J. Phys. Chem. 1988, 92, 925-931
925
Vibrational Coupling in Inelastic Collisions with Neutral and Ionic Atom-Molecule Systemst F. A. Gianturco,* Department of Chemistry, University of Rome, Cittri Universitaria. 00185 Rome, Italy
E. Semprini, F. Stefani, I.T.S.E., Area della Ricerca di Ron;a, CNR, C.P., 00016 Monterotondo, Rome, Italy
U. T. Lamanna, and G. Petrella Department of Chemistry, University of Bari, V. Amendola 173, 70126 Bari, Italy (Received: April 10, 1987)
The possibility of computing relaxation rate constants, vibrational and/or rotational, for gaseous molecular systems rests on our capability of evaluating the corresponding cross sections, as a function of relative energy, for the various inelastic processes. In the present paper we show how the latter quantities can be obtained with some level of confidence only after a way is found to evaluate the dynamical coupling between internal coordinates (molecular) and collision coordinate. Some examples of atom (ion)-diatom molecule interactions are chosen, and a characteristic coupling parameter is obtained for each of them in order to qualitatively understand the ensuing cross-section behavior yielded from quantum mechanical, coupled-channel calculations.
I. Introduction The collision dynamics of diatomic and polyatomic molecules in the gas phase is a major area of modern physical chemistry. One of the reasons for this wide interest comes from the variety of topics that can be studied under the above general heading. For example, internal energy transfers and relaxation processes in molecular gases can be fully understood only when the detailed dynamics of the inelastic encounters is worked out for all the relevant partners. Moreover, experimental data on molecular photodissociation and predissociative fragmentation of weakly bound complexes or the general interpretation of reactive phenomena which go through the formation of a loosely defined “transition state” all require knowledge of the structural features which are coupled together by dynamical effects and which ultimately control the final product distributions into the exit channels. This is clearly described in a recent review,’ which shows how infrared (IR)lasers have enabled a new class of experiments to be performed on the photodissociation of van der Waals (vdW) c o m p l e x e ~ ,where ~ . ~ the quantity of interest is the lifetime of the complex upon absorption and internal redistribution of the IR photon energy. The enormous subject of chemical reaction kinetics, where rate coefficients as a function of temperature are often the prime concern, continues to see new developments in the study of highly reactive species often involving free radicals or simple ion^.^^^ The study of kinetics with state-selected species, for example, is fundamental in improving existing chemical lasers and in developing new ones6 but often finds its progress hampered by the limited knowledge that one has on the interactions between partners at various relative distances. Finally, a perusal through the recent literature makes one realize fairly quickly that the whole field of molecular collision dynamics is continuously stimulated by the urgent need for data on molecular processes required from areas such as interstellar, atmospheric, plasma, and combustion chemistry.’-” On the theoretical side, a great deal of progress has been made in dealing with the quantum equations of motion and in finding efficient numerical algorithms which can be used to solve them,I2 as we will discuss in more detail in the following section. Any final comparison with experiments, however, is crucially dependent on the accuracy of the potential energy surface (PES) being used and, for polyatomic problems, it is first of all necessary to obtain an appropriate vibrational force field for the isolated molecule Dedicated to the memory of Prof. Massimo Simonetta.
0022-3654/88/2092-0925$01 .50/0
so that normal modes can be defined. For a small number of relatively simple polyatomics these force fields have been extracted from experimental data or from empirical ah initio calculations but are not generally available for most systems. Therefore, computational models have to rely on empirical procedures with a highly questionable level of ~onfidence.’~Even harder is the problem of determining reliable intermolecular potentials which, during the collision process, are effectively coupling internal motion with translational motion, be it done classically, semiclassically, or quantally. As systems become more complicated, the corresponding accuracy normally drops very rapidly and only ad hoc prescriptions can be used to provide some small region of the PES and its coupling with the molecular force field.14 In the present work we therefore try to show how a general prescription can be worked out, once the PES is reliably known, for the simple cases of atom (ion)-diatom collisions and that the qualitative behavior of the coupling with internal modes becomes a useful tool for interpreting the general features of the energytransfer scattering process. Moreover, the present formulation allows one to generate in a consistent way the orientation and distance-dependent coupling between force field and relative motion for cases where no a b initio calculations are available. Differences of behavior between weakly interacting neutral systems like Ar-HCl (or Ar-HF) will be analyzed in comparison with strongly interacting ionic partners like H+-HF and H+-C02.The general theory is described in section I1 while section I11 reports our computational findings. 11. Dynamical Models IZ.1. The Scattering Equations. In the usual treatment of the
collision problem in quantum mechanical terms,I2 one wishes to (1) Miller, R. E. J. Phys. Chem. 1986, 90, 3301. (2) Levy, D. M. Adu. Chem. Phys. 1981, 47, 323. (3) Janda, K. C. Adu. Chem. Phys. 1985, 60, 201. (4) Howard, M. J.; Smith, I. W. M. Prog. React. Kine?. 1983, 12, 55. (5) Bernsteih, R. B.; Zewail, A. H. J. Phys. Chem. 1986, 90, 3467. (6) Gross, d\ W. F.; Bott, J. F.,Eds. Handbook of Chemical Lasers; Wiley: New Yo\k, 1976. (7) Rice, S.A: J. Phys. Chem. 1986, 90, 3063. (8) Leone, S. R. Adu. Chem. Phys. 1982, 50, 255. (9) Bersohn, R. J. Phys. Chem. 1984, 88, 5145. (10) Simons, J. P. J. Phys. Chem. 1984,88, 1287. (11) Moseley, J. 3T.Adu. Chem. Phys. 1985, 60, 245. (12) For example! see: Gianturco, F. A. The Transfer of Molecular Energy by Collisions; Springer-Verlag: West Berlin, 1979. (13) Clary, D. C. J. A m . Chem. SOC.1984, 106, 970. (14) Clary, D. C. J. Chem. Pkys. 1985.83.4470.
0 1988 American Chemical Society
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The Journal of Physical Chemistry, Vol. 92, No. 4, 1988
evaluate a total wave function \k with fixed total angular momentum J , and one therefore starts with an ansatz given by an asymptotic, coupled channel expansion of the form
Gianturco et al. potential coupling part in the C matrix. For instance, if one writes the full PES a,s the usual multipolar expansion over cos 79, with 6 = arccos ( R - i ) ,then V(R,r) = Cv~(R,r)P,(COS79)
x
where the f are unknown translational functions that depend on the distance R describing the collision coordinate and (gi)is a suitably chosen set of internal basis functions of the isolated partners. They are often constructed from the rotational-vibrational states of the molecule or molecules participating in the collision. For the cas eof a structureless atom and a rotatingvibrating diatomic molecule the g l s are given by where the Xi(r) is one of the vibrational wave functions for the diatomic partner arid the 72 are angular functions describing the molecular rotor either in a space-fixed (SF) or body-fixed (BF) frame of reference, a choice which will change the meaning of the extra channel index a. The index j refers to the rotor state of the target. Substitution of expansion 1 into the Schrodinger equation yields a set of coupled equations for the radial coefficients (3)
which can be solved numerically in either the S F or the BF reference frames. Application of the usual boundary conditions to the numerical solutions of eq 3 yields the familiar S matrix from which the absolute squares of its matrix elements, Si/, give the probability for the transitions between the two channels li) li?. By repeating the computations for many values of J, the corresponding state selected integral cross sections U,-~,(E)can be obtained for the chosen collision energy E . Furthermore, by performing the calculations for an appropriate range of energies, we obtain the temperature-dependent rate coefficients
-
K(i-i‘;T) =
I-
d E uj+,(E) y ( E , T )
(4)
The y’s are suitable distribution functions for the initial velocities in the gas and are often given by simple Maxwell-Boltzmann distributions. As matrix manipulations are involved, the computed time depends roughly on Nm2 and several hundred channels can now be treated on the latest-generation supercomputer^'^ when one wants to reach very high numerical convergence in solving equations. The latter method is usually called the close coupling (CC) technique. The entire structure of partners and dynamical effects are stored in the matrix elements Cii, on the right-hand side of eq 3. For the case of atom-diatom (rovibrator) and within the SF representation, one can write more specifically the contributions to the various Cii,:
CL‘(R)= VJ(R)- K
+L~/RZ
(5)
where all the terms on the right-hand side of eq 5 are defined in several reviews (e.g. see ref 12). In particular, each matrix element
describes how inelastic transitions arise through the spatial range and strength of the interaction potential which couples different vibrational states and which also couples, through its anisotropic part, different components of the rotor included in the 7 coefficients. l 2 One clearly sees that the final flux distribution into the various accessible exit channels depends on the spatial extension of the (15) Schwenke, D. W.; Truhlar, D. G. Theor. Chim. Acta 1986.69, 175.
(7)
Hence, the corresponding coupling in eq 7 over angular functions more explicitly depends on the relative importance of each V, coefficient as each channel solution in eq 5 propagates to its j ’ rotational excitations asymptotic form, since the various j are controlled by the order of the specific P A polynomial that couples the J ’ S . ’ ~ At the same time, the vibrationally inelastic processes require knowledge of the PES behavior as a function of the internal coordinate since the latter feature controls the importance of the various vibrational final states through the performance of the r integration within the coupling in eq 6 . We will discuss this aspect more in detail below as it constitutes one of the main points of the present work. ZZ.2. The Sudden Approximations. Because of the rapid proliferation of the CC equations (5) that are coupled together by the C matrix, it is often necessary to introduce approximations which reduce the number of equations that need to be solved. The calculations reported in this article involve a set of such theories which use the sudden approximations. In the centrifugal sudden approximation (CSA)” the centrifugal terms of eq 6 are drastically reduced by setting all the diagonal terms of the L matrix equal to a fixed, arbitrary 1 value which can be chosen according to several prescriptions.12 The essentialy physical approximation implied by the CSA reduction is that, within the range of action of the PES considered, interference effects due to different turning point dephasings are being disregarded since essentially one, arbitrary, average turning point is chosen for all trajectories. For vibrational relaxation, the CSA has been s h ~ w n ’ ~to. ’ be ~ very accurate for H e + H2 and was extended successfully to He + C02.20 One further step to simplify the structure of the C matrix is provided by modifying the structure of the wavenumber matrix K and by disregarding the energy spacings either of the rotational levels only or of both the rotational and vibrational levels. Because of the usually large differences between those two types of spacings and because one usually requires cross sections for Eovalues which are not larger than a few electronvolts, to disregard vibrational spacings is seldom realistic while the choice of an arbitrary wavenumber value, kl, within the whole K matrix for the full rotational manifold corresponds to a more popular use of the energy sudden approximation (ESA)?l The main advantage of this procedure is that it becomes necessary only to solve coupled equations for fixed values of the polar angle 8;hence, expansion 1 for atom-diatom vibrational relaxation becomes
-
PJ = w;J(R;cS)xu(r)
(8)
U
which now constitutes the vibrational coupled-channel 10s approximation (VCC-IOSA), since the combined use of CSA and ESA provides the well-known 10s approximation.22 The main assumption of expansion 8 is that the vibrating molecule does not have time to rotate during the collisional interaction. This may be particularly appropriate for polyatomic targets with larger moments of inertia but must be used with great care for vibrational problems involving light, diatomic targets.’* In any event, even if the VCC-IOSA procedure can be considered physically realistic, the problem remains to evaluate correctly, or as correctly as possible, the coupling integrals appearing in the contribution (6)to the C matrix. In this last approximation they can be written as follows: (16) Lester, W. A., Jr. In Modern Theoretical Chemistry; Miller, W. M., Ed.; Plenum: New York, 1976; Vol. 1 . (17) McGuire, P.; Kouri, D. J. J . Chem. Phys. 1974, 60, 2488. (18) McGuire, P. Chem. Phys. 1974, 8, 231. (19) Flower, D. R.; Kirkpatrick, D. J. J . Phys. B 1982, 15, 1701 (20) Banks, A. J.; Clary, D.C . J . Chem. Phys., in press. (21) Pack, R. T. Chem. Phys. Lett. 1972, 14, 393. (22) Tsein, T. P.; Pack, R. T. Chem. Phys. Lett. 1970, 6, 54.
The Journal of Physical Chemistry, Vol. 92, No. 4, 1988 927
Atom-Diatom Inelastic Collisions [ V(R;S)l~,Uv, = 2pSdr
x*&)
V(R,r;cos 8 ) x,(4
(9)
Hence, the coupling between vibrational channels depends on the PES dependence on the internal coordinate r for fixed values of relative distance and orientation. How to obtain such a dependence will be the subject of the following subsection. 11.3. Tke Vibrational Coupling. The term “interaction potential” as used in this paper refers to the energy of the complete system computed as a function of nuclear coordinates within the usual Born-Oppenheimer (B-0) approximation. The molecular energy of the isolated target at each bond distance r has been subtracted out. In most model calculations of vibrational relaxation it is assumed that short-range forces in the interaction are dominating the energy-transfer process as partners need to get close to each other for an effective coupling to occur.23 Hence, a familiar form of coupling is written asz4 V(R,r,cos 19) = A(r,tl)) exp[a(R,G)(r - r,)]
(10)
where one assumes that, for weakly interacting systems, the slowly varying attractive part of the potential affects transition probabilities mostly through changing the relative velocity of approach. For cases where the multipolar coefficients of expansion 7 are known, one can also rewrite the above form as V,(r,R) = A,(R) exp[a,(R)(r - r,)l
(11)
where the parameters A , and a, need to be determined as functions of relative distance R . If one defines now another quantity, obviously related to the bond stretching, Le., the parameter [ = I(r - req)/r,], then an alternative way of writing eq 10 is V(R,r,S)= A(R,8)CCk(R,ts)tk
(12)
k
which, for low-lying vibrational states or for f ritten in a truncated form
i
'7
?
m
90 y (degrees)
135
180
Figure 8. VCC-IOSA partial cross sections at different collision energies and as functions of relative orientation for the Ar-HCl system.
certainly in keeping with what one expects from a qualitative analysis of the anisotropic PES. The matrix elements of eq 6 were also computed over rotovibrational wave functions of the HCl target expanded'on Morse oscillators and are shown in Figure 7 for the most efficient direction of approach. One clearly sees that Au = 1 selection rules appear to dominate since the V2, element is much smaller that
I
I
I
u)
2.0
30
E(eV1
Figure 9. Integral partial inelastic cross sections for the Ar-HCl system as a function of collision energy. The arrows connect initial and final vibrational states of the target.
the other two elements, and this is true up to the u = 8 vibrational state examined in the present calculations. When VCC-IOSA cross sections are computed, it is instructive to see how the various contributions from all orientations vary as t9 changes. Thus, Figure 8 shows such contributions at various collision energies. When E is low, as seen from the two bottom curves, the outer, strongly anisotropic part of j3 is dominant, and therefore orientational effects are rather marked. On the other hand, as the energy increases, one finds that log IS+, changes sign and the efficiency of the energy transfer is less sensitive to the details of the PES since the dynamical coupling prevails over the potential couplings of the Cmatrix in eq 5: one goes then from a potential-dominant (PD) situation to a more dynamical-dominant (DD) scattering situation, as frequently discussed for neutral systems.12 The integral relaxation cross sections, for the first three levels of HC1 are shown in Figure 9. They were computed by solving VCC-IOSA equations which included seven vibrational target states to reach numerical convergence of -2% for the cross sections. The usual rapid rise. from low collision energies is present in this case, and the dominance of one-step processes is also shown; relaxation probabilities of one vibrational quantum being exchanged during collisions are markedly larger than those where a "double jump" occurs during encounters. In conclusion, we have shown with a few examples of radically different systems that the generation of coupling parameters by eq 15, 16 and 18, 19 is a useful tool for analyzing qualitatively the expected collisional behavior of vibrational relaxation processes and a possible numerical aid for implementing VCC-IOSA calculations for a wide variety of simple systems. A recent extension to polyatomic targets3*also showed reasonable results, although the lack of RR interactions that could be used with confidence. there is the main drawback for a reliable treatment of complex molecules. (38) Gianturco, F. A.; Lamanna, U. T.; Petrella, G. Nuouo Cimento SOC. Ital. Fis., D 1983, ZD,731.
