Faraday Discuss. Chem. SOC., 1982, 73, 325-338 Relaxation Channels of Vibrationally Excited Van der Waals Molecules BY GEORGE E. EWING * Department of Chemistry, Indiana University, Bloomington, Indiana 47405, U.S.A. Received 9th February, 1982 Energy flow patterns of vibrationally excited Van der Waals molecules, such as Iz* * * * He, HCl* * - - Ar, N,O* - H20, CzH4* * * - CzH4 and HF* * - * HF, are reviewed. These complexes are described as A-B * * * C, where A-B* is a vibrationally excited chemically bonded molecule attached by a Van der Waals bond to C , an atom or molecule. Relaxation of the initially prepared excited complex can proceed by at least four channels: (a) A-B* . * * C -+ A-B + C + AEV-T (b) A-B* ( c ) A-B* * * * C + A-B + C* + AEv-v (d) A-B* * . * C + [A-B * .. C]*. * * C -+ A-Bt -t Cf + AEV-T,R In (a), energy from the excited chemical bond breaks the Van der Waals bond and A-B (now relaxed) and C fly away with translational energy AEV--T. If A-Bf- and CJr contain rotational energy as in (h), the relaxation is by the vibration-translation rotation channel. Vibrational excitation of the fragments, such as C* in (c), appears in the vibration- vibration channel. Finally, in (d) energy initially localized in A-B* flows throughout the complex by exciting isoenergetic internal modes without breaking the Van der Waals bond. This is the intra- molecular relaxation channel which produces [A-B - * * C]*, which may some time later dissociate to yield A-B + C with translational, rotational and possibly vibrational excitation. Taking recently studied vibrationally excited Van der Waals molecules as examples, simple theoretical models are applied to assess the relative importance of these four relaxation channels.Our results are summarized with a set of propensity rules. This is the vibration-translational channel. 1. INTRODUCTION The spectroscopy of vibrationally excited Van der Waals molecules is beginning to reveal intricate and sometimes confusing energy-flow patterns. These patterns include the breaking of bonds and production of rotationally and translationally hot frag- ments. Interchange of vibrational energy is also a possible outcome of the energy flow of Van der Waals molecules. It is the purpose of this paper to provide propensity rules for the possible relaxation channels.We shall develop these rules by examining some experimental studies of vibrationally excited Van der Waals molecules. The theoretical models we offer for the energy-flow processes are simple and, although the numerical results for relaxation rates are not quantitatively meaningful, we hope that the physical insight provided will serve as a qualitative guide to those designing and interpreting experiments. We begin by developing a vocabulary for the relaxation channels possible for vibrationally excited Van der Waals molecules. The simplest system we may imagine is the complex A-B - - - C. The chemically bonded molecule is A-B attached to an atom C by a Van der Waals bond. We might take I2 - - He or HCl - - - Ar as examples of this type of Van der Waals molecule.On vibrational excitation of the * Supported by the National Science Foundation.326 R E L A X A T I O N C H A N N E L S O F V I B R A T I O N A L L Y EXCITED M O L E C U L E S chemical bond within the complex we produce A--B* * * * C. Energy can flow from the energetic chemical bond to the weak Van der Wads bond and cause its rupture in time 2 : A--B* - - * C -L A--B --I C -+ AE, (1) The fragment A-6, now vibrationally relaxed, and C then fly apart with relative kinetic energy AE. An early proposal to study the vibrational predissociation of the type shown in eqn (1) and shown schematically in fig. 1 was ofkred by K1empcrer.l The system he suggested was F-H* - - F-H. These experiments have recently been perfor~ned.~*~ 5000 4000 3000 13 ' 2000 -..- 3 g 1000 A c OJ G -1000 -2000 ...- A - B-* C FIG. 1,-Energy states for vibrational predissociation of A--B* - - - C. The energy is scaled for the complex F-H" * * * F-H. Adapted from ref. (1). There is now a sizeable number of other experimental and theoretical studies of vibrationally excited Van dcr Wads molecules which has been recently re~iewed.~ In understanding the processes of vibrational predissociation we need to re-examine eqn ( I ) in more detail. If the fragments A---B or C contain neither rotational nor vibrational encrgy, thc kinetic encrgy must reside solely in translational motion. This will then be called the vibration-translation (V-7') relaxation channel. We can rewrite eqn (1) to emphaske this channel: A--R* - * - CA A--6 + C + (2) An alternative channel places vibrational energy in the fragments.Here if C is taken to be a diatomic or polyatomic molecule it may appear vibrationally excited as C* after rupture of the Van der Wads bond: A--B* - * C -L A--B -1- C* + AEv-v. (3) This is called the vibration-vibration (V-V) relaxation channel and the energy left over in translational motion, after brcaking the Van der Waals bond and depositing vibrational energy to produce C", is idcntified by AEvmv. If A-B is a polyatomic molecule, it is also possible to produce A--B* as a fragment in a low-lying excited vibrational mode. Alternatively, the A--B* * * - C system may approach ergodicity. If the Van der Wads molecule has polyatomic components, A-B and C, energy initially placed in a localized chemical bond may flow throughout the complexG .E . EWING 327 populating a variety of resonant chemical/Van der Waals vibrational modes. describe this intramolecular vibration-vibration energy flow by: We may A-B* - * * C -L [A-B - - - C]*. (4) The system [A-B * - * C]*, with its delocalized energy, will eventually break its Van der Waals bond and produce fragments. Finally, the fragments A-B and C (if it is a molecule) may be produced in rotation- ally excited states which accompany translational motion. We indicate this as the vibration-translation, rotation (V-T,R) relaxation channel A-B* * . * C -* A-Bt $- Cf + A E V - T , R ( 5 ) where t indicates rotational excitation. It is likely that for the V-V channel the fragments will also contain some translational and rotational compcnents to their kinetic energy.We shall consider the V-T, V-V and V-T,R relaxation channels in Sections 2, 3 and 4. Finally, we shall provide in Section 5 the propensity rules for locating efficient relaxation channels of vibrationally excited Van der Wads molecules. 2. THE VIBRATION -TRANSLATION CHANNEL Early discussions of vibrational predissociation of Van der Waals molecules considered only the vibration-translation channel.'-' The complex may exist in a variety of vibrational states u, = 0, 1,2,3 . . . , representing stretching motions against its Van der Waals bond. In fig. I , A-B* - - - C in its u = 1 vibrational state of the A-B* chemical bond and the u, = 0 vibrational state of the Van der Waals bond crosses over into the continuum of translational states of the fragments A-B (now in its u = 0 state) and C.Another view of this same process is shown in fig. 2. The intermolecular potential for A-B* -1 C is displaced above the potential for A-€3 + C. The displacement is WA-B; the vibrational energy lost by A-B* when it relaxes The transition from a discrete state isoenergetic with a continuum as in fig. 1 naturally suggests the calculation of the vibrational preciissociation rate 7-l by the Golden Rule or its equivalent: to A-B. (6) ( 0 ) 2. The initial state describing A-B* - - - C is given by v/Ao). The final state of the fragments A-B + C is approximated by v$). The fragments have final relative velocity ZI,,,. The coupling term VcouOling connects the vibration of the chemical bond with the motions of the Van der Waals bond.z-' = (4/h2un1)/<~$)l Vcoupling l ~ n >I The initial-state wavefunction needed in eqn (6) may be approximated by y/p =- r - 'R,p)Yl(x). (7) Vibrational displacement of the A-B chemical bond is defined by the x coordinate. Vibration of A-B* for the u = 1 state is given by the harmonic oscillator function ql(x). The separation of A-B and C is given by the intermolecular radial r coordin- ate. We obtain RV,(r), which describes vibration of A-B* - - C against the Van der Waals bond, after specifying the form of the intermolecular potential, which we will approximate by the Morse function : 1 (8) V(r) = DC[e-2"(r-re) - 2e-Q(r-re)328 RELAXATION CHANNELS O F V I B R A T I O N A L L Y E X C I T E D MOLECULES where D, is the well depth measured from the bottom of the intermolecular potential where A-B* - * C is separated by distance re.A measure of the steepness of this surface is given by the range parameter a. The analytical form of the RUY(r) Morse oscillator wavefunction is given el~ewhere.~ An example of R,(r) (i.e. for the v, = 0 level) together with V(r) for a typical A-B* * - - C complex is shown in fig. 2. One 400C 2000 rl I . 3 2 0 0 0 -2000 , , I ] , ] , , , !I w, - i I AEV-T -. I A - B t C e I I I I I I I I I 2 4 6 8 44 FIG. 2.-Potential-energy surfaces, wavefunctions and energies of A-B* - * * C. have been scaled for uv = 0 of F-H* * * F-H and F-H + F-H by the V-T channel. These quantities Adapted from ref. (8). can see that R,(r) resembles an harmonic oscillator wavefunction which is localized near the Van der Waals bond separation of re.The vibrational energy levels of the Van der Waals bond stretching mode relative to the bottom of the V(r) well are expressed by W., = - [ah(2d - 1 - 2vv)l2/8pV + D, (9) in terms of the dimensionless parameter d = (2pvD,)”/ah and the reduced mass of the Van der Waals molecule:G . 8. EWING 329 Since the parameters D, and c1 of the potential function in eqn (8) are usually only approximately known, we lose little useful information by rounding off d t o an integer. In this case the Van der Waals bond vibrational quantum number takes on any of the values uv = 0, 1 , . . . ., d - 1. The parameter I! then gives the number of bound u, statcs for thc potential surface. The final state wavefunction is where yl,(x) describes the relaxed, A-B fragment.The continuum function R,,,(r), see fig. 2, describes the A-B + C fragments flying away over the lower Morse poten- tial surfxe, again given by eqn (8) with the same parameters as the upper, A-B* + C , surface. and is the difference between the vibrational energy, WA-B, given up by A--B* during its relaxation and the energy needed to break the Van der Waals bond from its v, level, D, - W.,. This translational kinetic energy is thus The final relative translational energy of A-B 4- C is and it appears in the useful dimensionless quantity 9 m = (2PvAEv-TYla~. (14) The analytical form of R , ( r ) is given elsewhere.' It can be seen, in comparison re, it with Ro(r), to be a rapidly oscillating function. At large separations, Y appears as a plane wave of sinosoidal form with a de Broglie wavelength b where p is the fragments' relative momentum.see that On comparing eqn (14) and (15) we 271 4m = - aA* This allows the interpretation of qm as the number of de Broglie waves in a distance 271/a. (Since typically a 2 A-', this distance is ca. rc A.) Near the potential minimum, r re, the wavelength is shortened as the fragments, A-B + C, experience the attractive potential of the Van der Waals bond and increase their relative velocity. Finally, at a shorter separation, r c. r,, the wavefunction abruptly fdls to zero as the fragments fcel the repulsive portion of the intermolecular potential and can approach each other no closer. Thc term Vcouplins of eqn (6) couples the vibrational motion of A-B with motions involving changes in the Van der Waals bond length.Following Herzfeld and Litovitz,' we assume that the interaction is modulated by the A-B vibration only through changes in the nearest neighbour B - - - C distance. Because of the rapid decrease in interaction with distance, vibrational displacement of A is ignored. The result is Vcoup,ing -Z V(Y - XSX) - Y(r) E -CLYX[~ V/dR] (17) which is equivalent to saying that an increase in vibrational displacement, x, of B toward C produces the same energy change as a decrease by the same amount, Ar,330 RELAXATION CHANNELS OF VlBRATlONALLY EXCITED MOLECULES of the Van der Wads bond length. The portion of the coupling term of eqn (17) which is a function of r is the derivative 1. (18) dJ//dj- ~ -2111D,[eLZ"('-'c) - e-fl(r-rt) The fraction of the vibration of B toward C is weighted by the masses through a = m,l(m, + m,).(19) The angle A-B makes with the axis connecting the centre-of-mass of A-I3 with C is 0. The component of the vibrational displacement x of A--B pointing toward C depends on this orientation and is given by the steric factor s = coso. (20) The matrix element which we need to evaluate is the two-dimensional integral where for harmonic oscillator vibrations of A-B we have with the reduced mass of A-€3. ally 5 * 7 and the final form for the vibrational predissociation rate becomes Fortunately the radial portion of the integral of eqn (21) has been solved analytic- 1 a s a D,<x)z [ u,!(2d - LI, - I ) ! (2d - 217, - 1) - 1 - fi-t4=2 2 2 2 TVy d n= 1 x c r " - 3Y + qA1> x (t(d - 4 - +Y 3.q;1/2d)2exp(--xq,) 124) with d an integer. This equation can be solved in a few minutes with an electronic hand calculator. There are only two intermolecular parameters, the well steepness given by a and its depth D,, needed to evaluate eqn (24), The initial orientation angle of A-B* with respect to C in A-B* a - C is contained in s = cos0. The only other physical quantities involve masses and the vibrational frequency of A--B*. Lifetimes for a variety of A-El* * * - C complexes axe presented graphically in fig. 3. The numbers are not to be taken literally. In the first place the relaxation channel for these calculations is restricted to V-T. Secondly, the lifetimes depend sensitiveIy on the (not well known) steepness parameter a.Uncertainties in a by several per- centage points can often result in lifetime changes by orders of r n a g n i t ~ d e . ~ , ~ Finally, the shapes of the A--B* t C and A-I? + C surfaces may be significantly different contrary to our assumption. The Iifetirnes are seen to be correlated with (2pvAEv-T)+/uh, which is proportional to the translational momentum of the fragments. We shall calf this the mornentum-gap correlafion.10 It is the effectiveness of the overlap [through the coupling term of eqn (17)J between the &(r) continuum wavefunction with the R,(r) bound-state wavefunction that determines the vibration predissociation rate in the Golden Rule expression of eqn (6). The rate then depends on a type of Franck-Condon factor in the radial coordinate.If the fragments fly away at a high momentum the de Broglie wavelength of Rm(r) QuaIitativeIy, a pattern does nevertheless emerge from fig. 3 .G . E . EWING 331 will be short and the many oscillations of this wavefunction will nearly cancel with the slowly changing Ro(r) wavefunction, as in fig. 2. By contrast, a long de Broglie wavelength for R,(r) when the fragment momentum is low can allow constructive interference with R,(r) and rapid predissociation will ensue. Franck-Condon-type factors in the radial coordinate are then responsible for the momentum-gap correlation. 1 O'O 1 o5 -! .C E U I O - ~ do 5 10 15 20 25 30 h % ~ v - d+lah FIG. 3.-The momentum-gap correlation for vibrational predissociation lifetimes for t', = 0 of A-B* - * - C by the V-T channel calculated from eqn (24).Adapted from ref. ( l o ) , (8), (16) and (22). For 1," - a He with ma11 reduced mass (i.e. small pv) and low vibrational frequency [i.e. small AE,,, by eqn (13)] the lifetime is short, ca. lo-" s. is in reasonable agreement with the experimental estimates of lifetime.12 By contrast, the theoretical calculation for HCI* - - * Ar predicts an exceedingIy long lifetime (years) by the V-T channel. The reason for this longevity is that both the reduced mass and the vibrational frequency are large and consequently the momentum gap is large. The predicted lifetime for FH* - - - FH, N,O* C2H4 Fdll in the middle of the correlation for relaxation by the V-T channel. However, experi- ments 2 ~ 3 ~ 1 3 3 1 4 OR these excited complexes indicate lifetimes vastly shorter than those on fig. 3.It seems natural to turn to other relaxation channels to explain these results. This theoretical result * H20 and C,H4* 3 . VIBRATION-VIBRATTON RELAXATION CHANNELS We begin this section by comparing the V-T and V-V relaxation channels of Excitation of the chemical bonds is indicated by the In the experiments performed on the dimer the v3 If the excited complex relaxed by the V-T channel vibrationally excited (N,O),. usual notation:15 N Z O ( ~ 1 ~ 2 ' ~ 3 ) , mode at 2 200 cm-' was excited.13 we would write NIO(001) * * N20(000) 2 N,0(000) + N20(000) -t AE,-., (25) corresponding to a crossing from the first to the second column of fig. 4. The life- time estimated for this V-T channel from fig. 3 and detailed caIculations l6 is z % lo6 s.332 RELAXATION CHANNELS OF VIBRATIONALLY EXCITED MOLECULES 2000 d ' 1000 2 \ 8 h c: aJ 0 -1000 FIG. 4.-Energy states for vibrational energy flow of N20* * * NzO.Adapted from ref. (16) and (201. This long lifetime is a consequence of the large momentum gap. tation energy internally by the V-V channel. However, intepre- The momentum gap can be closed considerably if one of the fragments accepts of the experimental work l3 shows < T/S < lo-'. We can have, for example, (26) This channel is illustrated schematically by a crossing from the first to third columns of fig. 4. The momentum gap now is AEV-V = WA-B - 0, + Wvv - W, (27) where W, is the vibrational energy retained by C*, W, corresponds to v1 = 1285 cm-l.] s, a value consistent with the experiment.[For our example of N,O(IOO), A calculation by this channel l6 reveals z x The V-V channeI Iowers the momentum gap and consequently there is a better overlap of the R,(r) and R,(r) wavefunctions, and the radial Franck-Condon factor increases enormously. However, an additional Franck-Condon factor has been introduced. This is a vibrational factor analogous to {x) in the deactivation of A-€3" to A-B. It involves the transition matrix element for the excitation in eqn (3) of C to C*. A coupling term analogous to that of eqn (1 7) is needed. The forms that the coupling term may take and the corresponding matrix elements are considered el~ewhere.'~~'~ Another type of V-V channel may be important for excited Van der Waals molecules which preceeds predissociation as suggested by Gough." This is given in general by eqn (4).Consider again the example of N,O* - - - N20 laid out in fig. 4. Notice that the t', = 0 level of N,0(001) - * - N,0(000) is nearly isoenergetic with u, = 5 of N,O(OOO) - - - N,U(lllO>. We may represent this mixing 2o by [N,0(001) - * - N,O(OOO)],~=o +- [N20(OOO) * - * N20(1 110)]uy=5. (281G . E . EWING 333 This could be followed by the predissociation [N20(000) * - - N20( 1 1 = &- N,O(OOO) - 1- N20( 100). This predissociation is expected to be much faster than that by eqn (26) since the u, = 5 dimer has 5 nodes in the Van der Waals wavefunction R5(r) and a favourable overlap with &(r) will result. The mixing of eqn (29) is just one example of many possible.There are three low-frequency (ra. 10 cm-') bending modes of (N20), against the Van der Waals bond l7 and many near-coincidences with [N,0(001) . * . N,O(OOO)J,y = are expected. [Considering all the levels populated in (NJ)), at low temperatures, the density of states gives rise to lo4 overlapping spectral lines in a band- width in only 1 cm-'.l7] The picture that emargcs from this process is that energy flows throughout N20* * a * N,O involving a variety of near-isoenergetic vibrations of both chemical and Van der Waals bonds. One of these energy states, which has an efficient relaxation channel (e.g., [N20(000) - N,0(1 l'O)]vv = 5 } , then flies apart to produce the fragments. One might object l o mechanisms of this type in other molecules because it requires too many coincidences to bring the interacting levels into resonance.However, the more atoms a complex has, the more likely it is that reson- ances will occur. While V-V relaxation is a likely channel for many excited Van dcr Waals molecules, it is closed for complexes with no accessible internal vibrational levels. The V-T,R channel remains as the final possibility for relaxation of vibrationally excited com- plexes. 4. V I B R AT1 0 N- ROTATI ON RE LAX AT10 N C H ANN E LS The importance of the V-T,R channel was first explored numerically for HCl* - . . Ar and presented at the last Faraday Discussion dcaling with Van der Waals mole- cules.2' For this relaxation the most efficient channel places H-Clt into thc J - 15 rotational level. End-over- end tumbling of the fragments, H-Clt t- Ar, about their common centre-of-mass at an angular momentum state of 15 but directed in an opposite sense to the J - - I5 state of the H-Clt molecular rotation satisfies conservation rcquircments.The life- time cstimate by the V-T,R channel is z = I s, in contrast to that given in fig. 3 for the V-T channel of z M 10" s. The reduction in lifetimes on going from the V-T to the V-T,R channel is a consequence of the lower momentum gap and a more favour- able Franck-Condon overlap in the radial coordinate. The translational momentum of the fragments is lowered because is reduced to AEv-T,R by the rotational energy to place HCI into i t s J = 15 level. However since changes in angular momen- tum must occupy the relaxation of eqn (30) angularly dependent Franck-Condon factors cornc into play, which tend to increase lifetimes.The most cfficient channel for relaxation is then a compromise between Franck-Condon factors in both radial and angular coordinates. We then find that fragments fly away both translationally and rotationally hot. For a fragment with a small moment of inertia (ix. HCI), large rotational energy may be accepted with only modest changes in angular momentum and we expect that the relaxation channel may be nearly pure V-R (as for HCI* . - - Ar). However, for a fragment with a large moment of inertia (i.e. I,), it is difficult for it to accept much rotational energy without large changes in angular momentum. We then expect relaxation to proceed through a nearly pure V-T channel (as for 12* - - - He).The magnitude of the anisotropic intermolecular forces is also an important334 RELAXATION CHANNELS OF VIBRATIONALLY EXCITED MOLECULES consideration in these calculations. While H, or D, within (H2)2 or (D,), have small moments of inertia, the weak anisotropic forces binding the complexes provides little coupling for changes in rotational energy of the vibrationally relaxed fragments. Consequently the V-T,R channel tends to be suppressed and estimates of predissoci- ation based on the V-T channel in fig. 3 are qualitatively correct. While the aniso- tropic forces binding (N2)2 are relatively more important than those of (H2)2 or (D2)2, the greater moment of inertia of N2 tends to reduce the importance of V-T,R relaxation. Again the estimate of fig.3 for a slow relaxation by a dominantly V-T channel is probably correct. In calculations of vibrational predissociation one must take care in specifying the intermolecular surfaces of both the initial, A-B* * * * C, and final states, A-B + C. It has usually been assumed that they have the same shape and are merely displaced in energy, as in fig. 2. For the V-T,R channel this may be an exceptionally bad assumption for some complexes. Consider, for example, the case of C2H4* . - C2H4 vibrational predissociation. In the experiments l4 on (C2H4)2 the complex is excited into the v7 monomer mode near 1000 crn-l. During vibrational predissociation after the breaking of the Van der Waals bond there is not sufficient energy to excite internal vibrational modes of the fragments.The V-V channel is then closed and so only the V-T or V-T,R channels are open. For the V-T channel the fragments will be produced rotationally cold corresponding to the J = 0, K = 0 state for each monomer, using the usual n0tati0n.l~ The C2H4" - - - C2H4 complex then begins in the u, = 0 level and crosses isoenergetically above the surface labeled ( J = 0, K = 0) + (J = 0, K = 0) for C2H4 + C2H4. The minimum for this surface corresponds to a geometry (see upper portion of fig. 6) of (C2H4)2 locked by anisotropic intermolecular forces with a Van der Waals bond of ca. 400 cm-I and a separation of 4.2 The predissociation lifetime for this channel is estimated 22 to be z M The suggested l4 lifetime for The intermolecular potential surfaces for (C2H4)2 are illustrated in fig.5. s, as shown on fig. 3. C H * . . . C2H4 is, however, z < s. We now consider the V-T,R channel : C,H4* * * C2H4 L CzHjT + C2H4t + AEv-T,R. (31) Here the fragments are to be found in high J , K states. The intermolecular potential surfaces are shifted up by the total rotational energy in the fragments. For some cases the fragments are rotating so rapidly that we imagine the anisotropic forces which are responsible for most of the intermolecular bonding to be averaged out. The well depth is consequently much shallower than for the ( J = 0, K = 0) + ( J = 0, K = 0) surface. In order to experience this type of rotation the molecules must be pushed away from each other in order for their hard-core surfaces, as defined by their Van der Waals radii, to clear.This is shown in the lower portion of fig. 6. The separation of the rotating cores is now 5.4 A, an increase of 1.5 A over the locked geometry configuration. The consequence of a shallower well and greater separation is that the surface for rotationally hot fragments can cross the initial surface of C2H4* - - - CpH4. Since there are a variety of V-T,R channels the fragments may arrive in a number of J , K states, each corresponding to a particular intermolecular potential surface. The crossing surface for the channel ( J = 4, K = 4) + ( J = 4, K = 4) is shown, for example, in fig. 5. (Each C,H4 in the J = 4, K = 4 state is spinning about the C=C axis, which allows the light hydrogen atoms to take up considerable amounts of kinetic energy at a relatively small angular momentum.) With curve crossing of intermolecular potential surfaces, the Golden Rule calcu-G .E. EWING 335 I000 800 6 00 400 3 I -\ e 2 200 5 0 -200 -400 I I I I 3.0 4.0 5.0 6.0 r/A FIG. 5.-htermolecular potential surfaces for C2H,* - - - C2H4 and CZH4 4- C2H4. Adapted from ref. (22). lation of eqn (6) and the corresponding radial Franck-Condon arguments are in- appropriate. We turn toward the approximate Landau-Zener method of calculating the transition rate for a molecule from one surface to a n ~ f h e r . ~ ' , ~ ~ For the V-T,R channel, where curve crossing occurs near u, = 0, the rate of jumping from one curve to another is given by (32) z-' _- - vP,-, where v = W,/h is the zero-point osciIlation frequency characterizing motion against the Van der Waals bond.The probability that a transition wilI occur from the upper to the lower curve is PI-, = (271/hl~,)v21"/ls, - &I. (33) The slopes of the upper and lower curves at the crossing point are S,, and S,. The effective velocity of motion against the Van der Waals bond near the crossing at separation r, is I),, obtained from +pYrc2 = Wo. According to the Landau-Zener picture vibrational predissociation depends on the frequency v with which A---B* - - - C on the upper curve crosses the lower curve and the probability Pl-u that each crossing can result in a transition to A--Bt + Ct. A high probability requires a slow passage c, at the crossing point and compatible slopes S1 and S, The translational and vibrational coordinates must also be effect-336 RELAXA'TLON C H A N X E L S OF VIERATIONALLY EXCITED MOLECULES FIG.6.-Geometry for locked CZH4 - C2H4 (upper part) and free rotor C2H4 + CZH4 {lower part). Adapted froin ref. (22) and (23). ively mixed through Vl-u. The Golden Rule description, eqn (6), on the other hand, depends essentially on a quantum-mechanical overlap of wavefunctions in the radial coordinate. The matrix element mixing the two curves at the crossing point is VIu = <V;')I vcoupling(r = ~ c ) ] ~ I p ) > - (34) Vl"' - VI(X)@O(~) (35) The upper-curve wavefunction is given approximately by where q1(x) describing A--B* is defined earlier and Q(0) is the orientational wave- function of A--B against C within the complex. For dimers with a locked geometry like (C2H412 or (HF),, @,(O) describes a ground-state bending motion against an anisotropic intermolecular force.The lower-curve wavefunction is written Y 4 O ) V O ( W J ( 0 ) (36) where again po(x) describes the relaxed A--B fragment. For high J states the wave- function mJ(0) gives the near free-rotor bchaviour of the spinning fragments A-Bt + The coupling term of eqn (34) resembles eqn (17) except that it is evaluated at the ct. crossing point and becomes Ycoupling(P. = r,) = --MSXS,. (37) The entire matrix element now is with the angular portion. K u = - ~<S>J.O(X>&, (S>J,O = S ~ J ( ~ > S ~ ) , ( ~ > c - l ~ Estimates for { s } ~ , ~ may be foundG. E . EWING 337 Numerical evaluation 22 for C2H4* * - C2H4 predissociation by the V-T,R channel leading to C2H4 (J = 4, K = 4) + C2H4 (J = 4, K = 4) gives z w 10-lo-lO-ll s.This is clearly a favoured channel over the V-T process of fig. 3 which gave T M s. We must realize that there are a great variety of V-T,R channels possible, many crossing near the u, = 0 level. The overall relaxation lifetime may then fall in the picosecond time scale. FH predissociation. The surfaces appropriate to the V-T channeI are those given in fig. 2. The lower surface misses the upper surface near ZJ, = 0 by 0.4 A. Imagine the surface crossing at this point where the fragments now are FHt + FHt, in excited rotationaI states. The strongly anisotropic hydrogen bond averaged out by the fragments rapidly spinning with respect to each other would be responsible for the curve crossing. Rapidly spinning HF moIecules approach the behaviour of isoelectronic Ne atoms in their intermolecular interactions. A shallow shifted intermolecular potential surface of FHt + FHt crossing the initial FH* * - - FH surface is expected.A numerical estimate 22 of relaxation by the V-T,R channel yields a lifetime of 7 z lo-'' s, a figure more in accord with recent rnea~urements.'~~ In considering Van der Waals molecule intermolecular potential surfaces, the nature of the anisotropy must be considered when evaluating the importance of curve crossing to vibrational predissociation lifetimes. Many Van der Waals moIecules have dominantly isotropic intermolecular surfaces. Anisotropic terms for HC1 * * - Ar, for example, make only a 20% contribution to the total intermolecular surface.26 Thus while a V-T,R channel is important to the relaxation process 21 for this complex, in order to lower the momentum gap, curve crossing will not occur and consequently long reIaxation times are expected.Likewise, curve crossings for vibrational pre- dissociation in (D2)2r (NJ2 or I, - - * He are not expected. For some molecules with strong anisotropic forces the V-T,R channel may still not be important. Consider (N20)2 locked into a parallel geometry by quadrupole forces. Curve crossing may result for fragments placed in rotational states. How- ever, since the moments of inertia of the monomers are large, the J states needed to average out the anisotropic portion of potential functions will need to be very high. Consequently the angular portion of the matrix element of eqn (39) wilI be very small and reduce the relaxation rate considerably.For N,W - - N,O relaxation, the V-V channel probably dominates. We may also apply the curve-crossing model to FH* - 5 . PROPENSITY RULES The guide to locating efficient vibrational predissociation pathways of Van der Mraals molecules is obtained by learning which relaxation channels can most effectively lower the translational momentum of the fragments. The guide essentially parallels that already known for relaxation pathways of vibrationally excited molecules by collisions in the gas phase.27 (1) The V-V channel: When either fragment, A-B or C , can accept energy into its internal vi brationaI levels during vibrational predissociation, an efficient relaxation pathway will result.(11) The V-T,R channel: If the V-V channel is closed, then the fragments may accept energy into their rotational motions. This can be efficient for a fragment with a small moment of inertia since useful amounts of rotational energy may be accepted with only modest changes in angular momentum. When anisotropic forces binding the complex are dominant, then curve crossing of the initial and final intermolecular surfaces is possible, providing an unusuallj open relaxation channel. Here then are the propensity rules:338 RELAXATION CHANNELS OF VIRRATIONALLY EXCITED MOLECULES (111) The V-T channel: When the moment of inertia of each fragment is large and the anisotropic forces binding the complex are small, the V-T,R channel closes and the products of vibrational predissociation will tend to be translationally hot.W. Klemperer, Ber. Bunsenges. Phys. Chem., 1974, 18, 128. T. Ellenbroek, J. P. Toennies, M. Wilde and J. Wanner, J. Chem. Phys., 1981, 72, 3414. J. Jortner and A. Beswick, Adv. Chem. Phys., 1981, 47, 363. C. A. Coulson and G. N. Robertson, Proc. R. SOC. London, Ser. A , 1974, 337, 167. J. Beswick and J. Jortner, Chem. Phys. Lett., 1977, 49, 13. G. Ewing, Chem. Phys., 1978, 29, 253. G. Ewing, J. Chem. Phys., 1980,72, 2096. K. F. Herzfeld and T. A. Litovitz, Absorption and Dispersion of Ultrasonic Waves (Academic, New York, 1959). ’ J. Lisy, A. Tramer, M. Vernon and Y. T. Lee, J . Chem. Phys., 1981, 75, 4733. lo G. Ewing, J . Chem. Phys., 1979, 71, 3143. l1 J. A. Beswick, G. Delgado-Barrio and J. Jortner, J . Chem. Phys., 1979, 70, 3895. ’’ K. E. Johnson, L. Wharton and D. H. Levy, J . Chem. Phys., 1978, 69, 2719. l3 T. E. Gough, K. E. Miller and G. Scoles, J. Chem. Phys., 1978, 69, 1588. l4 W. R. Gentry, M. Hoffbauer and C. Giese, Symposium on Molecular Beams, 1979, Trento Italy; M. P. Casassa, D. S. Bomse, J. L. Beauchamp and K. C. Janda, J. Chem. Phys., 1980,72, 6805. l5 G. Herzberg, Iafiared and Raman Spectra (Van Nostrand, Princeton, 1945). l6 D. Morales and G. Ewing, Chem. Phys., 1980, 53, 141. l7 L. S. Bernstein and C. E. Kolb, J. Chem. Phys., 1979, 71, 2818. J. A. Beswick and J. Jortner, J. Chem. Phys., 1981, 74, 6725. l9 T. Gough, personal communication. ” G. Ewing in Potential Energy Surfaces and Dynamics Calculations, ed. D. G. Truhlar (Plenum, New York, 1981), p. 75. M. S. Child and C. J. Ashton, Faraday Discuss. Chem. Soc., 1976, 62, 307. ” G. Ewing, Chem. Phys., 1981, 63, 411. 23 Ad van der Avoird, P. E. W. Wormer, F. Mulder and R. M. Berns, Top. Current Chem., 1980, 24 L. D. Landau and E. M. Lifshitz, Quantum Mechanics (Pergamon, Oxford, 1977). 25 H. Eyring, J. Walter and G. Kimball, Quantum Chemislry (Wiley, New York, 1957). 26 S. L. Holmgren, M. Waldman and W. Klemperer, J. Chem. Phys., 1978, 69, 1661. 27 J. D. Lambert, Vibrational and Rotational Relaxation in Gases (Clarendon Press, Oxford, 1977); Y . T. Yardley, Znfroduction to Molecular Energy Transfer (Academic Press, New York, 1980). 9 3 , l .