trends without explicitly stating that those trends can be explained in terms of the atomatom interactions or encounters.) This also explains why a relatively small number of trajectories are required for qualitative convergence in and hence in ( A E 2 ) . Each collision consists of many individual atom-atom encounters. Thus, the AE value from any one collision (trajectory) represents a kind of ensemble average over many individual atom-atom encounters. Hence a relatively small number (ca. IO2) of trajectories is sufficient to sample the carbon/bath gas and hydrogen/bath gas interactions. Conclusions
Our new technique of obtaining energy-transfer parameters from classical trajectory simulations has been applied in a study of collisional energy transfer between vibrationally excited azulene and the series of monatomic colliders from helium to xenon. The data, analyzed by using the theory of paper 1, indicate that the second moment of the energy-transfer rate is easily obtained from such trajectory simulations. The average energy transferred per collision (AE) can then be obtained by using the results of paper I . These calculations represent essentially an exact classical trajectorl evaluation of ( AE2) and (AE)and are the first properly converged simulations on such a large molecule. The new method (42) For example: Tardy, D. C.;Rabinovitch, B. S. Chem. Reo. 1977, 77, 369.
Lim and Gilbert obviates the problem of eliminating the "elastic peak" which has caused difficulties in the past in many trajectory of this type of energy transfer. The results of the calculations show that simulations using intermolecular LJ potentials predict larger energy-transfer quantities ( ( A E 2 ) etc.) than those using potentials with softer repulsive walls, e.g., exponential repulsion. The classical trajectory simulations predict that energy-transfer quantities decrease through the series from helium to xenon. Both of these predictions are consistent with the recent calculations of Bruehl and S ~ h a t z ~ ~ on CS2/He, CS2/Ne, and CS2/Ar. The decrease of ( A E 2 )(or, equivalently, ( AE)and ( AEdom)) for azulene in progressing from helium bath gas to xenon bath gas is contrary to the experimental trend observed by both H T and BG. The comparison between theory and experiment suggests that, for all except the lightest bath gases, the commonly assumed interaction potential functions (combined with the methodology used here) enable energy-transfer quantities to be evaluated with quite sufficient accuracy for use in predicting and fitting falloff curves. This is an important and useful conclusion, since no other reliable a priori method for this has existed hitherto. We have exploited this new technique to predict and/or interpret energytransfer data on a number of systems.@ However, for light bath gases (He and Ne), the comparison between theory and experiment strongly suggests that the common assumption of atom/atom Lennard-Jones repulsions for the interaction potential is wrong and that the true interaction is much softer. Reliable a priori evaluation of energy-transfer quantities for these light bath gases must await further data on the repulsive part of the interaction potential; such data could be obtained from quantum chemical calculations and from spectroscopic studies of van der Waals molecules. The exact calculations of this paper clearly illustrate that the mechanism for energy transfer depends primarily on the repulsive interactions or encounters between the individual atoms of the reactant and the individual atoms of the bath gas. This mechanism qualitatively explains the results of some experimental and theoretical energy-transfer investigations and will be discussed further in a later paper within the framework of the biased random walk model. Irrespective of agreement with experiment, the results reported here arise from exact classical mechanical simulations f o r the assumed potential surfaces. As such they can serve as a benchmark for testing models for energy transfer which make assumptions about the dynamics involved in the energy-transfer process.
Acknowledgment. The financial support of the Australian Research Grants Scheme, and stimulating interaction with Dr. Andy Whyte, are gratefully acknowledged. K.F.L. gratefully acknowledges the support of a Commonwealth Postgraduate Research Award. We much appreciate Dr. Horst Hippler and Professor Jurgen Troe for communicating their results prior to publication, and Professor George Schatz for preprints of his work. Registry No. Ne, 7440-01-9; Ar, 7440-37-1; Kr, 7439-90-9; Xe, 7440-63-3; He, 7440-59-7; azulene, 275-5 1-4.
(43) Brown, N. J.; Miller, J . A. J . Chem. Phys. 1984, 80,5568. (44) Whyte, A. R.; Lim, K. F.; Gilbert, R. G . ;Hase, W. L. Chem. Phys. Left. 1988, 152, 377; Whyte, A. R.; Gilbert, R. G . Ausf. J . Chem. 1989, 42, 1227.
