The Journal of
Physical Chemistry
0 Copyright, 1983, by the American Chemical Society
VOLUME 87, NUMBER 12
JUNE 9, 1983
Very-Low-Energy Collision-Induced Rotational Relaxation: A Theoretical Analysis V. Sethuraman, C. Cerjan, and Stuart A. Rice' The Lbpaftment of Chemistv and The James Franck Institute, The Unlversity of Chicago, Chicago, Illinois 60637 (Receive& November 16, 1962)
Rotational relaxation induced by very-low-energy collisions is examined within a closed-coupled scattering formalism. Three-dimensional quantum mechanical calculations, which employ a sum of pairwise Morse interactions to model the potential surface, are reported. A vibrationally averaged potential is assumed to be a valid description of this system, so that rotational effects are thereby decoupled from vibrational processes. Total cross sections for elastic and inelastic processes are given as a function of increasing translational energy for three different Morse parameter choices. The trends seen in the calculations support the experimental observations for rotational relaxation in the Iz-rare gas systems studied by Tusa, Sulkes, and Rice.
I. Introduction It is now well established that very-low-energy collision-induced vibrational relaxation is, typically, very efficient.' For example, the study of collisions of He atoms with 12!3n&) by Tusa, Sulkes, and Rice,2 leads to the conclusion that the vibrational relaxation cross section is comparable to the collision cross section over the range of energy -0-3 cm-l, and very small for energies in excess of -3 cm-'. This conclusion, and its qualitative interpretation, has been fully confirmed by detailed closedcoupling calculations by Cerjan and Rice.3 Both the magnitude of the very-low-energy vibrational relaxation cross section and its rapid falloff with collision energy are consequences of scattering under the same potential that account for the bound state energies of the He-I, van der Waals molecule and the rate of predissociation of that van der Waals molecule. Much less is known about very-low-energy collision-induced rotational relaxation than is known about the corresponding vibrational relaxation. Tusa, Sulkes, and Rice have studied rotational relaxation of Iz(3n&) seeded in He, Ne, and Ar supersonic free jets? The experimental method (1) D.H.Levy, Adu. Chem. Phys., 47,(1981).
(2)J. Tusa,M. Sulkes,and S. A. Rice, Proc. Natl. Acad. Sci. U.S.A., 77,2367 (1980). (3) C. Cerjan and S. A. Rice, J. Chem. Phys., submitted for publication.
is based on the direct measurement of the rotational temperature of the seeded I2 as a function of position along the jet axis, and the assumption that the rate of rotational-translational equilibration satisfies a simple relaxation equation where 7-l = T U ~ ~ ( U , ~with ) , p the carrier gas density, (urel) the seeded molecule-carrier molecule relative velocity, and aaRZ the cross section for rotational relaxation of the seed molecule through collisions with the carrier gas. (In this paper we will denote the cross section u, in contrast to the usage of ref 2.) The adaptation of eq 1 to the conditions in the free supersonic jet is made via the transformation d/dt u dldx where u is the axial flow velocity and x the position along the jet axis. Tusa, Sulkes, and Rice find that uR2increases as the relative kinetic energy of collision decreases (Figure 1). However, they argue that this energy dependence of uR2is a feature of the ensemble of rotational levels involved in the jet experiment and that, since the rotational energy of the molecules decreases as the relative kinetic energy decreases, the increase of uR2 for small (Etrms)derives from the decrease in spacing of the rotational states at the corresponding lower (Erot).Indeed, they find that uRZ(Erot) shows little or no dependence on (E,,,,) within the experimental error (Figure 2). Thus, unlike the case of very low energy collision induced vi-
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completely described by the motion of the atom-diatomic molecule system on a single, nondegenerate, electronic potential energy surface, which is represented by a painvise sum of Morse functions. When the center-of-mass motion is removed, the Schroedinger equation for this atom-diatomic molecule system is
\
Flgure 1. Tusa-Sulkes-Rice experimental observation of the rotational retaxation cross section mro? as a function of increasing relative kinetic energy.
look
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Etrans(c6') Figure 2. Corrected rotational relaxation cross section, mR2( E,,,), as a function of relative kinetic energy (Ebans).
brational relaxation, there is no evidence for unusual enhancement of the rotational relaxation process at very low collision energies. In this paper we present an analysis of very-low-energy collision-induced rotational relaxation in the system He-12(311&). In order to establish a basis for this analysis, a type of "infinite order sudden" approximation*-' is employed. The model proposed replaces the vibrational couplings via the He-I, interaction potential with an averaged potential interaction. In this manner vibrational deexcitation is uncoupled from the rotational channels, since the scattered He atom is only influenced by an effective rigid-rotor potential. The other H e I , interactions can be adequately modeled by a pairwise sum of atomatom Morse functions. Similar calculations of rotational effects in the predissociation of He-1, indicate that this choice of model will be sufficiently accurate for our purposes.8 The purpose of the work we report is to test our understanding of very-low-energy collision-induced rotational relaxation by comparing the cross sections calculated within the theoretical framework described above with the Tusa, Sulkes, and Rice experimental observations. In section I1 we present the formalism defining our analysis and the computational procedures used; section I11 contains the results of these computations. In section IV we discuss the relationship between the results of the calculations described in this paper and general aspects of very-low-energy collision-induced relaxation.
11. General Formalism A. Theoretical Considerations. In the derivation of the equations relevant to collision-induced rotational relaxation, it will be assumed that the scattering event can be (4)D.Secrest, J. Chem. Phys., 62, 710 (1975). (5)L. W.Hunter, J. Chem. Phys., 62, 2855 (1975). (6)L. Eno and G. G . Balint-Kurti, J. Chem. Phys., 71,1447 (1979). (7)D. J. Kouri in "Atom-MoleculeCollision Theory",R. B. Bemstein, Ed., Plenum Press, New York, 1979. (8) (a) J. A. Beswick and J. Jortner, J. Chem. Phys., 69,512 (1978); (b) J. A. Beawick, G. Delgadc-Barrio, and J. Jortner, ibid., 70,3895(1979); (c) J. A. Beswick and G. Delgado-Barrio, ibid., 73, 3653 (1980).
where R is the vector displacement between the He atom and the center of mass of the I, molecule; r is the internal coordinate vector of the I,; R and r are their associated magnitudes; B is the angle between R and r; m is the reduced mass of the I2molecule and p is the He-12 reduced mass; j is the diatomic angular momentum operator and 1 is the H e I , angular momentum operator; VI&-)describes the isolated diatomic potential energy curve and VHe12(R,r,B) is the He-I, interaction potential. A convenient choice of frame for the closed-coupling scattering equations is the helicitf body-fixed frame. The rotationally coupled wave function in this coordinate frame is
where 3 labels the projection of the total angular momentum along the R-axis; n labels the radial quantum state of the I, molecule; R and i are the unit direction vectors for R and r, respectively; and the X/f(R,i) are the symmetric top eigenfunctions of total angular momentum J. The Xnj(r) are chosen to satisfy the isolated molecule Hamiltonian
In this coordinate frame, the action of the angular momentum operator l must be expressed in terms of the total angular momentum J and the diatomic angular momentum j, as l2 = (J - j),. The expectation values between different symmetric top eigenfunctions become [12]J& = (XJ,Mll21XJ,M)= h2[J(J + 1) + jo' + 1) - 2321 (5) [121J#,*, = (XJ?ll21XJ&d = -h2[J(J+ 1) - 3(3 f 1)]'/21i0' + 1) - 3(Q f l)]'/, (6) where all other possible 3 couplings vanish. Proceeding with the usual construction of the closed-coupled equations in this coordinate frame, we eliminated the r dependence in eq 2 by multiplying by xn,y and X y g on the left-hand side and then integrating over r, yielding the standard formlo
(7) (9)M. S.Child, *Molecular Collision Theory", Academic Press, New York, 1974,section 6.4. (10)P. Villareal, G.Delgado-Barrio, and P. Mareca, J . Chem. Phys., 76,4445 (1982).
The Journal of Physical Chemistry, Vol. 87, No. 72, 1983 2023
Collision-Induced Rotational Relaxation
where It is clear from eq 7 that the only off-diagonal vibrational coupling terms arise from the interaction described by eq 8. If we assume that the He atom only senses the vibrationally averaged potential of the I,, or equivalently, that the gradient of the interaction potential with respect to the magnitude of the internal coordinate r is small, the full interaction VHe12(R,r,e)may be replaced by its value at r
-- req
VH~I~(RJ&') VHe-12(R,req,e)
(9)
where rw is the equilibrium internuclear separation in the vibrating diatomic molecule. If the orthonormality of the Xnj(r) is used, the vibrational coupling term (8) vanishes identically, and the resulting closed-coupling equations become
(10) All possible rotational couplings are retained in these equations, but these couplings are restricted to one vibrational manifold. After eq 10 was integrated the standard asymptotic analysis of the wave function is performed in the space-fixed frame by applying the appropriate transformation matrix to the body-fixed frame wave function. The solution of the coupled equations is made more tractable by application of parity decoupling and the decoupling of even and odd rotational states, which are valid for the particular nondegenerate electronic state of I, studied. These simplifications substantially reduce the computational effort required to solve eq 10. Upon completion of the asymptotic analysis of the wave function, the resulting T-matrix elements provide the cross sections for the energetically allowed rotational transitions. The cross sections are given by the usual expressiong
where the T$)qr,niL are the T-matrix elements for different 0'1) transitions wthin a fixed vibrational state n, with total angular momentum J, and where K t = 2p(E, - Enj). B. Computational Procedures. Since an accurate ab initio potential energy surface for the He-1, system does not exist, it is necessary to use the extant experimental data to construct such a surface.l' A qualitatively correct choice for the form of the He-I, potential energy surface, valid for the low-energy collisions of interest to us, is a sum of pairwise Morse interactions VHe-Iz(R,r)= V H ~ - I ( ~-kI ZV) H ~ I ( ~ I ~ (12) ) where rlz and r13 are the vector displacements of the He atom from each of the iodine atoms. The Morse form for this atom-atom interaction is VHeI( IrIjl) = D [& % I r ~ j I - r e ) - 2e-U(lrijl-rd],=2,3 . (13) where D is the well depth parameter, 0is the characteristic inverse length, and re is the separation at the minimum (11) G. Herzberg, "Spectra of Diatomic Molecules", Van Nostrand, New York, 1966.
Figure 3. Energy dependence of the total cross section for two deexcltation processes In the n = 0 Vibrational state, u(O,Oc0,2) and u(0,0+0,4), using parameter set (a) of the text.
of the potential. Three different sets of parameters were chosen for the H e 1 interaction: (a) D = 7.0 cm-', 0 = 1.24 A-l, re = 4.0 A; (b) D = 18.5 cm-', 0 = 1.14 A-l, re = 4.0 A; (c) D = 7.5 cm-l, 0 = 1.20 A-l, re = 4.6 A. The first two parameter sets correspond to those values used in a previous vibrational predissociation calculation,&and the last parameter set is that used in a previous vibrational relaxation cal~ulation.~ The I, vibrational spectrum for this excited electronic state is reproduced by the Morse potential VI2(lrz31)= D[e-28(lrzsl-r*)- 2e-@(hI-re)] (14) with parameters D = 4.9119 X lo3cm-', 0 = 1.772A-l, and re = 3.0157 A. In order to evaluate the He-I, interaction potential matrix elements, we must obtain the radial wave functions for the isolated I, molecule, the Xnj(r) of eq 4. These wave functions were determined by expansion in terms of Hermite polynomials and checked by comparison of the calculated energies with the energies observed experimentally. The convergence of the expansion used was confirmed by varying the number of functions included until the results became insensitive to further variation. The matrix elements in (8)were then calculated, using this basis set, by Gaussian quadrature. No explicit expansion and truncation of the interaction potential was made, only that induced by the number of quadrature points taken. Once these matrix elements have been obtained, the differential eq 7 must be solved. The propagation of the solution was carried out by R-matrix techniques using a hybrid log derivative', and VIVS integrati~n.'~ This choice permits utilization of the computational advantages of both the wave-following (log derivative) and potential-following (VIVS) integrators. The parameters governing the use of these algorithms were varied to obtain maximal efficiency and stability. Likewise, the number of rotational channels was increased until numerical stability was achieved. The total angular momentun J was continually incremented until the resulting cross sections became insensitive to further increase for the lowest rotational state. 111. Results
Calculations of the rotationally inelastic cross sections and u(25,0+25j) were carried out in the manner outlined above. In addition, we have calculated the elastic component of the rotational relaxation cross sections, u(O,OcO,O) and a(25,0+25,0). Figures 3 and 4 display the variation of the two deexcitation processes u(O,O+Oj)
(12) (a) B. R. Johnson, J. Chem. Phys., 67,4086 (1979); (b) ibid., 69, 4678 (1978). (13) G. A. Parker, T. G. Schmalz, and J. C. Light, J. Chem. Phys., 73, 1757 (1980).
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Flgure 4. Energy dependence of the total cross section for two deexcitation processes in the n = 0 vlbratlonal state, u(0,0+0,2) and u(0,0+-0,4), using parameter set (b) of the text.
+ Tusa
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-Sulkes-Rice Data
u125,0+-25,2) u125,Ot-25,4)
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Flgure 5. Energy dependence of the total cross section for two rotational deexcitation processes in the n = 25 Vibrational state, u(25,0+25,2) and u(25,0+-25,4), using parameter set (c) of the text. The Tusa-Suikes-Rice uncorrected cross section is displayed for comparison.
a(O,OC0,2) and u ( 0 , 0 4 , 4 ) with respect to increasing translational energy of the He atom. For these calculations parameter sets (a) (Figure 3) and (b) (Figure 4) were used. Figure 5 displays the variation of 0(25,+25,2) and u(25,0+25,4) with respect to increasing translational energy of the He atom; for this calculation we used parameter set (c). Figures 6-8 present the three elastic cross sections and the associated excitation transitions for the parameter sets used. Convergence of these calculations was achieved by inclusion of all rotational channels with 0 I j 5 8 for the cross sections displayed in Figures 3 and 4, and 0 5 j 5 10 for the cross sections displayed in Figure 5. In all cases the sum over total angular momentum, J , in eq 11 was carried to J,, = 8. The general behavior of the inelastic cross sections as a function of energy is independent of choice of parameters over the range sampled. The relaxation cross sections are quite large for the translational energy range studied, with the larger well depth parameter values (Figures 4 and 5) yielding larger cross sections than those with D = 7.0 cm-' (Figure 3). A comparison of the n = 25 results with the n = 0 results implies that it is the depth of the well, and not the vibrational state studied, which affects the magnitude of the relaxation cross section. It should be emphasized that in the region 1-3 cm-' the inelastic cross sections do not vary rapidly with energy.
IV. Discussion The vibrationally averaged potential model for collision-induced rotational relaxation has been described
1
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Figure 7. Energy variation of the total rotationally elastic cross section, u(O,O+O,O), and the two excitation cross sections u(0,2+-0,0) and u(0,4-0,0) using parameter set (b) of the text.
2000,, ,
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Flgure 8. Energy variation of the total rotationally elastlc cross section, u(25,0+25,0). and the two excitation cross sections u(25,2+25,0) and u(25,4+25,0) using parameter set (c) of the text.
within the closed-coupling formalism. This approach is related in spirit to the analysis based on "nuclear diabatic"
Collislon-Induced Rotational Relaxation
states that has been used by Beswick and J ~ r t n e r ' ~in~ ' ~ the study of vibrational predissociation of He-I2 van der Waals, and to the rotational infinite order sudden approximation. Since these other methods have been shown to give qualitatively valid results for several problems we believe that our analysis can be used with confidence to interpret the data of Tusa, Sulkes and Rice. The method used by Tusa, Sulkes, and Rice to infer the collision-induced rotational relaxation cross section in very-low-energy He-12(311&) collisions has several deficiencies. The most important of these are the following: (i) only a thermally averaged cross section can be measured, (ii)the initial rotational states from which relaxation occurs vary; (iii) the cross section measured refers to the relaxation of Trotto T,,,,, and not to state-to-state processes. Accordingly, it is not possible to make a direct comparison of the state-to-state rotational relaxation cross sections we have calculated and the cross sections defined in the Tusa-Sulkes-Rice experiment. Nevertheless, some qualitative comparisons between theory and experiment are worthwhile. These comparisons are sharpened by considering, first, the elastic component of the rotational scattering (Figure 6). The calculations reveal a relatively strong variation of the cross section at very low energy. This behavior is consistent with the energy dependence of the vibrational relaxation cross section, which also has a peak at very low energy, and with the suggestion that resonances can play an important role in determinging the magnitude of the vibrational relaxation cross section. In contrast, the calculated inelastic rotational relaxation cross sections, particularly for parameter sets (b) and (c), show a gentle energy dependence, without evidence for striking enhancement at specific low energies; this behavior is similar to that inferred from the experiments of Tusa, Sulkes, and Rice. We also note that the magnitude of the calculated inelastic rotational relaxation cross section is (14)J. A. Beswick and J. Jortner, J. Chem. Phys., 68, 2277 (1978). (15)G. N. Robertson, J. Chem. SOC.,Faraday Trans. 2, 72, 1153 (1976).
The Journal of Physical Chemlstty, Vol. 87, No. 12, 1983 2025
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similar to that deduced from experiment, namely, 100 au2. Because the initial rotational state from which relaxation occurs varies with position along the jet axis, the energy dependence of the temperature relaxation cross section inferred by Tusa, Sulkes, and Rice from their experimenta is a complicated convolution of variation of state-to-state cross section, for fixed initial rotational state, with collision energy, variation of state-to-state cross section, for fixed collision energy, with initial rotational state, and thermal averaging over the distributions of collision energies, which are different at each point along the jet axis. It is plausible that the correction to the observed relaxation cross section suggested by Tusa, Sulkes, and Rice, namely, multiplicaovercompention of the observed cross section by (Erot), sates for the variation in unit energy transfer with initial quantum state. Certainly this will be the case if the total cross section, representing the sum of all state-to-state processes, increases as the initial value of the rotational quantum number increases. This argument leads us to expect the energy dependence of the state-to-state relaxation cross section to lie between the curves shown in Figures 1 and 2, probably closer to the curve in Figure 1. We show in Figure 3 some of the "uncorrected" cross sections of Figure 1. Error bars of f20% are shown on the data points. Clearly, the shape of the curve of temperature relaxation cross section as a function of energy is similar to that calculated for rotational relaxation in the n = 25 vibrational state of 12(3n&). (The experimental data refer to n = 28 in the same electronic state.) Given all of the uncertainties in the interpretation of the experimental data this similarity can only be viewed as consistent with other aspects of the interpretation of the experiments. Viewed overall, we believe the calculations presented in this paper further buttress the previously suggested interpretation of very-low-energy collision-induced relaxation processes.
Acknowledgment. This research has been supported by grants from the AFOSR and the NSF.
