978
The Journal of Physical Chemistry, Vol. 83,No, 8, 1979
J.
C. Polanyi and N. Sathyamurthy
Rotational Energy Transfer (Theory). 3. HCI 4- HCI J. C. Polanyl" and N. Sathyamurthyt Department of Chemistry, University of Toronto, Toronto M5S 1A 1, Canada (Received July 14, 1978)
Results of a quasi-classical trajectory study are reported for rotational energy transfer in HC1-HCl collisions for two different initial vibrational states and various rotational states at two different collision energies. Apart from computing integral inelastic cross sections, results are interpreted in terms of the first moments of the rotational energy transferred, and also in terms of the "C value" obtained by fitting the integral inelastic cross sections to an exponential equation for R T transfer probability, proposed in earlier work. While the various results seem to be essentially independent of the initial vibrational state of the rotor, they are shown to depend on the initial rotational state of the primary collision partner (HC1 molecule number l),Jli.The C value decreases with increase in J;. With increase in collision energy rotational excitation becomes dominant over rotational deexcitation in qualitative accord with experiment. Quantitatively, however, the average C value computed from theory deviates considerably from the experimental value most probably due to the inadequacy of the short-range anisotropy of the potential used in this study. Results of HC1-HC1 rotational inelastic scattering are compared with those of HCl-Ar. Though the total integral inelastic cross section doubles as the collision partner is changed from Ar to HCl (perhaps due to increase in collision diameter) first moments of the rotational energy transferred, and the C value for J1'= 8, are virtually indistinguishable for the two systems since T R transfer is the dominant process at enhanced collision energies (15-30 kcal mol-l) employed in this study.
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1. Introduction Despite numerous theoretical studies of rotational energy transfer for diatom-atom collisions, AB + C,I little has been done in the area of diatom-diatom, AB CD, rotational transfer. The quantum mechanical formalism for scattering between two rigid rotors is well nonetheless only two accurate close c ~ u p l i n and g ~ ~a ~few approximate calculation^^@^^ have been reported. When the collision partner is also diatomic (or polyatomic) additional inelastic channels open up, and the complexity of the close-coupling calculations increases enormously. Even semiclassical and classical trajectory calculations have been few in The classical trajectory treatment for diatom-diatom systems are significantly more expensive than for the diatom-atom case but this has probably not been the principal impediment, The obstacle has been the nonavailability of intermolecular potentials other than rotationally averaged potentials (e.g., ref 20). The number of geometric variables to be included increases from two to four when the collision partner is changed from an atom to diatom. This means that the fitting of an intermolecular potential becomes a difficult task. As a result, reliable ab initio potentials exist only for HF-HF21-23and for C02-H2.24 Even adequate (semi)empirical potentials for four-atom systems are rare. T o our knowledge, they are available only for H2-H2 and HCl-HC1.11,17,25-28 Apart from its interest as a test of the interaction potential, a study of rotational energy transfer in diatomdiatom collisions poses the interesting question: how is the amount of energy transferred in a collision affected by the ability of the collision partner to take up energy as internal excitation. In an attempt to gain insight into this, we have undertaken a study of rotational energy transfer in HC1-HC1 collisions and compared the results with those of our previous study on HC1-Ar and HC1-He systems.l In this work, we have used the quasi-classical trajectory (QCT) method to follow HC1-HC1 collisions on the semiempirical potential of Raff et al.27as employed by Bass and Thompson.28 Bass and Thompson used this potential to study vibrational energy transfer; their results, obtained
+
'Department of Chemistry, Indian Institute of Technology, Kanpur, India 208016. 0022-3654/79/2083-0978$01 .OO/O
by means of a QCT study, were in good accord with high temperature experimental results. This suggests that the short-range repulsive interaction on their hypersurface was satisfactorily represented. It does not, however, shed much light on the dependability of the anisotropy of the surface, which is of prime importance in connection with rotational-energy transfer. Our earlier study1 indicated that, for the collision energies employed in the present work, the long-range forces were of secondary importance. Hence it was hoped that the present study would provide information regarding the adequacy of the short-range anisotropy of the HC1-HC1 potential. We have computed state-to-state integral inelastic cross sections and first moments of rotational energy transferred for various initial vibrotational states, a t collision energies of 15 and 30 kcal mol-l. The initial rotational state (Jli) of the primary molecule was varied from 0 to 8. The vibrational state was varied for the case of Jli= 8 from u = 0 to u = 1 in order to examine the role of vibrational energy in rotational energy transfer. It is common practice to make use of a vibrationless rotor (Le., an initially nonvibrating molecule that can be vibrationally excited during the course of the co1lision)l or rigid rotor8-l9 in studies of rotational energy transfer. Pattengil130 has recently demonstrated for the case of H2-He collisions that the inclusion of the vibrational motion does not alter the rotational inelasticity significantly. We find the same to be true for the case of HCl-HCl rotational transfer. The inelastic cross sections computed in the present study were fitted to the exponential equation for R 6 T energy t r a n ~ f e r ' ~in*order ~ ~ ~ ~to obtain the value of the constant C in the exponential term. This treatment of the data has its parallel in surprisal a n a l y s i ~ (see ~ ~ -section ~~ 4.1). The results, namely, the integral inelastic cross sections, the first moments, and the C values, were then compared with those for HC1-Ar to ascertain the effect of changing the collision partner from an atom to a diatom, in rotational relaxation of an HC1 molecule. In addition, the mass of the halogen atom in the collision partner was altered from 35Cl to that of a hypothetical lz7Clin order to isolate the mass effects in HC1-HC1 rotational energy transfer. Finally, we have computed the population changes in a thermal distribution of rotational states of the primary 0 1979 American Chemical Society
The Journal of Physical Chemistry, Vol. 83, No. 8, 1979
Rotational Energy Transfer Theory
979
molecule due t,o collision with a thermal distribution of rotational states in the collision-partner Jd, at an enhanced collision energy. The C value obtained by fitting the results to the exponential equation was compared with the fluorescence-depletion-beam (FDB) results of Ding and P ~ l a n y i .The ~ ~ change in C with altered collision partner from Ar to HC1 as obtained from theory was compared with the experimental. f i n d i ~ ~ g . ~ ~ ! ~ ~ Section 2 discusses certain aspects of the intermolecular potential employed, while section 3 outlines the computational method. Slection 4 presents the results and discussion, and section 5 summarizes the findings.
2. Potential-Energy Surface The semiempirical potential-energy surface employed in the present study is based on the valence-bond formulation of Raff et al.27 The relevant formulae along with the potential-energy surface parameters have recently been given by Bass and Thompson,28and hence not reproduced here. The anisotropy of the potential is illustrated by a series of contour diagrams in Figure la-d for four different in-plane orientations of the HC1-HC1 pair, each molecule being at its equilibrium separation. The coordinates R and I9 correspond to the center-of-mass separation and the angle between R and one HCl bond. The angle 6 between R and the other HC1 bond 1i3 kept fixed a t 0,90,180, and 270°, respectively in Figure la-d. In the case of an atom-diatom system, the potential is often expanded in terms of Legendre polynomials. It then becomes easy to separate the isotropic from the anisotropic part of the potential and to plot the coefficients in the expansion as a functiion of R. In the case of a diatomdiatom system, on the other hand, an expansion is needed in three angular variables (e,$, and the out-of-plane angle). Such an expansion is cumbersome. Even when it is carried out, as was done for HF-HF by Alexander and De Pristo,2' it is difficult to picture the anisotropic part of the potential. The short-range anisotropy of the HCl-HC1 interaction as can be seen from Figure 1 is qualitatively similar to that of HCl-Ar (cf. the potential designated GGK in ref 1). The long-range interaction also resembles the HC1-Ar potential in that the most stable conformation is on the H side of the molecule (I9 < 90°) for all four orientations considered. The HC1-HC1 long-range attractive well, however, is nearly 4 kcal mol-I deep in contrast to the shallow 0.4 kcal mols1 well for the diatom-atom counterpart. This is likely to be connected with the dipole-dipole attraction for the HC1-HC1 pair. The uncertainty in the description of the long-range interaction is probably unimportant for energy transfer at the enhanced collision energies employed in the present study.'lz8 3. Computational Method The QCT method can be found described elsewhere.37 The diatom-diatom coordinate system used in the present study, and the Hamiltonian equations of motion were the same as those employed by Raff, Thompson, Sims, and Porter.3s Fifth-order predictor sixth-order corrector Adams-Moulton i n t e g r a t i ~ nwas ~ ~ utilized, with a step size of 3.23 X s. Conservation of total energy and total angular momentum, back integration, and step-size reduction were employed to check the accuracy of the integration. Initially R was set a t 15.0 bohr radii. The collision energy (Ti)was fixed a t 15 or 30 kcal mol-l. Both HC1 molecules were treated as rotating Morse oscillators. The initial rotational quantum number for the "primary" molecule (J1')was set to be either 0 or 8 or selected randomly from a Boltzmann distribution at 300
dI
$:
270'
Figure 1. Intermolecular potential for HCI-HCI in kcal mol-', for four different orientations. Dashed lines are the atomic sizes of chlorine and hydrogen atoms. Distances are in A.
or 750 K, while that for the secondary molecule (J2')was selected randomly from a Boltzmann distribution at 300 K. These initial conditions were employed since they correspond to those used in an experimental study performed in this l a b ~ r a t o r y The . ~ ~ primary rotor was in its ground or first excited vibrational state (ul = 0 or 1)while its collision partner was always taken to be in its ground vibrational state, v2 = 0. The vibrational phase (and hence internuclear separation) for each molecule was selected randomly, according to the prescription given by Porter, Raff, and Miller.40 The orientation angles and the rotational phase (8, 6, y of ref 28) were selected randomly from the appropriate distributions. The initial rotational angular momenta jli and j ; were taken to be j,i = [Jli(J1i + 1)]1/2h (la) jzi= [Jzi(Ji + 1)I1I2h Ob) Although the orbital angular momentum could also have
980
J. C. Polanyi and N. Sathyamurthy
The Journal of Physical Chemistty, Vol. 83, No. 8, 1979
TABLE I: Integral Inelastic Cross Sections in A ' for HC1(ul,J l i )t HCl(u, = 0, JZi= (300 K } ) Collisions at Two Different Collision Energies (Ti)and Various ( u , , J,' ) Combinationsa Jl
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17
U, = 1,J,' = 8 , Ti = 1 5 kcal mol-' 0.16( 0.16) 0.63(0.25) 1.04(0: 3 2 ) 2.91(0.57) 3.5 8( 0.62) 4.75(0.74) 5.37(0.80) 11.9761.18
U , = 1,J , ' = 8 , Ti = 30 kcal mol-'
U , = 0, J,' = 8 , Ti = 30 kcal mol-'
0.32( 0.18) 1.06(0.33) 1.92(0.45) 1.92(0.45) 3.30(0.58) 3.62( 0.61) 6.7 1(0.82) 10.22(0.99)
0.16( 0.12) 1.03(0.37) 2.34( 0.60) 2.53( 0.62) 2.93(0.62) 4.35(0.82) 5.28(0.88) 10.21(1.22)
8.87(1.02) 5.70(0.74) 3.93(0.60) 2.47(0.43) l.gO(0.40) 1.93(0.42) 1.01(0.30) 0.41(0.17) 0.35( 0.15)
9.15(0.94) 5.96(0.77) 3.5 1(0.60) 3.41( 0.59) 2.98(0.55) 2.87(0.55) 0.96(0.32) 0.53( 0.24) 1.70(0.42) 0.11(0.11)
12.03( 1.34) 4.47(0.77) 4.24(0.73) 4.04( 0.67) 1.62(0.39) 2.14( 0.47) 0.71( 0.25) O.gg(0.30) 0.75(0.26) 0.95(0.29)
30.41(1.87) 26.57(1.62) 56.98( 2.47) - 0.53( 0.04) 0.82(0.07) 0.29(0.08) - 1.55(0.18) 0.65(0.19)
29.07 (1.71) 31.18(1.77) 60.25(2.46) - 0.49( 0.04) 1.1O( 0.09) 0.61( 0.10) -2.24(0.25) 0.58( 0.19)
1000
1000
28.83(2.05) 31.94(2.01) 60.77(2.87) - 0.51( 0.05) 1.07(0.10) 0.56( 0.11 ) -2.10( 0.28) 0.45(0.22) 800
18 CJ
U + 0
(AE - ) ( A E') (A E)
(A E ")/(AE-) C, kca1-I mol no. of traj
U,
= 1,J,' = 0,
Ti = 30 kcal mol-' 17.11(1.16 10.96(1.05 7.44( 0.88) 10.37(1.01 8.24(0.91) 5.91(0.77) 5.09(0.68) 4.57(0.62) 3.56(0.52) 2.24(0.40) 1.66(0.33) 1.79(0.34) 1.13(0.24) 1.08( 0.24) 1.1O( 0.25) 0.98( 0.24) 0.40( 0.14) 0.13( 0.09) 83.76(2.73) 83.76(2.73) 1.28( 0.07 ) 1.28(0.07 ) 1.51(0.27) 1200
a Error estimates in parentheses are for the 68% confidence level. First moments are in kcal mol-' and C values in kcal-' mol.
been quantized, it was taken to be a continuous variable since it was obtained from the impact parameter b which was selected by the method of stratified sampling, with b,, = 11.0 bohr radii for all cases except for Jli= 0 for which b,, had to be 12.0 bohr radii. For each Jli,Jzi,and 3.0-bohr radii wide impact parameter stratum, 200-250 trajectories were computed. The exact numbers of trajectories are given along with the results reported in section 4. The final rotational states Jlf and Jzf for both molecules were computed from the corresponding final rotational energies and hence the final rotational angular momenta jlf and j2f,respectively, using the analogue of eq 1; the numbers were then rounded off to the nearest integer. Inelastic cross sections were computed from the distribution of Jlfand Jzf.
4. Results and Discussion 4.1, Selected JliStates. For the initial rotational state Jli= 8, inelastic cross sections were computed for u1 = 1, u2 = 0,JZi= (300 K) at Ti = 15 and 30 kcal mol-l. For TI = 30 kcal mol-l, the effect of changing u1 to 0 was exkept constant; the results are to amined, with u2 and JZi be found in Table I. Included in the table are the total deexcitation (6) and excitation (a+) cross sections, and the total inelastic cross section (u): u- =
c u(J1'cJ1'
J11
(24
Jf
= u-
+ u+
c u(J1' (a+ ) = c Jj> (AE-) =
J~
