Collision-induced vibrational energy transfer in small polyatomic

J. Phys. Chem. 1987, 91, 6 106-6 119

6106

at 385 nm provides strong evidence that the lowest energy excited states in these species may be formulated as [Ir1v(ppy)2(bpy-)]+ and [IrIv(bzq),(bpy-)]+. This formulation is consistent with prior cyclic voltammetric studiess which indicate that the first reduction wave in these species arises from addition of an electron to a n* orbital of bpy rather than a n orbital of ppy or bzq. Formation of a MLCT excited state, *Ir(ppy)3, arises from promotion of an electron from Ir(II1) to a n* orbital of ppy and yields a species which might be formulated either as IrIv(ppy-’i3), (delocalized excitation) or as Ir1V(ppy)2(ppy-)(localized excitation). Although the degree of coupling of A* orbitals of adjacent ppy ligands is expected to be influenced strongly by the M - C bonds in this species relative to the M-N bonding in complexes such as Ru(bpy),,+, the present data do not provide a basis for assessment of the degree of delocalization of excited electron density in *Ir(ppy),. The absorption of *Ir(ppy), is distinguishable from the absorption

associated with *Ir(ppy)2(bpy)f and with *Ir(bzq),(bpy)+, indicating that excited-state absorption techniques are capable of differentiating between charge transfer to a chelating ligand and charge transfer to an ortho-metalating ligand in these mixed ligand complexes. The breadth and structure of the bands in each of the three absorption regions suggest that more than a single electronic transition may give rise to each of these bands.

Acknowledgment. Funding of travel expenses for this research was provided by the NSF/DOE United States-Japn Program of Cooperation in Photoconversion and Photosynthesis. R.J.W. and K.A.K. were supported by the Office of Basic Energy Sciences, United States Department of Energy, Project DE-AT0378ER70277. K.I. and T.K. were partially supported by a Grant-in-Aid for Special Distinguished Research (56222005) from the Ministry of Education, Science, and Culture of Japan.

FEATURE ARTICLE Collision-Induced Vibrational Energy Transfer in Small Polyatomic Molecules Brian J. Orr School of Chemistry, University of New South Wales, Kensington, N.S.W . 2033, Australia

and Ian W. M. Smith* Department of Chemistry, University of Birmingham, Birmingham B15 2TT, England (Received: September 2, 1986; In Final Form: July I , 1987)

Recent theoretical and experimental progress has helped to clarify what determines the rates at which small molecules relax from vibrationally excited states via intermolecular vibration-rotation, translation (V-(R,T)), and intramolecular vibration-vibration (V-V) energy transfer. In this article, we examine how the influence of some of these important factors can be identified by comparing experimental data for similar molecular systems and can be understood on the basis of simple theoretical models. The role of molecular rotation in collision-inducedvibrational energy transfer is considered by comparing experimental results for V-(R,T) relaxation for non-hydride and hydride molecules. It appears that in hydrides rotation can quite strongly accelerate vibrational energy transfer not only by absorbing significant amounts of energy in rotation but also by increasing the effective impact velocity in collisions. This latter effect can be reproduced reasonably well by using theoretical models based on first-order perturbation theory. We also consider, in some detail, what effect the mixing of rovibrational states by Fermi resonance or Coriolis coupling has on energy-transfer rates, especially of intramolecular V-V processes. Collisional processes involving CO,, OCS, HCN, and H2C0 are considered. It is shown that, contrary to what has often been assumed, coupling between vibrational states will not invariably lead to efficient V-V transfer between them because of the possibility of quantum mechanical interferences associated with the necessary orthogonality of molecular eigenstates.

Introduction The transfer of vibrational energy in molecular collisions has been studied for over 50 years, and it is impossible to review the huge literature of the subject in a short article. Therefore, we have set ourselves more limited objectives: namely, to examine a few of the fundamental factors whose role in determining the pathways and rates of collisionally induced vibrational energy transfer has been clarified by recent experiments, some of which have been carried out in our own laboratories. It is useful to begin by relating the phenomenon of energy transfer to that of chemical reaction. The connections are several and close: inelastic collisions can redistribute energy into those 0022-3654/87/2091-6106$01.50/0

reagent motions that promote reaction; they also serve to equilibrate products which are initially formed in excited molecular states as chemical energy is released in an exothermic reaction. At a more fundamental level, a detailed theoretical understanding of reactive collisions is likely to be preceded by a similar understanding of nonreactive, inelastic collisions. The latter presents the theoretician with a more tractable problem, because the “natural” coordinates used to describe the system are obvious and remain the same throughout the collision. Experiments on collisional energy transfer and bimolecular reactions have also undergone a similar evolution. Until quite recently, quantitative information (relaxation times and rate constants) could only be obtained about systems that were at, or 0 1987 American Chemical Society

Feature Article close to, Boltzmann equilibrium. In the case of vibrational relaxation, ultrasonic measurements’ showed that the frequency dependence of the overall heat capacity of most systems is characterized by a single relaxation time, related to the rate at which molecules are transferred between the lowest excited vibrational level (or group of levels) and the vibrational ground state. No information is obtained about molecules in more highly excited levels, and frequently the range of gas mixtures that can be examined is limited. Increasingly in energy-transfer studies, as in experiments on chemical reactions, the emphasis has shifted to more detailed measurements. The ultimate goal is to find a matrix of rate coefficients for “state-to-state” processes. In many experiments, infrared lasers are used to prepare molecules in a specific rovibrational state and the subsequent evolution of excited-state populations is inferred from time-resolved measurements of infrared fluore~cence.~.~ Despite its many successes, this technique has important limitations, resulting from the relative insensitivity of spontaneous infrared emission as a diagnostic tool. Firstly, no information can be obtained about the kinetics of molecules in specific rotational levels: usually rotational equilibration is much faster than the transfer of population from the excited vibrational state, so kinetic data for vibrational energy transfer relate to a rotational distribution at the ambient temperature. Secondly, laser-induced vibrational fluorescence (LIVF) experiments can rarely determine the pathways for vibrational relaxation, that is, the states to which population is transferred as the system returns to thermal equilibrium. However, these states can sometimes be identified in double resonance experiments: where a second laser is used to measure the populations in several vibrational levels as a function of time after the initial perturbation. Developments in experimental technique have been paralleled by progress in theory, which can now treat inelastic collisions with relatively few approximations. In this article, we concentrate on the relaxation of small polyatomic molecules from levels in the low-energy portion of the vibronic level manifold where the density of vibrational levels is relatively sparse. Moreover, only self-relaxation and relaxation by rare gas atoms are considered explicitly. Chemical forces and electronically nonadiabatic effects5 are then absent, and quite accurate intramolecular and intermolecular potentials can sometimes be calculated. In addition, and of special importance in the context of this article, reasonable approximations to the true nature of the vibrational eigenfunctions can be computed either from accurate spectroscopic data or through ab initio calculations. This information is crucial when one examines theoretically the transfer of molecules between vibrational levels formally associated with different harmonic oscillator modes, what is commonly referred to as intramolecular vibration-vibration (V-V) energy transfer. One of our major purposes is to examine what influences the rates of such processes in molecular collisions. It has long been recognized that coupling of the normal vibrational modes of isolated polyatomic molecules, either by Fermi resonance or by Coriolis mixing, may enhance the rates of collisional transfer between the perturbed levels.6 Although this is generally true, there are subtle quantum mechanical effects which must be recognized if one is to obtain a reliable theoretical estimate of the V-V transfer rates. We begin by considering simplified theoretical descriptions of ( I ) Herzfeld, K.; Litovitz, T. A. Absorption and Dispersion of Ultrasonic Waves; Academic: New York, 1959. (2) Yardley, J. T. Introduction to Molecular Energy Transfer; Academic: New York, 1980. (3) Bailey, R. T.; Cruickshank, F. R. Spec. Period. Rep.: Gas Kinet. Energy Transfer 1978, 3, 109. Weitz, E.; Flynn, G. In Photoselective Chemistry, Part 2; Adv. Chem. Phys., Vol. 47; Jortner, J., Levine, R. D., Rice, S. A,, Eds.; Wiley: New York, 1981; p 185. Smith, I. W. M. In Lasers as Reactants and Probes in Chemistry; Jackson, W. M., Harvey, A. B., Eds.; Howard University Press: Washington, DC, 1985; p 373. (4) Steinfeld, J. I.; Houston, P. L. In Lasers and Coherence Spectroscopy; SteinTeld, J. I., Ed.; Plenum: New York, 1978; p I . (5) Smith, I. W. M . In Non-Equilibrium Vibrational Kinetics; Capitelli, M., Ed.; Springer-Verlag: West Berlin, 1986. (6) See, for example: Yardley, J. T.; Moore, C. B. J . Chem. Phys. 1967, 46, 4491.

The Journal of Physical Chemistry, Vol. 91, No. 24, 1987 6107 V-(R, T) energy transfer between vibration and the rotational and translational degrees of freedom, first in collisions between diatomic molecules and atoms and then between simple polyatomic molecules and atoms. This provides much of the framework for discussing collision-induced intramolecular V-V energy transfer. Although the factors which are discussed in this article will also influence the probabilities of intermolecular V-V energy exchange, we only consider such processes in the self-relaxation of COz.

Some Basics Low-energy collisions between diatomic molecules (BC) and rare gas atoms (A) are remarkably ineffective at inducing V-(R, T) energy transfer. To understand this, it is useful to begin by considering collinear collisions in which only V-T transfer is possible. The approach of A perturbs the vibration of BC because the intermolecular forces do not depend entirely on the distance X but are partially localized, repulsion between A and B being greater than that between A and C. The result is shown on the (idealized) potential energy surface portrayed in Figure 1: the path of minimum energy curves toward smaller rBc as X decreases. However, the curvature is very slight at low energies, indicating that the V-T coupling is weak and that calculations based on first-order perturbation theory should be valid. First-order theoretical treatments have been reviewed frequently, notably by Rapp and Kassal’ some years ago. Two main approaches have been adopted. In the semiclassical method, one considers how the oscillator states are perturbed by the timedependent potential, V’(t),caused by the approach and subsequent separation of A and BC. V’(t) is derived from a classical analysis of the relative translational motion. In the alternative distorted wave approach, the translational motion is treated by considering the waves associated with streams of particles moving under the influence of potential curves representing different vibronic states. In both cases, a necessary assumption is that the amplitude of vibration of BC is small compared with the range of the repulsion between A and B. In both the semiclassical and distorted wave treatments, the intermolecular potential is taken to be a function of X, the distance between A and the center of mass of BC, and x = rBc - re,BC,the displacement of the B-C internuclear distance from equilibrium. The potential is often assumed to take the form V(x,X) = Vo exp[-(X - y x ) / L ] (1) where y = mc/(mB mc) so that yx is the displacement of atom B from its equilibrium position. Assuming yx to be much smaller than L, solution of the classical equation for relative translation motion yields’

+

Vo exp(-X/L) =

y2~Gizsech2 (iiit/2L)

(2)

where M is the reduced mass of the system (A + BC) and iii is its initial relative translational velocity. This result is obtained by referring to the turning point of the motion, where all the initial is converted to potential energy. It is now kinetic energy, ‘/2/*Gt, feasible to implement time-dependent first-order perturbation theory, to find the probability of transfer between initial and final vibrational states li) and If)

(3)

Pif = h-2JUif(2 j/2M5i2$_:sech2 (Git/2L) exp(iwfit) dt

(4)

Uif has been written for the dimensionless vibrational matrix element

Yf =

(tr(x)lexP(rx/L)lti(x))

(7) Rapp, D.; Kassal, T. Chem. Rev. 1969, 69, 61

(4a)

6108 The Journal of Physical Chemistry, Vol. 91, No. 24, 1987

Orr and Smith

1

#

X

Figure 1. Potential energy surface for a collinear collision between A and BC, where X,as shown, is the distance from A to the center of mass of BC, rBc is the BC internuclear distance, and yrec is the distance of B from the center of mass of BC.

and hwn = AEfiis the difference in energy between initial and final states. Equation 3 rests on the assumption that the coefficients in the basis set expansion representing the vibrational states are at all times close to their initial values, Le., one for li) and zero for all other states. Moreover, no allowance has been made for the fact that a change in the internal state causes the final relative velocity, Ef, to differ markedly from its initial value, ai. Comparisons' with accurate quantum calculations suggest that the use of a mean velocity )/,(ai

+ a,)

= B,[l

+ (1 + 2hwfi/paiz)]

(5)

I

I

30

35

I

xlA

LO

I LS

Figure 2. Wave functions J ( X ) and f r ( X ) associated with relative translational motion of He + N2at fixed total energy on parallel potential energy curves. Both potentials are of the form V, exp(-X/L), with X as in Figure 1 and L = 0.2 A, and are plotted against an ordinate calibrated in units of N, vibrational quanta (2330 cm-I); the upper and lower potential curves are separated in energy by one such quantum. The one-dimensional relative velocity ai is set at 1.2 X los cm s-], which is the mean value corresponding to 1000 K. The uppermost curve shows the quantity exp(-X/L)fi(X) f&Y) as in eq 9.

Equation 9 was first solved by Jackson and Mott,* who found 1 [exP(ei) - ex~(-ei)l[exP(@f)- ex~(-ef)l lTif12= - (ei - e f ) 2 472 [exp(Oi) + exp(-Oi) - exp(8,) - exp(-ef)I2

(10)

Usually, for "downward transitions", Of >> Bi >> I , exp(-Of) and exp(-ei) can be ignored, and combining eq 8, 9, and IO yields an expression for Pif identical with the semiclassical result, eq 6. Before proceeding, it is instructive to examine predictions based on eq 6. The numerator in the last term, i.e., exp(Oi - Of), usually plays the dominant role in determing Pif,and the exponent is given by ej - e, = 2 7 p ~ ( -~a,)i / h

in eq 4 is satisfactory. The integration over time is then straightforward and yields

= - ( 2 ~ L / h ) ( 2 p E j ) ' / ~ ( (+ l AEfi/Ei)'12- 1)

where t9i = (2apLDi/h) and 8, = (2apLGf/h). In order to make a comparison with results obtained in thermally equilibrated systems, it is necessary to average over initial collisional conditions: at least, the relative translational energy, impact parameter, and orientation. Before considering how this is normally done, we give a brief outline of the distorted wave theory. In the distorted wave approach, one considers the full Schradinger equation and seeks solutions of the form

where Ei = 1/2pfii2is the initial collision energy. Large values of Pifare favored by large Ei, small A&, small L, and small p. In diatomics, fundamental transitions between neighboring states are strongly favored over overtone transitions, and this preference is enhanced by the vibrational matrix elements. The usual procedure for averaging over ai (or Ei) and impact parameter b parallels the method used in the simple collision theory of bimolecular reactions. That is, for 0 I b I d, Pifis independent of b for collisions with the same component of relative velocity directed along the line of centers of the colliding species. This averaging is represented by

(1 1)

( P i f )= S m 0 ? r d 2 P j Pexp(-piii2/2kT) i d B i / S0m r d 2 a jX As before, this implies that the amplitude of vibration must be small relative to the range of repulsive forces. Using the potential represented by eq 1 and the assumption of small transition probabilities, a first-order perturbation treatment yields an equation for the amplitude, A? of the reflected wave on the vibronic curve associated with state If) and Pif = IAfI2V'f/Gi= (16/h2GjGf)(Ui,(21Tif12

(8)

if the initial incident wave has unit amplitude. lUif12is again defined by eq 4a, while lTif12can be regarded as the square of a "translational matrix element" defined by

or as the almost completely destructive overlap off;(x) and&(x) on the potential Vo exp(-X/L) as represented in Figure 2.

exp(-pGi2/2kT) dL',

= (p/kT)LmPjPiexp(-pCi2/2kT) dei

(12)

When a steric factor, Po, is included to allow for orientational effects, substitution of eq 6 into eq 12 yields the usual first-order expression for the thermally averaged probability of V-T energy transfer between nondegenerate vibrational levels. (Sometimes a multiplicative factor exp(e/kT) is included to take account of attractive intermolecular forces accelerating the collision partners toward one another.) Schwartz, Slawsky, and H e r ~ f e l d ' (SSH) .~ (8) Jackson, J. M.; Mott, N. F. Proc. R . SOC.London,A 1932, 137, 703. (9) 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. 7 6 7 .

Feature Article recognized that the integrand in eq 12 peaks quite sharply around a “most effective” value of Bi at a particular temperature, in cases where lei - 8,4 >> 1. They performed an approximate integration around this point. Assuming that yx > k T and the first term in the exponent is dominant, this equation for a given suggests that In ( P l o )should vary linearly with T1i3 collision partner and with p 1 / 3for a variety of collision partners at the same temperature (assuming L to be the same for each pair of species). Although this expression has been of great value, eq 12 should be evaluated by numerical integration when 10, - 0,l is not much larger than one, as in many cases of intramolecular V-V energy transfer. The discussion to this point has ignored several important features of real molecular systems or at best treated them simplistically. These factors include the existence of molecular rotation, the three-dimensional character of molecular collisions, the effects of vibrational anharmonicity, and the orientational dependence of the intermolecular potential. For diatomic molecules, the main effect of anharmonicity is to reduce the spacing between successive pairs of vibrational levels, with the result that Pupl should increase more rapidly than for a harmonic oscillator, for which = V P ~ ,The ~ . steric factor Po is often taken to be ‘/3, the average of a (cosine)2 factor. Rotation of BC can affect the values of P,,in two ways: by absorbing some of the vibrational energy released in a collision, thereby decreasing (6, - Efl and increasing Pi,, or enhancing the velocity at which the impacting atoms collide. In the case of parallel vibrational modes in linear molecules, the “mode matching” is best and lU,,12largest, when the collision partner approaches BC collinearly. However, the motion of B due to rotation will then be perpendicular to the relative translational motion and therefore ineffective. First-order calculations which take account of molecular rotation, mode-matching effects, and different orientations at impact are discussed in the next section, but such calculations do not appear to have been carried out for inelastic collisions involving diatomic molecules. The modematching effects, but not those of molecular rotation, are allowed for in the more sophisticated VCC-10s (vibrational close coupling-rotational infinite-order sudden) calculations of Clary.loJ1 The first-order treatment that has been outlined in this section can be carried over to treat the problem of collision-induced vibrational energy transfer in polyatomic molecules.12J3 The vibrating molecule is considered to be a “breathing sphere”, that is, a structureless species whose size is modulated at the frequencies of the normal modes and by an amount which depends on the vibrational displacements of the “surface atoms” from their equilibrium positions. It is then possible to replace the operator ...) of exp(yx/L) in eq 4a by a dimensionless function V(Ql,Q2, the normal coordinates, Ql, Q2, ..., describing the surface atom displacements. Then, carrying forward the expressions that have been derived previously

The Journal of Physical Chemistry, Vol. 91, No. 24, 1987 6109 where

I = (4~L’pAEif/h~)’(p/kT) X Vi exp(Bi - 8,) exp(-&/2kT) eXP(€/kv [ 1 - exp(8, - tYf)lz

1

diii (15a)

and gf is the degeneracy of the final energy level. For the breathing sphere model to be correct, it would be necessary for the characteristic time scale for rotation (Trot) to be much shorter than those for vibration (Tvib) and relative translation ( T ~ ” ) .In practice, T , , ~is often much greater than Tvlb and T ~ and ~ the ~ assumption , of equal vibrational displacements irrespective of orientation is a poor one. Consequently, the breathing sphere model cannot be expected to predict absolute transition probabilities, although it can be useful-and has been extensively used-to rationalize ~ ~ , sphere trends. With hydrides where T , , ~ T ~ the~ breathing model is inadequate and a different first-order perturbation theory approach must be adopted (see below). As already indicated, the factor I decreases steeply with increasing Mif.Now, for polyatomic molecules, near-resonant (A& = 0) intramolecular V-V energy transfer may be possible, but at the simplest level such processes may involve considerable changes in the quantum numbers of states li) and If) and consequently a small value of IUifI2. The preferred pathway for collision-induced transitions from vibrational states above the lowest excited level usually involves a balance between these two factors. The separation of terms depending on the vibrational and translational coordinates implies weak coupling between these motions and is a common feature of any approximate treatment. In processes involving polyatomic molecules, the existence of vibrational anharmonicity and molecular rotation can lead to effects additional to those already mentioned for diatomic molecules. Now, the harmonic oscillator, normal-mode states only serve as a basis for the true molecular eigenstates. The resultant mixings are especially significant in the presence of strong anharmonic or Coriolis perturbations. It is then necessary to evaluate the square of the vibrational matrix element, Le., IUifI2,with particular care. When states of the same vibrational symmetry would, in the lowest approximation, lie close in energy, the anharmonic mixing can become very strong-we speak of Fermi resonance-and the vibrational quantum numbers conventionally attached to the molecular levels are only a crude guide to the true nature of the state. Reasonable approximations to the true eigenfunctions \k(Ql,Q2,...) can then be formed by taking linear combinations of products of harmonic oscillator basis functions +l(Ql), + 2 ( Q 2 ) , .... The coefficients of terms in the expansion indicate the harmonic “purity” of the true eigenstate. Coriolis coupling introduces the additional feature that rotational, as well as vibrational, basis sets need to be mixed to describe the molecular eigenfunctions. As will be discussed later, this raises the possibility that there might be rotationally specific channels which permit particularly efficient collision-induced intramolecular V-V transfer. The effect of anharmonic and Coriolis mixing on intramolecular V-V energy transfer is a major theme of this article. That such mixings may enhance the rates of such processes is widely recognized but not always quantified correctly. Specific examples are examined later. For the moment, the general point can be illustrated by considering a transition between two eigenstates 11) and 111) which can be expressed as simple mixtures of the same two harmonic oscillator states IA) and IB), that is

-

91= CaIA) *II

Cb(B)

(16a)

= CbIA) - CaIB)

(16b)

f

With such wave functions, the vibrational matrix element is (10) Clary, D. C.; Drolshagen, G . Chem. Phys. Lett. 1981,82,21. Clary,

D.C. J . Chem. Phys. 1981, 75, 2899. ( 1 I ) Clary, D. C. Chem. Phys. Lett. 1980,74,454. Clary, D. C. J . Chem. Phys. 1981, 75, 209. Clary, D. C. Mol. Phys. 1981, 43, 469. Clary, D. C. Chem. Phys. 1982, 65?247. (12) Tanczos, F. J . Chem. Phys. 1956, 25, 439. (13) Stretton, J. L. Trans. Faraday SOC.1965, 61, 1053.

whereas the corresponding matrix element for a transition between eigenstates which are free from any anharmonic or Coriolis perturbations is

6110 The Journal of Physical Chemistry, Vol. 91, No. 24, 1987

Vi, = (A’IVB’)

(17a)

The latter matrix element influences the rate of intramolecular V-V transfer in the unperturbed situation. In the presence of strong anharmonic or Coriolis mixing, however, this term is reduced in importance as cb2- Ca2in eq 17 approaches zero and the contribution of ( ( A I V A ) - (BIUB)) becomes more pronounced. The negative sign which separates the above two terms diagonal in IA) and IB) is a direct consequence of orthogonality of the eigenfunctions for 11) and 111). This quantum mechanical interference means that the two diagonal terms cancel, at least partially, and prevent Vi, from attaining the magnitude which might be anticipated on intuitive grounds. Pursuing this matter, we return to the breathing sphere model for a polyatomic molecule and specify the potential V(Q,,Q2,...) in its c ~ s t o m a r yform ~~~~ V(Q11Q29...) = ~ x P [ ~ ( ~+I aQ2 QI2

+ ...)I

(18)

where Qj (with j = 1, 2, ...) are dimensionless, mass-weighted normal coordinates defined such that, for the eigenstate Iuj) of a harmonic oscillator, (ujlQ;luj) equals uj + Equation 18 describes a multidimensional exponential repulsive potential between surface atoms of the polyatomic molecule and its collision partner, with 2ajQj taking the place of yx/L in the corresponding one-dimensional case. The coefficients aj relate the normal coordinates to displacements, Ar, of the surface atoms from their equilibrium positions (e) via the equation

= (a h r / a ~ ~ ) ~ / 2 ~

(18a)

On expanding the potential in eq 18, one obtains V(Q1,QZ,...) = 1

+ 2CajQj + 2COljakQjQk

...

(18b)

It follows15that only quadratic and higher order terms in Qj make a nonzero contribution to ( ( A l V A ) - (BIVB)] in eq 17, since (AIlIA) = (BIlIB) = 1 and (AIQjlA) = (BIQjlB) = 0 for a harmonic oscillator. It is therefore not appropriate to assume (as in the derivation of eq 14) that Vis linear in the displacement coordinate. Rapp and SharpI6 have given expressions for the matrix elements U,, based on harmonic oscillator wave functions and an operator V(Ql,Q2,...) of the form given in eq 18, so that there is no need to rely on an expansion of the exponential as in eq 18b. Nevertheless, such an expansion is useful in assessing the possible extent of quantum mechanical interferences in eq 17. To be more specific, we consider the relatively simple situation where the harmonic vibrational states IA) or IB) are expressed respectively as lua) or Iub), with ua or u b quanta in normal modes a or b and zero quanta elsewhere. (AITIA) and (BIVB) can then be evaluated by considering only the quadratic contributions to V(QI,Q2,...) in eq 18b, yielding I(AlVA) - (BIVB)J = {(Val V Q I , Q ~ , . . . ) I U-~ )

I

VQi, Q z * . , . ) I ~ ~ )

= 2{(Ya2ua- a b 2 u b )

(19)

where the extent of cancellation will depend on the relative magnitudes of aa2and qb2, which are in turn determined by the vibrational force field of the polyatomic molecule. A commonly encountered instance of the above situation is when ua = 1 and u b = 2. Then the tables of Rapp and SharpI6 yield

This result will be used later, in the context of the molecules CO, and OCS. It should in the meantime be noted that coefficients (14) The approximation that the same value of L characterizes the interaction between each surface atom and the collision partner is always made in simple theoretical models. (15) Laubereau, A.; Fischer, S. F.; Spanner, K.; Kaiser, W. Chem. Phys. 1978, 31, 335. Fendt, A.; Fischer, S. F.; Kaiser, W. Chem. Phys. 1981, 57, 55.

(16) Rapp, D.; Sharp, T. E. J . Chem. Phys. 1963, 38, 2641.

Orr and Smith aj are typically less than 0.1, so that the result of eq 19a can be approximated by 2(aa2- 2ab2), corresponding to the prediction of eq 19, on the basis of quadratic contributions only. The conclusion that the matrix element defined by eq 17 is not necessarily large is independent of the form assumed for the intermolecular potential. The dependence of any potential on the coordinates describing the displacements of the atoms in a polyatomic molecule can be expressed in terms of a Taylor expansion: V(Qi.Qz,..S) = 1 + C(aV/dQj)eQj + ‘/2C(a2V/aQj aQk)eQjQk + ... (20) j

ik

Once again, substitution into {(AlVA) - (BIVB)}will only yield a nonzero result through differences in the second- and higherorder terms. Due to cancellation of individual terms, Vi, is likely to be much smaller than might initially have been imagined, given strong mixing of the basis states contributing to 11) and (11).

V-(T, R ) Energy Transfer in Small Polyatomic Molecules Comparisons of relaxation data for pairs of related hydride and deuteride molecules gave the first indication that rotation might promote the transfer of vibrational energy in molecular collisions. In several such pairs, the hydride relaxes more rapidly despite the fact that vmin, the frequency of the lowest vibrational mode, is greater than in the corresponding deuteride. Correlations” of log ( P i * )with vminl* were found to differ depending on the numbers of H, and to a lesser extent D, atoms in the molecules. MooreI9 sought to explain the differences in these correlations for different classes of molecules in terms of rotational effects. H e pointed out that for many hydrides the average velocity of the peripheral hydrogen atoms, arising from overall molecular rotation, exceeds the average velocity associated with relative translation of the collision partners. Accordingly, he argued that the “rotational velocity”, rather than the translational velocity, should be used for such species in evaluating the SSH expression for ( P l o ) . Collisions were assumed to occur between a vibrating molecule and a collision partner rotating in the plane containing both species treated as pseudodiatomics. Although quite successful, Moore’s treatment onry included rotational effects: no allowance was made for simultaneous translational motion. Moreover, the rotational motion of the vibrating molecule was not included, so the method does not predict any effects of rotation in collisions between vibrating molecules and rare gas atoms. As we shall see, if this is included, it is necessary to account for mode matching: that is, the relationship between the motion of peripheral atoms as a result of vibration and the motion which brings the same atoms into contact with the collision partner and which results from the combined effects of molecular rotation and translation. In this section, we review some recent results on the vibrational relaxation of simple, linear, polyatomic molecules to see what can be inferred about the participation of molecular rotation in such processes. In addition, a summary is given of a relatively new theoretical model which includes molecular rotation.20 As was done in the second section, we begin by considering relaxation data for some diatomic molecules and their interpretation. The vibrational relaxation of C O has been studied over a wide range of both temperature and collision ~ a r t n e r . , I - ~ ~The probabilities of self-relaxation ( T > 1000 K) and of relaxation of C O by He ( T > 300 K) show the linear variation of In (Pia) (17) Lambert, J. D.;Salter, R. Proc. R. Soc. London, A 1959, 253, 277. (18) The SSH equation, eq 14, predicts a linear dependence on ( Y , , , , , ) ~ ! ~ in the case where the first term in the exponent is dominant and L and p are assumed to remain constant. (19) Moore, C. B. J . Chem. Phys. 1965, 43, 2979. (20) Miklavc, A.; Fischer, S. Chem. Phys. Lett. 1976, 44, 209. Miklavc, A,; Fischer, S.J . Chem. Phys. 1978, 69, 281. Miklavc, A. J . Chem. Phys. 1980, 72, 3805. Miklavc, A. Mol. Phys. 1980, 39, 855. Miklavc, A. J . Chem. Phys. 1983, 78, 4502. (21) Millikan, R. C.; White, D. R. J . Chem. Phys. 1963, 39, 3209. (22) Miller, D. J.; Millikan, R. J . Chem. Phys. 1970, 53, 3384. (23) Green, W. H.; Hancock, J. K. J . Chem. Phys. 1973, 53, 4326. Andrews, A. J.; Simpson, C. J. S. M. Chem. Phys. Lett. 1976,41, 565. Simpson, C. J. S. M.; Andrews, A. J.; Price, T. J. Chem. Phys. Lett. 1976, 42, 437.

The Journal of Physical Chemistry, Vol. 91, No. 24, 1987 6111

Feature Article with T1i3which is predicted by SSH theory. With collision partners heavier than He that are unable to undergo intermolecular V-V energy exchange, relaxation is too slow to measure at 300 K. Figure 3 shows a plot of log ( P l o )vs. p1I3for M = H2, He, CO, and Ar at 1000 K. All four data points fall quite close to a single line, suggesting that molecular rotation for M = H, or C O does not strongly enhance the rate of vibrational relaxation-at least at this temperature. The finding2)that D2 and He relax CO(v=l) by V-(R, T) energy transfer at virtually the same rate is consistent with this conclusion, although it has been shown23that p-H2 does relax CO(v=l) at twice the rate of normal H2. Figure 3 also shows data for the relaxation of HCl at 1000 K (obtained by a short extrapolation from Seery’s data24) and for HC1 and DC1 at 298 K.25 The results are qualitatively different from those for CO. Three features of these correlations are worthy of comment: (i) although AElois greater for HC1 than for CO, ( P l o )is larger at the same temperature; (ii) the variation of ( Plo) with k1i3is much less steep for HC1 than for CO; and (iii) the points for relaxation by H 2 and self-relaxation are no longer consistent with the correlation of data for relaxation by the rare gases. More than one factor may be responsible for differences between the relaxation in C O M and in HC1, DCl M collisions, but there seems little doubt that molecular rotation must make a contribution. At first sight this may seem surprising, since the motions of the H, D atom in HC1, DC1 produced by rotation and vibration are perpendicular to one another. In collinear collisions, it is difficult to see how the rotation could influence the probability of a vibrational transition. However, in collisions that are not collinear, rotation can cause a more impulsive impact between H, D and M and may thereby enhance the probability of energy transfer. There are major, qualitative differences between V-(R, T) relaxation in polyatomic molecules and in diatomics. Although the overall rate of relaxation is again usually determined by the rate of transfer between the ground vibrational state and the lowest excited level or group of levels, the excited levels involved are now generally at lower energy and are associated with low-frequency bending or torsional modes which may be strongly coupled to molecular rotation. For example, in the bending mode of a linear triatomic molecule, the terminal atoms move in a direction perpendicular to the molecular axis, as they do when the molecule rotates, and both types of motion are more likely to interact with relative translation during sideways approach of a colliding atom rather than in collinear collisions. The low frequencies of bending and torsional vibrations mean that V-(R, T) deactivation of the lowest excited states of polyatomic molecules can rarely be studied in direct LIVF experiments. There are no convenient pulsed laser sources with wavelengths greater than about 11 Mm, the Einstein A coefficients for spontaneous emission are small, and detectors are insensitive at these long wavelengths. Various indirect LIVF measurements have been made, and photoacoustic methods, using either modulated or pulsed sources, have recently yielded reliable rate data. Figure 4 compares data obtained for the relaxation of C02,26 N20,27and OCS28929for which the v2 bending mode frequencies are 667, 589, and 520 cm-I, respectively. The results with rare gases as collision partners are qualitatively consistent with predictions from SSH theory: log (Plo)falls approximately linearly with Moreover, the gradients of these lines decrease and the absolute values of ( P l o )increase as AElo falls in the sequence COz, N,O, OCS.

+

-31 -

-5

-

-6

-

-7

I

10

I

I

I

IS

20

25

(a)

m

Ip I amu I“3

- L3

+

(24) Seery, D. J. J . Chem. Phys. 1973, 58, 1796. (25) Chen, H.-L.; Moore, C. B. J . Chem. Phys. 1971, 54, 4072. Steele, Jr., R. V.; Moore, C. B. J . Chem. Phys. 1974, 60, 2794. Zittel, P. F.; Moore, C. B. J . Chem. Phys. 1973, 58, 2922. (26) Lepoutre, F.; Louis, G.; Taine, J. J . Chem. Phys. 1979, 70, 2225. Lepoutre, F.; Louis, G.; Manceau, H. Chem. Phys. Lett. 1977, 48, 509. (27) Rebelo Da Silva, M.; De Vasconcelos, M. H. Physica 1981, IOdC, 142. (28) Mandich, M. L.; Flynn, G. W. J . Chem. Phys. 1980, 73, 3679. (29) Smith, N. J. G.; Davis, C. C.; Smith, I. W. M. J . Chem. Phys. 1980, 80, 6122.

-4

HCI

(b)

L

- 5 1

-6t

I O

HCI 0

DCI :HZ

r

I

I

I

15

20

25

(ptamu~h

Figure 3. Thermally averaged probabilities of vibrational relaxation (a) for CO (0)and HCl(0) at 1000 K and (b) for HCI(0) and DCl(0) at 300 K, each with rare gas collision partners as indicated. The solid data points are for self-relaxation.

-3

-

-4 -

-5 r

-6

-1

I 15

20

25

30

(p/amu)’l3

Figure 4. Thermally averaged probabilities for vibrational relaxation at 300 K of COz (0),N 2 0 ( O ) , OCS (A), and HCN (V),each with rare gas collision partners. The solid data points are for self-relaxation.

For CO,, Lepoutre et a1.26have obtained data on relaxation at temperatures between 153 and 393 K. At low temperatures, (Plo)falls less steeply with k’I3 than would be predicted by the simple breathing sphere mode1.I2J3 The fit to the observed temperature dependence could be improved by reducing AElo due to a supposed change in the rotational quantum number.26 Much more detailed calculations on collisions between C 0 2 (01’0) and rare gases have been carried out by Clary.” Using

6112 The Journal of Physical Chemistry, Vol. 91, No. 24, 1987

+

Energy/ lo3 cm-’ 121 -

300 111 -

090 080 210 070 021 -

200 -

130 060 120 050 110 OLO -

I

030 -

100 -

020 010 -

(CN stretch)

Orr and Smith

IHCN bend)

I C H stretch)

Figure 5. Partial energy level diagram relevant to collision-induced vibrational transfer in HCN gas. The vibrational eigenstates are labeled in the conventional notation uluzuj(with vibrational angular momentum quantum number I omitted for clarity), with the three corresponding normal modes indicated under each stack of fundamental levels. Straight arrows indicate spectroscopic transitions accessible to a tunable optical parametric oscillator, and wavy arrows indicate corresponding LIVF emission.

the infinite-order sudden (10s) approximation, the orientation of the species is assumed to remain fixed during a collision (Le., no molecular rotation), but calculations of ( P l o ) are performed for a range of orientations and then the results are appropriately averaged. The vibrational states are accurately treated by a vibrational close coupling (VCC) technique. When accurate intramolecular and intermolecular potentials are used, these VCC-10s calculations give excellent agreement with observed data for COZ(Ol10) H e collisions at 300 K and above. Below 300 K, the calculated values of ( P l 0 )fall rather too steeply, and Clary attributes this to the lack of attractive terms in the intermolecular potential. With M = Ne, the agreement is again good at 300 K but there is a similar divergence of experiment and theory at lower temperatures. It should be emphasized that these calculations are performed without adjustable parameters and comparisons are made between absolute values of ( P l o ) . One conclusion to be drawn from the correlations displayed in Figure 4 and from Clary’s calculations is that molecular rotation probably plays no more than a minor role in V-(R, T) relaxation of non-hydride triatomics by noble gas atoms-at least at 300 K and above. However, the values of ( P l o )for self-relaxation of COz, N20, and OCS all lie appreciably above the lines correlating the data for relaxation by noble gases. It seems that the molecular rotation of the collision partner may be responsible for this enhancement. The only triatomic hydride molecule for which extensive relaxation data are available is HCN3&33for which an energy level diagram is shown in Figure 5 . Deactivation of the bending mode (v2 = 7 12 cm-l) has been studied32by exciting molecules to the (01’1) state and measuring their rate of deactivation to (OOOl). As the points in Figure 4 show, the measured values of ( P l 0 ) are 1-2 orders of magnitude greater than would be anticipated by extrapolating the data on OCS, N20,and C 0 2 to the larger value of AE in HCN. Moreover, the decrease with k1I3is less than might be expected. This comparison is reminiscent of that made between

+

(30) Arnold, G. S.; Smith, I. W. M. J . Chem. Soc., Faraday Trans. 2 1981, 77, 861. ( 3 1 ) Hastings, P. W.; Osborn, M. K.; Sadowski, C. M ; Smith, I . W. M. J . Chem. Phys. 1983, 78, 3893. (32) Cannon, B. D.; Smith, I. W. M. Chem. Phys. 1984, 83, 429. (33) Cannon, B. D.; Francisco, J. S.; Smith, I . W. M. Chem Phys 1984, 89, 141.

C O M and HC1+ M relaxation and suggests that once again rotation of the vibrating hydride molecule may enhance the probability of energy transfer. To interpret their data on the deactivation of the u2 mode in HCN, Cannon and Smith32adopted a theoretical modelZowhich incorporates the effect of molecular rotation into a first-order perturbation treatment of vibrational energy transfer. H C N was assumed to consist of three “embedded spheres”, and its rotation, as well as the translational motion, was assumed to contribute to the effective velocity at which the H and M atoms approach. The magnitude and direction of the relative velocity and rotational momentum were chosen by Monte Carlo methods, and for each choice, a transition probability was calculated-subject to the constraints of energy and angular momentum conservation. Calculations on a large sample gave a thermally averaged collisional probability. For HCN, the calculations gave thermally averaged probabilities approximately 5 times larger than the experimental values for all the rare gases and the variation of (PI,,) with k1/3was reproduced rather well. The same theoretical method has also been applied successfullyZoto the V-(R, T) relaxation of methyl halides.34 The position with regard to V-(R, T) relaxation of C2H2and C2D2is less satisfactory. Experiments on C2H2have used an indirect LIVF technique,j5 whereas a pulsed photoacoustic method has been used on mixtures containing C2D2.36 Both sets of measurements indicate little dependence of the relaxation rate on rare gas collision partner, suggesting that once again molecular rotation is important, at least in collisions involving the heavier rare gases. However, the values of (Plo)for C2H2are approximately an order of magnitude less than those for C2D2,whereas theoretical c a l c ~ l a t i o n predict s ~ ~ similar values. This discrepancy merits further investigation. This section has summarized some of the relaxation data that have been obtained recently for simple polyatomic molecules. The overall rate of V-(R, T) relaxation is determined very largely by relaxation from the lowest excited vibrational level or group of levels. We have emphasized that relatively simple theoretical calculations indicate that rotational effects are important in the relaxation of hydrides and deuterides. However, two important approximations are made in these calculations. First, the amplitude of hydrogen or deuterium atoms in low-frequency modes is usually rather large, violating the assumption that yx > 100 (assumed)

>>0.01 45 50.008 51 0.0055 52

IUir(l)12= [+1.14

X

10-2]2 = 1.3

X

The above results lead to three conclusions which, despite the crudeness of the breathing sphere model, we believe to be general. First, quantum mechanical interferences can reduce the overall matrix element for collision-induced transfer between Fermi53 coupled states to well below the values of individual terms in the expansion based on harmonic oscillator wave functions. Second, 120 f 50 (373 K) 0.005 50 E; IR-IR double the cancellations may be so extensive that it will be difficult to resonanceP calculate the matrix element accurately without very accurate 0.0004 11 T; S C F + 8.7 information about the intramolecular and intermolecular potenVCC-10s theory tials. (Note that, in the calculations above, U,, changed from 0.0003 52 E; IR-IR double -6 resonance -0.0113 to +0.0114 on including extra terms in the expansion.) Finally, it may be that the major contributions to the overall matrix 0.0027 46 T; S S H theory 25 element come not from the diagonal terms but from those terms 0.044 50 450 f 150 (373 K) E; IR-IR double which connect harmonic states differing only by one in one resonancee quantum number. 0.0007 52 E; IR-IR double -6 A useful comparison can be made with the comprehensive set resonance of calculations performed by Clary" for V-V transfer in C02-He 54 E; CO, laser gain 330 f 110 (400 K) and C02-Ne collisions. His calculations confirm that k , is not exceptionally large, relative to V-V processes which cannot be "Rates are for the forward reaction in eq 22 a t a temperature of 300 K, unless otherwise indicated. b E = experiment; T = theory. enhanced by Fermi resonance effects. In a study aimed at demInterconversion of V-V transfer rate constants k , , collisional efficienonstrating the role of Fermi resonance in V-V transfer, calculations cies (PI)), and collision numbers Z (=(P(]))-') is made in terms of the of cross sections for V-V transfer (according to eq 22 with M = following gas kinetic rates (in units of ms-' Torr-') for COz-M hardHe and a total collision energy of 0.66 eV) were performed with sphere collisions a t 300 K: 8.7 X lo3 for M = C 0 2 ; 20.1 X lo3 for M both harmonic and anharmonic intramolecular potentials. Much = He; 9.1 X lo3 for M = N,. d A very rapid decrease in observed more efficient V-V transfer was found when molecular anharsignal was attributed, probably erroneously, to the rate process of inmonicity (and hence Fermi resonance) was taken into account, terest. 'Calculated from the reported reverse rate k-', using detailed but nevertheless cross sections associated with Fermi-coupled states balance constraints. /Measured in a CO, laser gas discharge (He/ were not especially high. Indeed, the inclusion of anharmonicity CO2/N, = 12:4:1) at 270 Torr and 400 K. g k l is not necessarily fast. was found also to enhance dramatically cross sections for intrawith physically correct modeling. There is e v i d e n ~ e "that ~ ~ ~ ~ molecular ~ ~ ~ ~ ~V-V transfer out of the (OOol) level," despite the fact that it is not strongly affected by Fermi resonance. some estimates of k l were too large because of preconceived notions Similar considerations can be applied to other V-V transfer that energy transfer between Fermi-coupled vibrational levels channels in C 0 2 , such as the following processes (all of which are should necessarily be fast, leading to the adoption of kinetic shown with numerical labels 2-6 in Figure 6):55 modeling schemes in which k , was constrained to be one of the largest rate constants. It is therefore necessary to consider why C02(1000,0200)1 M -.% C02(0220) + M + 53 cm-I (24) such a preconception might be incorrect. A common feature of many simple theories is that the probability of collision-induced transfer, from an initial molecular state li) to a final state If), C02(1000,0200)~1 + M C02(0220) M - 50 cm-' (25) depends quadratically on the matrix element Ulf = ( q u i ) of an appropriately defined portion of the interaction potential. In the k4 present context we therefore use eq 21 to write (cf. eq 17) C02(0110) + C02(01'0) C 0 2 (10°0,0200), C02(OOO) - 53 cm-I (26) PI,(')a 1Uif(1)12 = 1((10°0,0200)I)I.1(1000,0200)11))2 570 48 f 8

+

+

2

-

= [0.486( 10°O(~lOoO) - 0 . 4 9 7 ( 0 2 ° 0 ~ ~ 0 2 0 0+)

0.044( 10°0~vJ0200)+

...I2

+ CO,(Ol~O) k,

C02(01'0)

+ C02(0110)

(23)

The extent of quantum mechanical interference between the two leading terms (diagonal in I 10°O) and 102OO), respectively) in this expression will depend on the relative magnitudes of quantities such as (d2V/dQIZ),and (d2V/aQZ2),, which appear in a general potential of the form of eq 20. The result of eq 23 therefore depends, in reality, on a subtle combination of the intermolecular dynamics and the vibrational force field of the C 0 2 molecule itself. However, Uif(')can be evaluated explicitly (but approximately) by adopting the breathing sphere model, assuming a repulsive potential of the form given in eq 18 with L = 0.24 A, cyl = 0.0592, cy2 = 0.0429, and a3 = 0.0230.6 We consider two

-

C0,(01'0)

+

+

C02(1000,0200)11 C02(OOoO)+ 53 cm-' (27)

k6

C02(0220)

+ C02(OOoO)- 0.4 cm-'

(28)

These processes of intramolecular V-V transfer (eq 24 and 2 5 , with M unrestricted) and intermolecular V-V transfer (eq 26-28) all have relatively small energy deficits and are important in the (55) We adopt the numerical labeling of rate processes depicted in Figure 4 of ref 52. This same reference also provides a useful survey of some of the gas-phase energy-transfer rates in C 0 2 .

The Journal of Physical Chemistry, Vol. 91, No. 24, 1987 6115

Feature Article mechanism of C 0 2 lasers (where M = C 0 2 , He, or N,), particularly with regard to collisional depletion of vibrational population in the lower laser levels. Kinetic modeling of recent experimentss2 has yielded, for M = C 0 2 , k2 = 160 f 20 ms-’ Torr-’ ( ( P i t 2 ) ) = 0.018), k4 = 45 f 15 ms-’ Torr-’ ((Pi+4)) = 0.005), and k6 = 800 f 200 ms-’ Torr-’ = 0.09); the rate constants k3 and kS can be inferred on the basis that states 11) and 111) are so alike that the ratios k 2 / k 3and k4/k5are determined simply by relative exothermicitie~.~~ It is interesting to note that the rate constants k2 and k3 are 3 or more times larger than k,, despite the fact that (0220) is relatively free from anharmonic perturbations and can reasonably be approximated as harmonic so that Pip

JUjf(2)12=

~((1000,0200),(~(0220))~2

+

= [0.681( 10°0(J10220) 0.720(02°01V10220) +

...I2

(29)

Here the second term should give the dominant contribution to Pit2),since it involves only a change of vibrational angular momentum, whereas the first term requires, in addition, the exchange of three vibrational quanta. Similar arguments apply to the probability Pit3),which should be of a similar magnitude to P,b2). In neither Pit2)nor Pit3)is there evidence of the type of quantum mechanical interference which reduces the magnitude this fact alone may account for the smallness of kl relative of P&’); to k , and k,, although it should be noted that processes 24 and 25 also have a smaller net energy deficit, +53 or -50 cm-I, than process 22, + l o 3 cm-I. In eq 26-28, both colliding molecules change their vibrational state, so that it is necessary to introduce a second portion V”of the interaction potential to deal with the collision partner. For the process represented by eq 26 Pir(4) a IUit4)I2= I ( (0 1‘0) I VI( 1000,0200)~) ( (0 1 ‘0)l V’l( 0000) ) 12

= ([0.681(01’0~~10°0) + 0.720(0110~J10200)+ ...I X ( 0 1‘01V’lOOoO) j2 (30) where it has been assumed that the state (01IO) is adequately described by the harmonic approximation. An analogous formulation applies to Pi,(s),which should be similar in magnitude to Pit4). For the process in eq 28

P i p ) cc

I

IUjf(6)12= ( (0 1‘0)l Vl(0220) ) ( (0 1‘0)

Iv’~(oo00)) 12

[ (01 ‘011/10220)(01 ‘01 v’loooo)] 2

(31)

(Here, I(Oll0)) and 101’0) represent the molecular eigenstate and harmonic oscillator basis state, which are, for the states considered, almost identical.) For Pit4),Pi,(s),and Pit6),the dominant contributions should come from products of harmonic matrix elements (where 1 = 0 or 2), which of the form (01’0~1/102’0)(0110~V’100%) in turn implicate first derivatives of Vand V’with respect to the doubly degenerate normal coordinate Q2. Again there is no sign of quantum mechanical interference. On the basis of eq 30 and 31 alone, k6 should be twice k4 (and also k s ) , whereas a factor of almost 20 is observed.s2 This discrepancy is satisfactorily reduced when it is realized that the energy deficit in eq 28 is very much smaller (-0.4 cm-I) than that in eq 26 and 27. Detailed calculations using theories invoking long-range48and ~ h o r t - r a n g e ~ ~ forces confirm this interpretation, predicting factors of 5 5 and 21, respectively, between k6 and k4.52 The effect of Fermi resonance on collision-induced intramolecular V-V transfer in C 0 2 can therefore be summarized as follows. Transfer from each member of a Fermi dyad to a third (approximately harmonic) vibrational level tends to occur at very similar rates and can be dramatically enhanced compared to what might be expected in the absence of Fermi resonance. Thus, processes 24 and 25, and processes 26 and 27, are all rapid since eigenstates 11) and 111) both contain equally large v2 character. In the absence of Fermi resonance, transfers between levels associated with the “pure” v2 mode would be rapid but those between these levels and the v l level would be slow. Situations involving V-V transfer between two members of a Fermi dyad are more complicated, however, owing to ubiquitous quantum mechanical

interferences which are an inevitable consequence of the orthogonality of the two Fermi-coupled eigenstates. Such considerations are neither novel nor confined to Fermi resonance in C 0 2 , which has merely been chosen for didactic purposes. Quantum mechanical interference effects are almost certainly widespread in processes involving mixed molecular basis states. For example, they have been clearly demonstrated and characterized in the contexts of vibrational population lifetimes for molecular liquids,I5 collision-induced rovibronic energy transfer,56 and intramolecular vibrational relaxation in large polyatomic molecules.s7 However, much of the literature of V-V energy transfer gives the general impression that Fermi (and Coriolis) resonance leads ineuitably to high V-V transfer efficiencies. To support our argument that this is not necessarily so, we consider the interpretation of experimental results obtained for OCS and H C N . Vibrational energy transfer in OCS has been studied in great detail by Mandich and Flynn.2a~SsAmong many other issues, they discuss the V-V transfer rate between the Fermi-coupled “2v2” and “vln vibrational eigenstates of OCS and report an experimentally determined probability of 4.2 X IO4, a factor of 500 greater than that predicted on the basis of SSH theory and a harmonic oscillator description of vibrational states.58 The process of interest is

0CS(“2v2”) + OCS

-

OCS(”vl”)

+ OCS + 188 cm-l

(32)

a situation which is closely analogous to that in C 0 2 as represented by eq 22. The relevant vibrational eigenstates for OCS are reported58 to be I“vl”)

= 0.971110°0) - 0.238102OO)

(33a)

+ 0.971102°0)

(33b)

( “ 2 ~ ~=” 0.238(10°0) )

from which it follows (in the manner of eq 23) that the V-V transfer probability should be

Pif

a

lUirl2 = ((“vInlJlu2v2”)l2

+

= [0.231( 10°O~JllOoO) - 0.231 (02°011/10200) 0.889 ( 1OoO( V102°0)] (34) where V represents the interaction potential. This result differs from that in eq 40 of ref 58 by including the potentially destructive interference between the contributions of ( 1 0 ° O ) ~ l O o O ) and (02°01V10200). Mandich and Flynns8 argue that the first term in eq 34 is much larger than the last two terms, so that Pif is determined primarily by (0.23 1( 10°O1 10°0))2. However, this is not borne out by substitution of the available potential parametersSaand eq 19a into eq 34, yielding Pif CT. lU# = [0.0018 - 0.0019

+ 0.0004 + ...I2 = 1.6 X

lo-’ (34a)

The level of cancellation between the two leading terms is comparable to that found for C02,and in eq 23a the ”unperturbed” term, ( 10°011/10200),survives as the dominant contributor to Uip The observed V-V probability for “vl” “ 2 ~ ~transfer ” in OCS/OCS collisions therefore remains several orders of magnitude larger than that predicted by SSH-style treatments using the wave functions in eq 33. However, as was pointed out when discussing energy transfer in C 0 2 , when Uif is small the value may depend sensitively on both the intermolecular potential and on the intramolecular potential which will control the true form of the wave functions for the vibrational eigenstates. H C N makes an interesting contrast to CO,, in particular, and to non-hydride linear triatomic molecules, in general. The two normal modes are CN and CH bond-stretching vibrations with

-

(56) Alexander, M. H. J. Chem. Phys. 1982, 76,429. Pouilly, B.; Robbe, J.-M.; Alexander, M. H. J. Phys. Chem. 1984, 88, 140. (57) Freed, K. F. Chem. Phys. Lett. 1984, 106, 1. (58) Mandich, M. L.; Flynn, T. W. J. Chem. Phys. 1980, 73, 1265. (59) Sharma, R. D.;Brau, C. A. J. Chem. Phys. 1969, 50, 924.

6116 The Journal of Physical Chemistry, Vol. 91, No. 24, 1987 v I = 2095 cm-I and v3 = 331 1.5 cm-I, respectively. In contrast to the situation in CO,, where v I is very nearly equal to 21, and there is strong Fermi resonance between the 110'0) and 102'0) terms of the harmonic oscillator basis set, in H C N the lower bond-stretching frequency is approximately 3 times the frequency (v = 712.0 cm-I) of the bending vibration. As a result, the nearby levels denoted by ( 10'0) and (03'0) have vibrational eigenfunctions of different symmetry and cannot be mixed with one another via anharmonicity in the intramolecular potential. Fermi resonance can only occur between more widely spaced zeroth-order states and is therefore likely to be of limited strength. Expressions for the vibrational eigenstates of H C N have not been published, but measurements of infrared absorption intensities6' confirm that there are no strongly mixed states below about 5000 cm-I. As well as the experiments on HCN(O1 11)32referred to earlier, measurements have been made on vibrational energy-transfer processes involving the (00°1),30 (00°2),31and ( 10°1)33states of H C N (see Figure 5). The probabilities of energy transfer between HCN(mn'1) and noble gas atoms at 298 K are given in Table 11. For the (00'1) state the probabilities show little dependence on the collision partner and are orders of magnitude larger than the probabilities of V-(R, T) relaxation of the hydrogen halides by rare gases. These facts suggest that HCN(oOol) is not transferred directly to the (00'0) vibrational ground state but relaxes by intramolecular V-V transfer to a level of similar total energy. Only the (12OO) and (05'0) levels are close in energy to (OO'l), and evaluation of the vibrational matrix elements ( 12°0)1/10001) and (05'0(1/100°1) on the basis of harmonic oscillator states indicates that the (1 2%) level is the more likely to be populated than (05'0) when rare gas atoms relax H C N (00'1). In discussing the process

+

-

+

HCN(OOO1) M HCN(12'0) M - 191 cm-I (35) Arnold and Smith30 pointed out that the (00'1) and (12OO) levels are both Z+ states and that there was some evidence62for mixing of these zeroth-order states in isolated HCN. However, they fell into the trap of assuming that if such mixing occurs it would necessarily cause an appreciable acceleration of V-V transfer according to eq 35. Moreover, their estimate of vibrational matrix elements based on harmonic oscillator representations of the two states was incorrect.63 Table I11 lists matrix elements and probabilities of V-V transfer according to the simple breathing sphere expression, eq 13, with two different assumptions: (i) by collisions with either the H or N atoms of the molecule so that the calculation of matrix elements are based on mean-square displacements of the two terminal atoms and (ii) only by collisions between M and the H atom of H C N . The estimated probabilities in the first row of Table I11 compare remarkably well with the observed probabilities listed in the first row of Table 11. In view of the assumptions in the breathing sphere treatment-especially in this case, that the atomic displacements due to molecular vibration are small relative to the range of the intermolecular repulsion-the quality of the agreement must be regarded as fortuitous. However, there is clearly no evidence from this comparison that mixing of the harmonic oscillator basis sets has any influence on the rate of collision-induced transfer between the (OOol) and (12OO) states of H C N . Relaxation of HCN(00°2) by noble gases has also been studied,31and the observed probabilities are given in Table 11. Since I(mn'21Vlm'n"1)(2 = 21(mn'lIVlm'nA'OJ2, it is, at first sight, surprising that molecules are removed from the (0O02) state more slowly than from (00'1). However, the equivalent process to (35), Le.

-

HCN(00'2) + M HCN(12'1) + M - 247 cm-I (36) is now less resonant. This reduces the term Z(AEfi,w,T,L,c)in eq 15, which allows for the "overlap" of translational wave functions (60) Smith, I. W. M. J. Chem. SOC.,Faraday Trans. 2 1981, 77, 2357. (61) Nakagawa, T.; Morino, Y . Bull. Chem. Soc. Jpn. 1976, 42, 2212. (62) Wang, V. K.; Overend, J. Spectrochim. Acta, Part A 1976, 32, 1043. (63) I. W. M. Smith is very grateful to Professor C. C. Davis (University of Maryland) for pointing out this error and for providing a correct analysis of the vibrational motions in HCN.

Orr and Smith TABLE 11: Thermally Averaged Probabilities for V-V Energy Transfer in Collisions between HCN(mn 1) and Noble Gases at 298

K" HCN(001) HCN(OO2) HCN(lO1) HCN(Ol1)

He

Ne

Ar

Kr

1.9 (-5)b 1.6 (-5) 0.95 (-4) 2.3 (-3)

2.9 (-5) 1.9 (-5) 1.32 (-4) 7.8 (-4)

2.9 (-5) 1.4 (-5) 1.27 (-4) 5.7 (-4)

3.2 (-5) 1.6 (-5) 1.24 (-4) 3.4 (-4)

Xe

2.8 (-5) 1.43 (-4)

The quoted probabilities are those determined experimentally for total transfer out of the specified state. Only in the case of HCN(Ol1) (001). is the state identified to which transfer takes place, Le., (01 1) 1.9 (-5) = 1.9 X probabilities are calculated from the equation (P) = k / ( r a Z ) C where , k is the observed rate constant and u and C are the collision diameter and mean relative velocity, respectively. The following collision diameters (in A) were used: H C N He, 2.84; Ne, 2.94; Ar, 3.26; Kr, 3.36; Xe, 3.58.

-

+

on the intermolecular potential. It is reasonable to suppose that the ratio of the thermally averaged probabilities should be given quite well by breathing sphere calculations, since some approximations in the simple model will than cancel. This ratio should then be given by ( PO02.l21) I( Po01, I 20) =

12/ l ~ o o, I, 2012)(~002,1 21 /I001 . I 201

~I~OO2,121

(37) Estimated values of (P002,121)/( are compared with experimental values in Table IV. The agreement is quite good, with the variation in relative probabilities with different rare gases reproduced extremely well. The experimentally determined values of (P) for the relaxation of HCN(10'1) show a similar lack of sensitivity to noble gas partner as those for transfer from the (OOol) state. This observation strongly suggests that molecules are transferred to a state with AEfi< kBT, and the experiments show that the excitation in the v3 mode is retained in the first collision-induced process. These two pieces of evidence clearly identify HCN(10'1)

+M

-

HCN(03'1)

+M +

-27 cm-'

(38)

as the rate-determining step. The rate could be enhanced by anharmonic mixing of the llOol ) and 102O1) harmonic oscillator basis states. However, such mixing is not required to explain the estimated by using observed results. The probabilities, ( P101,031), the breathing sphere expression (eq 13) with harmonic oscillator matrix elements, are recorded in Table 111and are in reasonable accord with the experimental values. Indeed, if mixing of the 110'1) and 102'1) basis states was at all marked, the relaxation rates of HCN( 10'1) would apparently be much greater than they are. In summary, it appears that the rates of transfer of H C N from levels (OO'l), (00°2), and (10'1) by noble gas atoms can be satisfactorily explained by calculations based on the breathing sphere model and harmonic oscillator descriptions of the vibrational states. There is no evidence, either from the energy-transfer data or from infrared band intensities, of strong mixing between the harmonic oscillator basis states of these levels. However, in assessing the accuracy of estimates of probabilities of V-V or V-(R, T) energy transfer in HCN, it has to be remembered that the vibrational amplitude of the H atom is large, especially in the v 2 mode. Enhancement o f Collision-Induced Intramolecular V-V Transfer by Coriolis Coupling

As already noted, anharmonic Fermi resonance perturbations are by no means unique in their ability (under favorable circumstances) to promote collision-induced intramolecular V-V transfer. Another class of such perturbations is that due to Coriolis coupling, which mixes rotational, as well as vibrational, basis states. A recent convincing demonstrationM6 of Coriolis-enhanced V-V transfer (64) Haub, J. G.; Orr, B. J. Chem. Phys. Left. 1984, 107, 162. (65) Bewick, C. P.; Duval, A. B.; Orr, B. J. J. Chem. Phys. 1985,82, 3470. (66) Haub, J. G.; Orr, B. J. J. Chem. Phys. 1987, 86, 3380.

The Journal of Physical Chemistry, Vol. 91, No. 24, 1987 6117

Feature Article

and (Plol,o,l) for HCN Calculated Using the Breathing Sphere Model, Eq 13, and

TABLE III: Thermally Averaged Probabilities, Harmonic Oscillator Vibrational Wave Functions

( POOl,l20 )

(9” (ii)b

L/A

Iu i d 2

PO

He

Ne

Ar

Kr

0.2 0.2

2.5 (-5) 3.8 (-4)

I13

2.1 (-5) 1.6 (-4)

2.5 (-5) 1.9 (-4)

2.9 (-5) 2.15 (-4)

2.8 (-5) 2.1 (-4)

LIA

IU;fP

0.2 0.2

3.8 (-5) 6.65 (-4)

Ar 3.1 (-4) 2.7 (-3)

5.0 (-4) 4.4 (-3)

,

I16

(p101,031)

(iii)“ (iv)*

I

1.1

Pll

He 3.6 (-5) 3.2 (-4)

If3 I16

Assuming a mean-square displacement for the H and N atoms.

Ne 1.25 (-4) 1 . 1 (-3)

Kr

Assuming a mean-square displacement for the H atom only. v’= 0

TABLE I V Comparison of Relative Probabilities of Relaxation of HCN(OOO2) and HCN(OOO1)with Estimated Values of (Poo2,12)/(PoO1.120) 1

M He Ne Ar

Kr

1~002,,2,12/

~002.121/

(~W2,121)/

(P002)l

I~001,12012a

~WIJ20

(P001.120)calcd

(~00l)SXPt

1.10 0.76 0.63 0.57

0.87 f 0.06 0.61 f 0.06 0.46 f 0.07 0.46 f 0.05

2.11 2.11 2.1 1 2.11

0.52 0.36 0.30 0.27

DzCO(U6=1;J,Ka,Kc)

+M

-

+

D~CO(U~=~;J’,K,’,K,’) M (39)

Here a D 2 C 0 molecule is prepared by pulsed infrared excitation into a specific rotational state (J,K,,K,) of the first vibrational level of its v6 (in-plane wag, 989 cm-I) mode where K, and K , are the customary asymmetric-rotor quantum numbers which refer respectively to the projection of angular ammonia J on the a and c inertial axes of the molecule. Collisions between the stateselected DzCO molecule and a collision partner M (which may be atomic or molecular) induce intramolecular V-V transfer from the initially prepared v6 mode state to particular rotational states (J‘,K,’,K,’) of the first vibrational level of the adjacent v4 (outof-plane bend, 938 cm-’) mode of D2C0, the process being monitored by vibronic laser-induced fluorescence (LIF). This V-V transfer is studied by recording a set of IRUVDR spectra with successively increasing values of pump-probe delay (and hence collision number) and watching the growth of spectral features associated with the v4 mode. Once the favored rovibrational channels of V-V transfer have been identified in this way, kinetic curves showing the growth and decay of particular IRUVDR features are generated by scanning the delay with probe wavelength fixed. In the case of self-collisions (M = D2CO), V-V collisional efficiencies (P(Vg-v4)) are found”@ to be as high as 4, indicating V-V transfer at longer range than that of hard-sphere collisions, and to exhibit clear-cut rotational propensity rules, not only with For instance, DzCO molecules respect to k,’ but also to prepared (by infrared excitation with the 10 R(32) COzlaser line) in the ( u p 1;11,4,7) rovibrational state are found to appear in v4 = 1, k, = 6 states with J’ = 9, 10, and 11 prior to any significant degree of collision-induced rotational scrambling. With atomic collision partners (M = Ar, He, ...) the V-V transfer rates are ~ m a l l e r , 6 ~but 3 ~still ~ comparable to that of hard-sphere collisions ( ( P) = l ) , and the channeling into specific rovibrational states is less pronounced. J‘.65368

(67) Orr, B. J.; Nutt, G.G.J . Mol. Spectrosc. 1980, 84, 272. Orr, B. J.; Haub, J. G.Ibid. 1984, 103, 1. (68) Bewick, C. P.; Orr, B. J., unpublished results.

x-i lL22 n m l

I

a Using formulas from Rapp and Sharp16 and mean-square displacements of the H and N atoms.

comes from time-resolved infrared-ultraviolet double resonance (IRUVDR) investigations of collision-induced rovibrational transitions in the molecules DzCO and HDCO. These results are now reviewed. The IRUVDR technique6’ employs the excitation scheme depicted in Figure 7 and enables detailed characterization of processes of the type shown in the following equation:

VISIBLE FLUORESCENCE

ULTRAVIOLET EXCIJATION TO A ‘A2 ELECTRONIC STATE

I

6:

PROBE

1366nml

1365 nm)

I N F R AR E D EXC ITAT I ON IN Z l A l ELECTRONIC GROUND STATE

+D\-

v 6 PUMP

+D’

+ .c = 0 V~lbll

-

Figure 7. Excitation scheme for time-resolved infrared-ultraviolet double v4 rovibrational energy resonance studies of collision-induced v6 transfer in D2C0 vapor. A C 0 2 laser pump pulse selects a single rovibrational state u6 = 1 (J,Ka,Kc).The population changes created by the pump radiation are then monitored by vibronically excited LIF, either v4 directly in the 366-nm 6; hot band (probe) or indirectly, after v6 transfer to the state u4 = 1 (J’,Ka’,Kc?, in the 365-nm 4: hot band (probe).

-

The above efficiencies for collision-induced mode-to-mode transfer are remarkably high, for typical V-V energy-transfer efficiencies usually lie in the range lo-’ to 10-4 per hard-sphere collision. This is borne out by V-V transfer processes discussed in earlier sections and by the following well-established examples:

-

CH,F(U~=~ +)CH~F(U=O) 2CH3F(u3=1), (P) HF(u)

+ HF(0)

+

HF(U-1)

+ HF(1),

0.3 (ref 69) (40)

(P) 0.08, u = 2, 3, 4 (ref 70) (41)

These processes involved near-resonant intermolecular V-V transfer and are regarded as “fast” by most gas-phase energytransfer standards, but they are invariably accompanied by rotational relaxation at least an order of magnitude faster than V-V transfer. The effect of this rotational ”scrambling” is to obscure rotationally specific channels of V-V transfer from one rovibrational state to another and to frustrate attempts to determine experimentally whether some of these channels are especially efficient. The above results for DzCO/DzCO collisions64966 therefore provide a rare opportunity to view rotationally specific V-V transfer channels and to gain insight into previously elusive aspects of molecular collision dynamics. (69) Sheorey, R. S.; Flynn, G. J . Chem. Phys. 1980, 72, 1175. (70) Copeland, R. A.; Pearson, D. J.; Robinson, J. M.; Crim, F. F. J . Chem. Phys. 1982, 77, 3974; 1983, 78,6344. Robinson, J. M.; Rensberger, K. J.; Crim, F. F. Ibid. 1986, 84, 220.

6118

The Journal of Physical Chemistry, Vol. 91, No. 24, 1987

Orr and Smith

The unusual efficiency of mode-to-mode transfer arising in selected molecule is referred, and p(u=O) is the “permanent” D2CO/D2C0collisions can be rationalized in terms of a semielectric dipole moment (2.35 D75) of the molecule. The offclassical collision theory,66based on long-range electric dipolediagonal nature of eq 42 with respect to K, (=4) and K,’(=6) dipole interactions. The essential mechanism depends on the strong is vital in ensuring that the transition moment in this case avoids Coriolis interaction between the u4 and V 6 vibrational modes of the type of destructive interference which has already been demD2C0, which mixes rotation-vibration basis states and yields onstrated in earlier sections. Such interferences are frustrated nonvanishing matrix elements of the permanent electric dipole in eq 42 by the K dependence of rotational matrix elements moment M between the u4 and V6 vibrational manifolds. Whereas (J,KI+lJ,K); in the absence of any such K dependence, the conventional long-range theories of V-V transfer depend on intransition moment would have been a factor of 50 smaller and teractions between oscillating electric dipoles (in addition to saved from annihilation only by the small vibrational-state decontributions from the short-range part of the intermolecular pendence of the electric dipole moment. Coriolis mixing by itself potential),2 it now becomes appropriate to use a collision theory cannot generate this off-diagonal character, for the a-axis Coriolis of the form normally applied to pure rotational relaxation, rather perturbation^^^ which cause the vibrational basis states I u 4 ) and than to conventional V-V transfer. In fact, the simplest interlug) to be mixed respectively into the u6 = 1 and u4 = 1 rovibpretation of eq 39 would be to suppose that the initial and final rational eigenfunctions are unable to mix different basis states eigenstates of the state-selected D 2 C 0 molecule belong to a single IJ,K). It is the asymmetric-rotor character of the molecule which “grand” (u4, v6) manifold of rotational states, within which colintroduces a variety of rotational basis functions, with K (or K’) lision-induced transfer can occur at the fast rates characteristic different from the value of KO(or K,? used to label the initial of rotational relaxation. However, such a simple interpretation (or final) rovibrational eigenfunction, and hence yields a relatively falls into the same trap as that already exposed in the case of Fermi large transition moment between K, = 4 and K,’ = 6. We resonance contributions to V-V transfer, for it neglects the therefore conclude that mixing of vibration-rotation basis funcpossibility of destructive quantum mechanical interferences and tions, by a combination of Coriolis and asymmetric-rotor perfails to account adequately for the collision-induced rovibrational turbations, causes D 2 C 0 to acquire a rovibrational transition propensity rules which are observed. moment which corresponds to almost 10% of the permanent Our theory66evaluates the collisional efficiency ( PiF)relatively electric dipole moment in the ground vibrational level (u = 0). crudely in terms of first-order perturbation theory as in eq 3, This leads in turn to an estimate of the efficiency of rotationally despite its general inferiority to the adiabatically corrected sudden specific V 6 u4 transfer in D2CO/D2C0collisions: (P(ug= 1;11,4 approximation or to fully quantum mechanical calculations in u4=l;11,6)) = 4.5 f 3.0, which compares favorably with that treating inelastic collisions between dipolar molecule^.^^ With observed (3.8 f 0.6).64366 this approach, it is possible to follow the course2 of earlier longThe proposed mechanism has been confirmed66 by several range perturbation theories of r ~ t a t i o n a l l yor ~ ~v ,i~b ~r a t i ~ n a l l y ~ ~ ~ additional ~~ examples (involving either D2CO/D2C0 or HDCO/ inelastic processes, It is found66 that a combination of Coriolis HDCO collisions) in which the contributing factors are less faand asymmetric-rotor perturbations generates nonzero matrix vorable than in the above instance and the observed V-V transfer elements of the permanent electric dipole moment p between rates are correspondingly smaller. It is concluded that only a different rovibrational eigenstates of the state-selected molecule; limited range of initially prepared rovibrational states of D,CO herein lies the major distinction between the present rovibrational or HDCO fulfill all of the prerequisites needed to achieve direct energy-transfer theory and earlier theories. Another significant V-V transfer with high collisonal efficiency. These prerequisites feature of the fast V-V transfer which we consider is its apparent comprise the following: (a) Coriolis coupling between the initial quasi-elasticity with respect to the state-selected molecule, such and final rovibrational eigenstates of the state-selected molecule, that its internal energy change is very much less than kT. There which enables use of the types of inelastic collision theory normally then exist many initial collision-partner states for which a corapplicable to pure rotational relaxation; (b) a close coincidence respondingly small energy change is probable, yielding an apin energy between those initial and final eigenstates, thereby proximately zero overall energy deficit. This provides “rotational enhancing the abundance of collision partners available to yield compensation” of rovibrational energy deficits which might otha zero overall energy defect for the pair of colliding molecules; erwise be relatively large on the basis of vibrational energy alone (c) the presence of a secondary intramolecular perturbation (here and contributes to large V-V transfer probabilities. due to the asymmetric-rotor character of the molecule) capable Such behavior is well illustrated by the above-mentioned exof frustrating quantum mechanical cancellations which otherwise ample of V-V transfer between the (u6=l;l1,4,7) and Iu4=1;11,6,6) tend to annihilate Coriolis-induced transition moments. The eigenstates of D2C0, which has been treated in detail elsecombined effect of (a) and (c) may at first seem surprising, for ~ h e r e . ~ This ~ f ’ collision-induced ~ transition is quasi-elastic, with it means that Coriolis-induced transition moments are most the energy change of the state-selected molecule as small as 3.5 pronounced at moderate values of the rotational quantum number cm-’, and the relevant rovibrational eigenstates can be expressed K, (-4), where both asymmetry and Coriolis perturbations are precisely in terms of products of vibrational basis functions, I u 4 ) simultaneously appreciable, rather than at higher values of KO and l u g ) , and symmetric-rotor basis functions, IJ,K). Explicit where Coriolis coupling is itself greatest; this is borne out in the e v a l ~ a t i o nof~ ~the , ~Coriolis-induced ~ electric dipole transition experimental results.66 moment yields The foregoing calculations are based solely on electric dipole-dipole interactions and appear to yield adequate agreement (u6 = 1;11,4,71plu4 = 1;11,6,6) = with experimental data for D2CO/D2C0 and HDCO/HDCO /.~(~=0)[0.185(11,61@~11,6) - 0.187(11,4~@~11,4) + ...I collisions. Nevertheless, there is an additional “background” contribution due to terms in the intermolecular potential which = p(u=0)[0.185(11,6~+~11,6) - 0.187(11,41@~11,4)+ ...I = are significant at short range. An indication of the magnitude O.O93p(u=O)( 11,4~+~11,4) (42) of some of these background contributions is available from recent experimental65368and the~retical’~ estimates of V 6 u4 transfer where @ is the direction cosine between the intra- and intermorates in D2CO/Ar collisions, where interactions between perlecular axes, to which the dipole-dipole interaction of the statemanent electric dipoles are impossible. Such rates are found to (71) Alexander, M. H.; DePristo, A. E. J . Phys. Chem. 1979, 83. 1499. be comparable to that of hard-sphere collisions, so that short-range (72) Oka, T.Ado. Ar. Mol. Phys. 1973, 9, 127. contributions are probably far from negligible. The recently (73) Gray, C. G.; Van Krandendonk, J. Can. J . Phys. 1966, 44, 2411. formulated approximate coupled-channel theory of Peet and Oka, T. J. Chem. Phys. 1967,47, 13. Murphy, J. S.; Boggs, J. E. Ibid. 1967,

- -

-

47,691. Rabitz, H. A.; Gordon, R. G . Ibid.1970, 53, 1815, 1831. Prakash, V.; Boggs, J. E. Ibid. 1972, 57, 2599. (74) Sharma, R. D. Phys. Reu. A 1970.2, 173. Dillon, T.A,; Stephenson, J. C . Ibid.1972, 6,1460; J . Chem. Phys. 1973, 58, 2056. Seoudi, B.; Doyennette, L.; Margottin-Maclou, M. Ibid. 1984, 81, 5649.

(75) Clouthier, D. J.; Ramsay,D. A. Annu. Rec. Phys. Chem. 1983, 34, 31.

(76) Peet, A. C.; Clary, D. C. Mol. Phys. 1986, 59, 529. Clary,.D. C. J. Phys. Chem. 1987, 91, 1718.

The Journal of Physical Chemistry, Vol. 91, No. 24, 1987 6119

Feature Article Clary76treats the vibrational and K , rotational degrees of freedom as quantized and makes a classical approximation for the J rotational motion; it clearly demonstrates Coriolis enhancement of the v6 v4 transfer rate for D2CO/Ar and D2CO/He collisions, as well as the need to take asymmetric-rotor effects into account. Significant contributions from Coriolis and asymmetric-rotor perturbations are therefore expected to be intrinsic to any theory, whether long or short range, capable of explaining fast v6 vq transfer in collisions involving D2C0.

-

-

Concluding Remarks Several themes run through our account of collision-induced vibrational energy transfer. The first concerns the nature of the intermolecular potential function and its manipulation into a form suitable for predicting rates of specific energy-transfer processes. Even though accurate intermolecular potentials are increasingly available for quite complex systems, a place remains for approximate dynamical theories which consider only that portion of the potential most likely to influence the process of interest. For processes of relatively low collisional efficiency, such as V-(T, R) relaxation, it is often reasonable to consider only the short-range portion of the intermolecular potential; this can conveniently be represented in the repulsive exponential forms of eq 1, for atomdiatom interactions, or eq 18, for the “breathing sphere” representation of processes involving polyatomic molecules. At the other extreme, the long-range part of the intermolecular potential tends to exert a significant influence on processes such as rotational relaxation or fast V-V transfer, for molecules having either permanent or oscillating electric dipole moments. Nevertheless, such a classification must be adopted with some caution, even in semiclassical collision theory, for there will always exist a level of accuracy beyond which the variety of contributing collisions (specified by a distribution of impact parameters, relative velocities, and orientational parameters) will require the full intermolecular potential. Another obvious theme is provided by the pervasive role of molecular rotation in influencing various forms of vibrational energy transfer. It seems well e~tablished’~ that rotational excitation tends to accompany vibrational equilibration of molecular products (such as diatomic hydrides including HCl, HF, and OH) formed in exothermic chemical reactions and that this manifests itself in the gain characteristics of chemical lasers operating on rotational transition^.^^ In this context, the dissipation of some of the excess energy into rotational degrees of freedom relieves the translational energy gap and thereby promotes vibrational relaxation. This is confirmed in the case of H F by infrared-infrared double resonance experiment^^^^^^*^^ and by theoretical calculations.81 A second, complementary proposition pursued in this paper suggests that rotational excitation of a molecule prior to vibrational relaxation can increase the V-(R, T) transfer probability, by enhancing the effective impact velocity of molecules undergoing noncollinear collisions. This proposition is supported qualitatively by correlations of V-(R, T) relaxation data for hydride molecules (e.g., HCl, DCl, HCN) with a variety of collision partners. Molecular rotation also plays a role in vibrational energy transfer when Coriolis coupling is implicated, as v4 transfer in demonstrated in the case of collision-induced v6 D2C0.64+6 Moreover, the previously elusive goal of specifying rotationally selective channels in V-V transfer now seems to have been attained,65through a combination of instrumental development and mechanistic subtlety.

-

(77) Holmes, B. E.; Setser, D. W. In Physical Chemistry of Fast Reactions; Smith, I. W. M., Ed.; Plenum: New York, 1980; Vol. 2, p 83. (78) Sirkin, E. R.; Pimentel, G. C. J . Chem. Phys. 1981, 75, 604; 1982, 77, 1314. Yang, X.F.; Pimentel, G. C. Ibid. 1984, 81, 1346. (79) Haugen, H. K.; Pence, W. H.; Leone, S . R. J . Chem. Phys. 1984,80, 1839.

(80) Robinson, J. M.; Pearson, D. J.; Copeland, R. A,; Crim, F. F. J . Chem. Phys. 1985, 82, 780. (81) Thompson, D. L. J . Chem. Phys. 1982, 76, 5947; 1983, 78, 1763.

Our third theme concerns collision-induced energy transfer involving coupled modes of vibration. Whether one is dealing with mixing of rovibrational basis states by Fermi resonance or by Coriolis coupling, the essential effect is as outlined in eq 17: the orthogonality of molecular eigenstates inevitably leads to a situation in which there is the possibility for cancellation (destructive interference) of individual terms which are diagonal in the basis states IA) or IB). This cancellation may be frustrated if the matrix elements (AlVA) and )BIVB) differ in magnitude; in instances involving Fermi resonance, this depends on the molecular anharmonicity, whereas the case of Coriolis coupling requires a second intramolecular perturbation, such as that due to the molecule’s asymmetric-rotor character. Such subtleties seem to have been overlooked in much of the literature, from which it is easy to gain the impression that Fermi resonance and/or Coriolis coupling between two vibrational states leads invariably to efficient V-V transfer between them. This conventional viewpoint has been reiterated in a recent review article,82as follows: “When, for instance, two nearly resonant levels are known, from spectroscopic observations, to be mixed by Fermi resonance, a greatly enhanced energy transfer is invariably found.” We maintain that this statement is not as universally valid as it appears to be. We therefore sound a general word of caution: the efficiency of collision-induced V-V transfer belween rovibrational eigenstates which can be represented in terms of the same basis functions, mixed by Fermi resonance or by Coriolis coupling, is not always much enhanced by such mixing, owing to the possibility of quantum mechanical interference effects associated with the orthogonality of those eigenstates. Our attention has been confined to energy transfer within the sparse low-energy region of the vibronic manifold. At the same time, our interests run parallel to the vast body of literature on nonradiative intramolecular energy transfer in congested rovibrational and rovibronic manifolds, including processes such as intramolecular vibrational redistribution (IVR), intersystem crossing, and internal conversion. These processes are controlled by an intramolecular Hamiltonian and tend to occur in the absence of collisions. On the other hand, the processes with which we are concerned in this paper require a collision partner to promote energy transfer and are influenced primarily by an intermolecular Hamiltonian, even though the physical processes monitored may occur within the one state-selected molecule. The interplay of such mechanisms has been discussed in more detail elsewhere.66 The topics discussed in this article have spanned a wide range of collisional efficiencies and molecular systems, but all have raised questions for which definitive answers still need to be sought. The maturity of the subject is therefore deceptive, for newly developed experimental and theoretical techniques continue to extend and challenge what might previously have seemed to be the ”final word”. It cannot yet be said that we fully understand or anticipate the processes by which molecular collisions transfer and dispose of vibrational energy, even in the sparse low-energy region of the vibronic manifold. If this article follows the tradition of numerous excellent earlier reviews, in updating aspects of physical insight in readiness for further advances, then it will have achieved its goal.

Acknowledgment. Many of the ideas presented in this article stem from helpful discussions and communications with our coworkers and other colleagues working the field of energy transfer. They are too numerous to list here; most of their names appear in the references quoted in this article. We gratefully acknowledge financial support for own research from the Australian Research Grants Scheme (to B.J.O.) and the U.K. Science and Engineering Research Council (to I.W.M.S.). In addition, we are grateful for the award of an SERC Senior Visiting Fellowship (to B.J.O.). The preparation of this article was begun during the period of the Fellowship. (82) Bondybey, V. E. Annu. Rev. Phys. Chem. 1984, 35, 591