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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 .

ISSN:0301-7249    

DOI:10.1039/DC9827300325

出版商:RSC    

年代:1982

数据来源: RSC

摘要:

Faraday Discuss. Chem. SOC., 1982, 73, 339-355 Predissociation of Weak-anisotropy Van der Waals Molecules Theory, Approximations and Practical Predictions BY ROBERT J. LE ROY, GREGORY C. COREY AND JEREMY M. HUTSON * Guelph-Waterloo Centre for Graduate Work in Chemistry, University of Waterloo, Waterloo, Ontario, Canada N2L 3Gl Received 25th January, 1982 Widths and energies for predissociating levels of Hz-inert-gas Van der Waals molecules are calculated by solving the close-coupling equations for accurate potential-energy surfaces. These exact results are used to test several approximate schemes previously employed for calculating level energies and widths. None of these is found to be entirely satisfactory, and a new method is proposed which gives more accurate results. In this new method (SEPTOC) the secular equations are solved in matrix form for the closed-channel manifold, and the open channels are treated by perturbation theory.The limited form of SEPTOC tested here gives very promising results. The level widths are found to be very sensitive to the potential used, particularly in the repulsive region. For Hz-Ar, two different potentials which both reproduce the level energies are found to give level widths which differ by ca. 30%. This is the first conclusive demonstration that measurements of predissociation widths can give potential information different from that yielded by the level energies. 1. INTRODUCTION Within the electronic Born-Oppenheimer approximation a polyatomic Van der Waals molecule can predissociate through one of two mechanisms.The simplest of these is the tunnelling-type " rotational predissociation " of a quasibound level which lies above the dissociation limit but behind a potential-energy barrier on the effective radial potential for the system. The second mechanism, the one of interest here, is associated with conversion of internal vibration-rotation energy of one of the mole- cules forming the complex into relative translational energy of the separating frag- ments. This process is induced by the same anisotropy and/or internal stretching coordinate dependence of the intermolecular potential which causes rotational and vibrational energy transfer in molecular collisions, and it has been suggested that it would provide a good way of determining those parts of the intermolecular potential which drive these inelastic processes.While detailed theoretical studies of this phenomenon were first reported some 15 years the first unambiguous observations of it did not come until a decade later.5-7 These experiments then sparked a rapid growth of theoretical work in this area *-I1 in which particular attention was paid to approximate schemes for predicting the energies and widths of predissociating levels and their use in qualitatively inter- preting and extrapolating beyond the growing body of experimental data. [Ref. (1 1) presents a detailed review of much of this work.] However, while use of these methods has provided substantial insight into the nature of the predissociation process, their reliability and the sensitivity of their predictions to the nature of the assumed potential-energy surface has not been fully explored.Moreover, since very little is known about the details of the potential-energy surfaces characterizing most systems * NATO Postdoctoral Research Fellow.340 PREDISSOCIATION OF V A N DER WAALS MOLECULES of interest, such studies can at most yield qualitative information about trends such as the dependence of predissociation lifetime on the “ energy gap ” (i.e, the amount of internal energy released) or on the vibrational quantum number. In spite of the interest in this predissociation phenomenon, only very limited predic- tions have as yet been reported 12*13 for the family of atom-diatom complexes whose potential-energy surfaces are most accurately the molecular-hydrogen- inert-gas (Ar, Kr and Xe) bimers.Moreover, most of those predictions l2 were not based on the best available potential-energy surfaces for these systems. These species are of particular interest because of the unique conjunction of three conditions: (i) accurate three-dimensional potential-energy surfaces have been determined for them from experimental data ; I4-l6 (ii) line broadening due to internal-rotational pre- dissociation has been observed; l7 and (iii) they are among the easiest systems to treat theoretically. This last condition facilitates the present use of close-coupling cal- culations to obtain accurate energies and widths of predissociating levels. These then allow comparisons with experiment to provide a quantitative test of the potential- energy surface used, and provide accurate results for testing various approximation schemes. The present paper reports calculations of the energies and widths of a variety of metastable levels of molecular-hydrogen-inert-gas complexes, with particular attention being paid to H,(v = 1, j = 2)-Ar.The spectroscopically observed l8 excited states of these species all correspond to metastable complexes formed from vibrationally excited hydrogen, and the dominant predissociation mechanism is BC(U = 1,j)-M + BC(V = 1,j’) + M (1) where j ’ = j - 2 for BC = H, or D, and j -.j’ = I and 2 for BC = HD. This process of ‘‘ internal-rotational predissociation ” is merely a special type of vi brational predissociation, the internal rotation of the BC diatom being in essence a very wide angle bending motion.Note that the process of internal-rotational predissociation defined by eqn (1) should not be confused with ordinary rotational predissociation, which has historically been defined l9 as the tunnelling type of predissociation men- tioned at the beginning of this section. The following discussion begins with a description of the potential-energy surfaces used in the present study. This is followed by a detailed comparison of a variety of methods for predicting the energies and widths of vi brationally and/or internal- rotationally predissociating levels of Van der Wads complexes. The final section then examines the potential dependence of predictions of this type, summarizes the information obtained about the molecular-hydrogen-inert-gas systems, and attempts to delineate the utility of experimental measurements of predissociation level widths, 2.POTENTIAL-ENERGY SURFACES FOR THE HZ-INERT-GAS SYSTEMS The calculations reported in the present work were all performed on potential- energy surfaces described by the BC3(6,8) function of Carley and Le R o ~ , ~ ~ , ~ ~ which is expanded as 3 where R is the distance between the atom and the diatom centre of mass, the diatomR. J . L E ROY, G . C . COREY AND J. M. HUTSON 34 1 bond length r is represented by the stretching coordinate 0.766 6348 A, and 6 is the angIe between R and the axis of the diatom. strength functions V&) all have the form : = ( r - ro)/r0 with ro = The radial D(R) = exp[-4(R,/R - for R < Ro = Rto = 1 for R 3 Ro (4) and the constants AAk and Cik are defined by the values of the depth Rtk of the minimum of T//k(R).definition of D, appearing there should read: and position {See eqn (74) and (75) of ref. (16); note that the D, F [dlfi(D(R))jdR]R=,~~.) The parameters Rtk, . . . etc.) defining the H,-Ar surface of Carley and Le Roy are listed in table V of ref. (16), and those for H2-Kr and H,-Xe are found in ref. (15). However, in practical calculations it is the vibrationally averaged (over c) forms of this surface (t.,jl V(R,<,O)[v',j'> which are relevant. Since the exponent parameters j ? ~ are independent of k, the resulting diagonal vi brationally averaged potentials are V(t.,jlR,O) = (t.,jl Y(R,<,O)[C,j) = FG(t.,jlR) + V2(Ll,jIR)P,(COS 0) ( 5 ) where the vibrationally averaged anisotropy strength functions V>.(u?jlR) have the same form as eqn (3) except that their linear scaling parameters are: for Q = A , C8 and C, where (~7,jlS~I~',j:: i s the expectation value of ck for the isolated diatom in vibration-rotation state ( u j ) .For molecular hydrogen these expectation values are readiIy calculated from the accurately known diatom potential-energy curves.2o The parameters AA, ci and ci defining the ensuing vibrationally averaged potentials for various types of H,(v,j)-Ar complexes are summarized in table I. The analogous parameters of the rigid-diatom potential defined by the k = 0 contributions to eqn (2) are also listed there. TABLE 1 .-PARAMETERS CHARACTERIZING THE VJBRATIONALLY AVERAGED CARLEY-LE ROY Ro = 3.5727 A BC3(6,8) POTENTIALS FOR Hz(v,j)-Ar; FOR ALL CASES Po = pz = 3.610 A-' AND 25 206 865 1 240 048 142 714 52.31 3.5985 9 177 581 539 933 15 797 7.59 3.843 25 065 503 1 234 302 142 185 52.19 3.5974 8 966 285 527 465 15 660 7.49 3.840 22 929 424 1 149 172 135 051 50.81 3.5772 5 452 225 13 706 320 085 5.42 3.768 22 802 769 1 143 965 134 534 50.68 3.5762 5 273 896 309 564 1 3 574 5.84 3.763 22 633 534 1138 176 134 500 50.87 3.5727 4 814 552 282 440 1 3 500 5.72 3.743342 PREDISSOCIATION OF VAN DER WAALS MOLECULES To facilitate comparisons, the parameters and Rt characterizing the depths and positions of the minima of these effective anisotropy strength functions are also listed in table 1.The changes in these quantities from one case to another, and in particular the dependence of E~ and R: on the vibrational quantum number u, provide a clear warning that this vibrational averaging must be done properly if predictions based on these surfaces are to have physical significance (see section 4A below).3 . CALCULATING THE ENERGIES AND WIDTHS OF METASTABLE LEVELS OF WEAK-ANISOTROPY ATOM-DIATOM COMPLEXES A. GENERAL CONSIDERATIONS In an exact treatment of predissociation, the energy and width of a resonance state must be deduced from the energy dependence of the total wavefunction (see below). However, the basis of most approximate treatments of predissociation phenomena is a partitioning of the full Hamiltonian for the system into two parts. The first of these, H,, provides a zeroth-order description of the dynamics of the system, and its discrete eigenvalues provide estimates of the energies of all bound and metastable states of the system.The remainder of the Hamiltonian, denoted H' = H - H,, couples the discrete eigenfunctions of Ho associated with metastable levels to the isoenergetic continuum wavefunctions associated with the open channels, leading to predissociation. Within an isolated-resonance approximation, the full width at half maximum of a predissociating level is given by the golden-rule expres- sion 21922 where tyb is the unit normalized bound-state eigenfunction of H, associated with the metastable state and tyi is a continuum eigenfunction of Ho associated with open channel i, normalized to a delta function of energy. When there is more than one open channel the total width is a sum of partial widths r i and the branching ratios are determined by the ratios of the contributing partial widths.In general there are three types of errors associated with this partitioning. The first of these arises when Ho does not provide a good description of the dynamics of the system; this is the dominant source of error in the SFD, BFD and BL approxim- ation schemes considered in sections 3C and 3D. The second is caused by neglect of the effect of the open channel(s) on the level energy; a way of estimating these level shifts is described in section 3F. The third type of error is the neglect of the effects of higher-order coupling via the open channels. A procedure for treating the second- order coupling among the open channels is described in ref.(1 1); however, it is not relevant to the present calculations of internal-rotational predissociation since the present model problem has only one open channel, B. EXACT (CLOSE-COUPLED) CALCULATIONS FOR A MODEL PROBLEM In the close-coupling formulation predissociating states are treated as scattering resonances and are located by solving the close-coupled scattering equations on a grid of energy values. Near a resonance, the phases of S-matrix elements change rapidly with energy. In the particular case where there is only one open channel, the asymp- totic phase shift increases by z over the width of the resonance, while in the many- channel case similar behaviour is shown by the S-matrix eigenphase sum.23 Only the single open channel case is considered here.R .J . LE ROY, G . C . COREY AND J . M. HUTSON 343 After separating out the motion of the centre of mass, the full Hamiltonian for an atom-diatom system may be expressed as : H(R,r) -(h2/2,L!)R-'( a2/ L'R2)R + 12/Z,UR2 + V(R,<,O) + f f d ( r ) (8) where R is the axis of the complex, Y that of the diatom, p = m,md/(m, + md) is the effective reduced mass associated with the interaction of an atom of mass ma with a diatom of mass md, l2 is, the angular-momentum operator associated with the rota- tion of the unit vector R and Hd(r) is the vibration-rotation Hamiltonian for the isolated diatom. The exact eigenfunctions of this Hamiltonian may be expanded as 1 " U" where J and M are the quantum numbers for the total angular momentum and its space-fixed projection, and O! is an index identifying a particular eigenstate of the system.? The functions {quJ(r)} are the wavefunctions for the st!etching motion of the diatom, (x;f,"(R)} are the radial channel wavefunctions, and (@i(R,E)} are a complete set of angular basis functions with quantum numbers collectively identifiedlas a.In the Arthurs and Dalgarno space-fixed r e p r e ~ ~ n t a r i o n , ~ ~ a = {j,l} and @i(R,t) is the total angular-momentum eigenfunction g;Y(R,E) which is defined as the appropria{e linear combination of the products of spherical harmonic functions Y i m j ( P ) Yrml(R) for mj + m, = M . Alternatively, in the body-fixed coordinate system, a = {j,Q}, the operator 1' is replaced by ( J - j ) 2 , and the angular basks functions are earity- adapted linear combinations of the products Y,,(E)DJ,M(R), where D ~ M ( R ) are the usual symmetric-top wavefunctions and Qh = (Klh is the magnitude of the pro- jection of the total angular momentum onto the axis of the complFx.In either case, substituting eqn (9) into eqn (8), premultiplying by [ ~ - ~ 4 3 , , ~ , ( y ) ~ ) ; i . ( R , f ) ] * and integrat- ing over r and the angular variables yields the usual set of coupled equations for an at om-d ia t om system, I- (h2/2p)d2/dR2 + U(d,a' ; v',a' ; J IR) -- EL}x!&(R) where Ei is the total energy of the system and U(u',a',"',a'' ; JIR) are the matrix elements of the angular and diatom-stretching wavefunctions with the operator U, defined as U(R,r) = Hd(r) + l2/2PR2 + v(R,<,o). (1 1) Note that in either angular basis the matrix elements of &(r) are diagonal and equal to the vibration-rotation energies of the free diatom (Ed(u,j)).Similarly, in the space-fixed representation the matrix elements of 12/2pR2 are diagonal in v',.j' and I' and equal to the simple centrifugal term "(1' + l)h2/2,uR2. The calculations reported in the present section describe the internal-rotational predissociation of the lowest (n = 0) vibrational levels of H2(1,2)-Ar and were per- formed using a realistic model potential subjected to three restrictions: (i) The matrix elements of U are assumed to be zero for u" f 0 ' ; perturbation-theory arguments show that this assumption does not significantly affect the predicted level energies and widths. (ii) The dependence of the vibrationally averaged potential onj' is neglected; the small differences between the ( u , .j = 0) and (21, j = 2) potentials seen in table 1 t In general, u is defined by the total energy of the system and its ( j i- 1 1- .I) parity; however, for weak-anisotropy molecules it may be associated with the set of " zero-coupling-limit " quantum numbers: v , j , n, 1 and J.344 PREDISSOCIATION OF VAN DER WAALS MOLECULES indicate that this too should introduce little error, (iii) The coupling to closed channels corresponding t o j >j,,, = 2 is assumed to be negligible; while this is not quite true (see section 4), it is perfectly legitimate to define a model problem in this way.As a result of these three restrictions, the potential contribution to the matrix element of Uis where 02(at,atf ; J ) is the matrix element of P2(cos 0) between the angular basis functions a;, and a;,.. For the potential-energy surface and Hamiltonian described above, even ( j + I + J ) parity metastable states associated with ( V J ) = (1,2) can dissociate into only one open channel (u = 1, j = 0, J = I ) . In this case the scattering S matrix consists of the single term S = exp[2iq(E)] where q(E) is the phase shift associated with the asympto- tic behaviour of the radial channel wavefunction for the open channel. In the present work, the coupled equations (10) were solved in space-fixed coordinates using the method of Sams and K o ~ r i , ~ ~ and the resonance energies and widths determined by fitting the resulting phase shifts to the Breit-Wigner line-shape plus a linear back- ground phase: The separations between the thresholds for the open and closed channels were taken as Ed(u,2) - E,(u,O) = 336.7 and 354.24 cm-I for u = 1 and 0, respectively, while the q(E) = b + cE + arctan [r/2(Er - E)].(13) TABLE 2.-cOMPARISON OF EXACT (cc) AND APPROXIMATE I2 = 0 RESONANCE ENERGIES OBTAINED F R O M ~ ~ ~ ~ = 2 CALCULATIONS ON THE HZ(1,2)-Ar SURFACE OF TABLE 1. THE ZERO OF ENERGY IS DEFINED AS T H E j = 2 ROTATIONAL ENERGY THRESHOLD. 1 J EYC/cm-’ y =SFD BFD BL SE SEPTOC 0 2 1 1 3 2 0 2 4 3 1 3 5 4 2 4 6 5 3 5 7 6 4 6 8 -23.238 -22.653 - 22.237 - 20.727 - 19.31 1 - 20.078 - 17.243 - 15.943 - 16.823 - 12.828 - 11.580 - 12.523 - 7.490 - 6.265 - 7.249 - 1.306 -0.113 -1.109 0.209 -0.018 0.111 - 0.074 -0.160 0.072 - 0.050 - 0.068 0.053 - 0.052 - 0.037 0.041 - 0.048 - 0.023 0.035 - 0.042 -0.017 0.028 3.453 2.412 5.780 1.265 7.990 1.157 10.074 1.076 - 0.074 - 2.461 - 4.694 a - 6.791 0.996 a -- 8.7 16 a a - 0.483 - 0.055 -0.358 - 0.005 0.41 7 0.027 0.292 -0.209 0.057 0.210 -0.158 0.053 0.150 -0.1 17 0.043 0.100 -0.083 -0.273 0.004 0.004 0.003 -0.009 - - 0.001 0.002 - 0.003 0.000 0.003 - 0.003 - 0.001 0.002 - 0.002 -0.001 0.001 - 0.002 0.000 0.002 0.029 0.002 0.023 - 0.002 0.035 0.01 9 0.001 0.032 0.01 7 0.005 0.029 0.01 5 0.006 0.028 0.01 1 0.006 0,022 0.008 mean deviation - 0.002 0.82 - 0.022 - 0.000 0.01 6 (10.082) (15.28) (f0.225) (*0.003) (&0.012) This level is incorrectly predicted to lie above the dissociation threshold.R .J . LE ROY, G , C . COREY AND J .M. HUTSON 345 values assumed for the physical constants are all incorporated in the factor h2/2p = 16,857 630/p cm-' A2, where p = 1.918 865 04 arnu for H2-Ar. The encrgics E:c and widths Ycc of the n : 0 metastable levels of H2(I,2)-Ar determined from the close-coupling (CC) calculations are tabulated in columns 1 of tables 2 and 3, respectively. Subject to the three approximations listed above, they TABLE 3.-cOMPARISON OI- EXACT (cc) WITH APPROXIMATE WIDTI-IS (IN C M - ') TOR THE H2( 1,2)-Ar ItESONANCFS CONSIDERED JN TABLE 2 1 J cc SFD BFD BL SE SEPTOC 0 2 1 1 3 2 0 2 4 3 1 3 5 4 2 4 6 5 3 5 7 6 4 6 8 0.0394 0.0530 0.0414 0.1 103 0.0235 0.0420 0.055 1 0.0237 0.041 1 0.0408 0.0219 0.0388 0.03 19 0.0191 0.035 1 0.0244 0.01 56 0.029 8 0.032 0.052 0.039 0.1 12 0.044 0.042 0.064 0.039 0.042 0.049 0.035 0,040 0.039 0.029 0.036 0.030 0.023 0.03 1 0.0 0.0 0.0 0.112 0.0 0.0 0.109 0.0 0.0 0.104 0.0 0.097 0.0 0.087 U U U U 0.001 8 0.036 0.0089 0.109 0.055 0.017 0.077 0.052 0.024 0.058 0.045 0.029 0.044 0.037 0.030 0.033 0.027 0.028 0.026 0.037 0.03 1 0.123 0.005 1 0.032 0.027 0.0072 0.034 0,022 0.01 1 0.033 0.01 3 0,0093 0.033 0.0087 0,0073 0.025 0.05 1 0.058 0,050 0.112 0.03 1 0.049 0.058 0.033 0.047 0.044 0.03 1 0.044 0.035 0.028 0.040 0.027 0.022 0.033 ~ ~~ a This lcvcl is incorrectly predicted to lie abovc the dissociation threshold.are exact results for these levels of the Carley--Le Roy BC,(6,8) potential-energy surface. The additional entries in these tables correspond to the resonance energies and widths predicted by the various approximate methods described below.Note that states of odd ( j -I-- 1 -1. J ) parity cannot predissociatc by internal rotation since there are no open ( v j ) = (1,O) channels of this parity. c. S E P A R A B L E WAVEFUNCTION APPROXIMATIONS The simplest approximate methods arc those in which Ho includes only the diagonal matrix elements of U, so that V(R,&U) is replaced by a set of channel potentials V(v,a; v,a; J[R) dcpending only on R. Jn either the spacc-fixcd or body-fixed rep- resentations, this approximation involves neglect of the coupling terms on the right- hand side of eqn (10) and hence causes the expansion of eqn (9) to collapse to a single term. If this is the only approximation made, the result is the " distortion " approx- imation of Levine et al.z-4 The resulting coupling operator H' then comprises the off- diagonal matrix dements of U appearing on the right-hand side of eqn (10).As might be expected, results obtained using these methods are quite sensitive to the choice of angular basis. This point is dramatically illustrated in tables 2 and 3, where the space-fixed distortion (SFI)) predictions are seen to be greatly superior to those obtained using the body-fixed distortion (BFD) approximation. In both cases, the dominant source of error is the neglect of the coupling among the closed channels346 PREDISSOCIATION OF VAN DER WAALS MOLECULES due to the off-diagonal matrix eIements of the operator U, and the reIative quality of the predictions may be understood from the following examination of the diagonal and off-diagonal matrix elements of this operator.In both space-fixed and body-fixed representations, the channel potentials appear- ing in Ho are the diagonal matrix elements of U, and it is the ratio of the spacing between these curves to the magnitude of the off-diagonal coupling functions which determines the relative accuracy of the various approximations. In the space-fixed description of the present model problem (with u = u’ = v” = I), the matrix elements of U are denoted U#’(JIR). For the J = 2 levels of H2(1,2)- Ar these matrix elements are plotted in fig. 1 . The upper portion of this figure shows 20 0 -20 3 I E !? 2 -LO G 10 0 -10 ’-1 7 / Ui’ 1 3.0 L.0 5.0 6.0 RIA FIG. 1 ..-Effective potentials and coupling functions for J - 2 states of H2(f ,2)-Ar in the space-fixed representation.The upper part of the figure shows the potential curves, and the lower part the coupling functions due to the potentia1 anisotropy. the functions U{{(J = ZIR), which are the effective radial potentials defining both the SFD level energies (the three upper curves) and the wavefunctions used in eqn (7); the curves in the lower portion of this figure represent the off-diagonal coupling among the closed channeIs due to the potential anisotropy. Similarly, the upper and lower portions of fig. 2 show the analogous diagonal and off-diagonal functions Uh&(JIR) associated with the body-fixed representation; in this case the off-diagonal coupling functions are solely due to the ( J - j ) 2 term in the total Hamiltonian.The results in fig. 1 show that in the SFD approximation the effective potentials are well separated relative to the magnitude of the relevant coupling functions. In contrast, fig. 2 shows that the BFD coupling functions are relatively stronger on theR. J. L E ROY, G . C . COREY AND J . M . HUTSON 347 20 0 - 20 3 I -... 2 -LO 2 C - --1 7.- 7 -- 1- -1 ---- - -. 1 ---T FIG. ?.-Effective potentials and coupling functions for J = 2 states of H2(1,2)-Ar in the body-fixed representation. The upper part of the figure shows the potential curves, and the lower part the coupling functions due to the Coriolis term in the rotational Harniltonian. interval between the classical turning points of the metastable levels, and the closud- channel potentials are much closer together; as a result the zeroth-order BFD eigen- functions provide a fairly poor description of the predissociating states of this system.Thus, the zeroth-order picture associated with the SFD approximation, that the diatom rotates freely within the complex, is much more realistic for weak-anisotropy Van der Waals complexes than the BFD assumption that the complex is a rigid symmetric top. Two other separable wavefunction approximations which have been applied to this problem are the '' central approximation " of Micha lh4 and the '' intermediate approximation " of Beswick and Requena.12 The former is effectively a crude version of the SFD approximation in which the potential appearing in Ho consists only of the vibrationally averaged isotropic term ro(v,jlR).For the H,( 1,2)-Ar levels of tables 2 and 3 it yields resonance widths roughly the same as the SFD ones, but its resonance energies are typically ca. 0.5 cm-l in error, as they neglect even the first- order splitting of the different J-states associated with a given value of 1. In general it has no advantages over the SFD method and its use should be discouraged. In the '' intermediate approximation " of Beswick and Requena,I2 the angular basis functions characterizing the closed channels for each j were expanded as linear combinations of the corresponding body-fixed basis functions, and defined by the requirement that they diagonalize the matrix Uhn((J1R) at some particular distance K = R"'. However, while this method yielded better results than the BFD approach,I2348 PREDISSOCIATION OF V A N DER WAALS MOLECULES its success depends critically on a propitious choice of the distance lP, and hence may not always be trusted.D, T H E BEST-LOCAL (BL) OR S L M P L E BORN-OPPENHEIMER APPROXlMATlON The most sophisticated of the decoupling schemes which have been applied to the present problem is the " best-local (BL) approximation, originally developed in a space-fixed formulation by Levine et ~ 1 . ~ -4 and rederived using body-fixed coordinates by Holmgren et a1.26 This approach differs from the separable approximations described above in thatfithe angular basis functions appearing in eqn (9) are parametric functions of R, @i(RIR,V"). functions are expanded in terms of a complete set of angular basis functions and are defined by the requirement that they diagonalize the operator U of eqn ( I 1).The resulting channel potentials, here denoted D#(JlR), are then the loci of the eigenvalues of U, By definition, U has no off-diagonal matrix elements in the best-local basis {@i(Rlk,P)}, and the coupling among the various channels arises from matrix elements involving these angular basis functions and the radial kinetic-energy operator ( -h2/2p)8/ aR2. The coupling functions have the form At each value of R these d dR f{j'(JIR) + gf:;'(JI.R) - and may be evaluated as described in ref. (27). In this approach, the level energies are obtained as the single-channel radial eigen- values of the closed-channel potential adiabats u#'(JJR) while the widths are obtained from cqn (7) using radial eigenfunctions associated with those effective potentials and with H' defined by eqn (14).Note that the diagonal potentials and coupling functions used in this approach are the same whether space-fixed or body-fixed basis functions are used to diagonalize U. Moreover, while the designation a = { j J } does not provide an exact description of the BL angular channels, it does suffice to identify them uniquely. For the present model problem, the energies and widths calculated in the BL approximation are listed in column 4 of tables 2 and 3. While they are distinctly better than the BFD predictions, it is perhaps surprising to see that this '' best " local approximation is inferior to the simple SFD method. However, this is readily explained by consideration of the effective potentials and coupling functions plotted in fig.3, which describe the same J = 2 case considered in fig. 1 and 2. The fact that the BL method effectively removes all coupling due to the interaction potential ensures that the adiabats o,jl'(JIR) shown in the upper segment of fig, 3 are well separated. However, the coupling functions ,f{{(JlR) [and also g{[(JJR), not shown] have large-amplitude oscillations near the point where the anisotropy strength function v2(v,jlR] changes sign. This reflects the rapid change in character of the functions (@i(RJR,P)} in the region where the preferred configuration of the complex changes from collinear to T-shaped. I n spite of the magnitude of these oscillations, the BL approximation yields reasonable level energies and widths for this system, because much of this extreme behaviour occurs at distances smaller than the classical inner turning points of the metastable levels.This argument suggests that the BL method should perform even better for the analogous states of H,-Kr and H,-Xe, since the change in sign of their V2(u,jlR) functions occurs at relatively smaller distan- ces than for H2--Ar,14.15 but it augurs ill for H,-Ne, where the sign change is expected to occur relatively farther out. Thus, the accuracy of the BL results shown in tables 2R . J . LE ROY, G . C . COREY AND J . M . HUTSON 349 3.0 4.0 5.0 6.0 N A FIG, 3,-Effective potentials and coupling functions for J : 2 states of H2(I,2)-Ar in the best-local representation. The upper part of the figure shows the potential curves, and the lower part the non-adiabat ic coup1 ing functions. and 3 is partly fortuitous, since it depends on a detail of the potential.Note too that this argument also partially explains the accuracy of the BL level energies obtained in ref. (3) and (4),* as in most of the model potentials used there V,(R) did not change sign. E. T H E SECULAR-EQUATION (SE) M E T H O D The secular-equation method of solving the coupled equations I h is in principle exact in that it chooses H, := Hand proceeds to solve the full set of coupled equations (10). However, the secular-equation method (SE) of Grabenstetter and Le Roy 28 for determining resonance energics and widths is lcss than exact because it represents the radial channel wavefunctions for both closed atdopen channels by a finite number of square integrable basis functions, These basis functions are defined as the exact eigenfunctions of some basis generating potential L’,,(R).When V,(R) is chosen appropriately a small number of these functions can provide an extremely good representation of both open- and closed-channel wavefunctions on the interval where the latter have non-negligible amplitude. 111 thc present work V,,(R) was defined in the same manner as in ref, (28), by modifying the effective isotropic potential by placing a n infinite wall at some distancc R, well outside the classically allowed region for the metastable level in question. Note that the coupling function used there for calculating the level widths is valid for the two-channel case only, and that its use in four-channel calculations [see eqn (1 I ) of r d .(4)] is incorrect. * I n ref. (3) and (4) the present BL method is called the adiabatic approximation.350 PREDlSSOCIATION OF V A N DER WAALS MOLECULES Since the ensuing representation of the open-channel wavefunction(s) cannot satisfy the appropriate scattering boundary conditions, the resonance energy and width cannot be determined in the usual manner from the asymptotic properties of the open- channel wavefunction(s). The SE method circumvents this apparent difficulty by using a Fano-type 22 decomposition of the eigenfunctions of the Hamiltonian matrix to define the metastable-level energy and width In this approach the wall position R, i s used as the variational parameter and the resonance position defined by the energy at which the amplitude or the closed-channel portion of the secular-equation eigenfunction has a local maximum, while the magnitude of this maximum is used to obtain an estimate of the resonance width.In principle the accuracy attained i n SE calculations depends on the number of radial basis functions used in the expansions for the {,y;lt(R)). In the present calcu- lations a basis set comprising the 20 lowest eigenfunctions of V,(R) was used for each case; this was more than enough to ensure convergence. However, in practice a single basis function associated with each channel usually sufficed to yield level. energies accurate to better than ca. 0.02 cm-’. The additional functions were inchded in the expansion in order to provide a realistic representation of the open-channel function, and hence of the width.The results in column 5 of table 2 show that the SE method is by far the most accurate of the approximate procedures for determining level energies considered here. However, the corresponding SE widths shown in table 3 are worse than those yielded by the SFD and BL methods, the latter both being on average some 20‘x too high while the SE widths are on average 40”/, low. The reason for this weakness is not yet clear. On the other hand, the SE method is the only one of those considered so far which mimics the pronounced /-dependence of the exact (CC) widths of levels associ- ated with a given value of J . Specifically, it predicts that the level I = J is usually much narrower than those for I : J 1).This probably occurs because the SE method provides a near-optimum representation of the closed-channeI contributions to the total wavefunction, a fact independently demonstrated by the accuracy of the level energies it yields. Thc procedure presented in the following section therefore attempts to combine this excellent representation of the closed-channel wavefunction with a better treatment of the coupling to the con- tinuum. 2 (the only exception occurs for J F. PERTURBATION-THEORY TREATMENT 0 1 7 T H E OPEN CHANNELS In the Feshbach formulation for compound state resonancqZ1 the presence of dissociative channels causes both broadening and shifting of a metastable level. Ignoring high-order coupling uia the open channels, the resulting width r and shift A are each a sum of A = 2 Re(Zi) where the index i runs over the open channels, r m raY x,(R) is the radial wavefunction associated with dosed channel c in the absence of dissociative channels, and Hire is a matrix element coupling open and closed channels. The outgoing Green’s function Gf(R,R’) is given explicitly by 29R .J . LE ROY, G . C . COREY AND J . M . HUTSON 351 = -n[x$')(R)x$')(R') + ix!')(R)x!,')(R')] , R > R' (18) where xio) and xi') are the regular and irregular solutions of the single-channel Schrodinger equation for (open) channel i, normalized to a delta function of energy. When eqn (18) is substituted into eqn (l7), eqn (1 5) collapses to the simple golden-rule level width expression of eqn (7) with Wb = 2 ~u'j'(r>Q~(R,?)xc(R> c and tpi a simple separable wavefunction for open channel i.Unfortunately, the expression for the level shift, eqn (16), does not simplify in a similar way since the real part of the Green's function is different for R > R' and R < R'. However, eqn (16) may be readily evaluated by numerical integration once the functions xio) and xj') have been determined. The present work used the uniform Airy approximation3' to represent these functions, but they could also be readily generated by direct numerical integration of the single-channel Schrodinger equation. Note too that since the present model problem has only one open channel, the sums over i in eqn (7) and (16) each collapse to a single term. Of the methods considered to this point, only the CC and SE procedures take account of the level shift due to the open channel(s).However, with the exception of the J = 0 case, for which there exists only one closed channel, use of eqn (16) to calculate corrections to the SFD, BFD or BL energies is not worth while, since their main source of error is the still-neglected coupling among the closed channels. On the other hand, as suggested at the end of the preceding section, a very promising approach would be to use an SE calculation on the closed-channel manifold to deter- mine an initial estimate of the level energy and the metastable state wavefunction tpb, and then calculate the level width and the shift due to coupling to the open channel(s) using eqn (7) and (16). This use of a secular-equation (SE) treatment of the closed channels plus perturbation-theory treatment for the open channels (PTOC) is the method identified here by the acronym SEPTOC. In the present SEPTOC calculations for H2(I,2)-Ar, tpb was expanded in terms of the SFD eigenfunctions, with the radial basis set restricted to the n = 0 level for each channel.Similarly, the regular and irregular open-channel wavefunctions xi') and 2%') were generated from the appropriate open-channel SFD potential. As is seen in the last columns of tables 2 and 3, this yields level energies which are significantly better than those obtained in the SFD, BFD or BLmethods, while the corresponding widths are distinctly better than those obtained in any of the other approximate procedures. A significant feature of the residual discrepancies in the SEPTOC energies is that they are virtually all positive.This occurs because the dominant source of error is due to the use of only one basis function in the expansion for each of the closed-channel radial wavefunctions. The improved radial behaviour obtainable in expanding these bases should also lead to improvements in the SEPTOC widths. However, this remains to be tested. G. CONCLUSIONS Several different approximate schemes have been used to calculate energies and widths for predissociating levels of H2( 1,2)-Ar in the preceding sections. The space- fixed distortion (SFD) approximation was the most successful of the simple decoupling approximations, while the SEPTOC (secdar equation with perturbation theory for open channels) method attempted to account for the neglect of coupling between352 PREDISSOCIATION OF V A N DER WAALS MOLECULES closed channels inherent in the separable approximations.The most promising method for future calculations appears to be an extension of the present SEPTOC procedure using a larger radial basis set to represent the Van der Waals stretching functions (xi:(R)). For H,-Ar, it was found that the SFD approximation was greatly superior to its body-fixed analogue (BFD), and this was rationalised in terms of the effective poten- tials and coupling functions shown in fig. 1 and 2. However, this situation will be reversed in strongly coupled Van der Waals complexes, where the body-fixed projection of the total angular momentum Q is very nearly a good quantum number. In Ar-HCI, for example, ignoring coupling for ACl + 0 gives good results for the energies of truly bound Ie~els.~’ The Born-Oppenheimer or “ best-local ” (BL) approximation gives energy levels that can be up to 0.5 cm-’ in error for H2(l,2)-Ar, since the non-adiabatic coupling functions are very sharply peaked functions of R.This reflects the rapid change in the preferred geometry of the H,-Ar complex from linear to T-shaped as the inter- molecular distance is decreased. i t should be stressed that the BL approximation is not always an improvement over the SFD and BFD approximations. The BL approximation is, however, much more accurate for systems where the preferred angular geometry is independent of R, and has also been successfully applied to strongly anisotropic systems.” Finally, it should be mentioned that all the methods tested here gibe good results for the J = 0 states of H2(v,2)-inert-gas systems; even the small errors in the SFD, BFD and BL energies for the J = 0 leveis of table 2 are almost completeIy removed if the level shifts are taken into account using eqn (16). For the type of model problem considered here, the J = 0 state has only two channels (one open, one closed), and the coupling between these channels is fairly weak.The J = 0 levels of this type of model problem have often been used in the literature for testing new theoretical methods; 3 , 1 2 * 1 3 9 2 8 , 3 2 however, as has been seen, obtaining accurate results for J = 0 states is very easy, and new methods should certainly not be tested on such states alone. 4.PRACTICAL PREDICTIONS FOR HY DROGEN-INERT-GAS VAN DER WAALS MOLECULES A. EFFECT O F DIATOM STRETCHING DEPENDENCE The least well characterized feature of most atom-diatom potential-energy surfaces is their dependence on the length of the diatom bond. Indeed, the hydrogen-inert- gas systems are the only ones for which accurate information of this type has been obtained. For H,-Ar the dependence of the potential on this degree of freedom is illustrated by the differences among the various potentiah listed in table 1 . As the diatom stretches (with increasing vibration-rotation excitation), both the isotropic and anisotropic parts of the potential become deeper and their minima are displaced to larger distances. The changes in the isotropic part of the potential affect mainly the level energies, while the much more pronounced changes in the anisotropy strength functions v,(u,jlR) affect mainly the level widths.Table 4 shows CC level energies and widths for H2(r7j = 2)-Ar obtained using potentials averaged over the diatom stretching motion in various ways. Since the widths of all IeveIs scale as matrix elements of v2(~i7jlR), the ratios of the widths for internal-rotational predissociation of divers levels should be approximately the same for any realistic model of the Hz(t!,-j = 2)-Ar potential. Relafire changes in level width due to a small change in the potential should be similar for all levels. Tn table 4 the properties of the (rz,/,J) = (0,2,0) level of H2(v7j = 2)-Ar are used toR.J . LE ROY, G . C . COREY AND J . M. HUTSON 353 TABLE 4.-CC RESULTS FOR THE (n,l,j,J) = (0’2’2’0) LEVEL OF DIFFERENT VIBRATIONALLY AVERAGED FORMS OF THE CARLEY-LE ROY l6 BC3(6,8) POTENTIAL FOR H2-Ar species E,/cm-’ T/cm - potential a Hz( 1’2)-Ar - 20.727 0.1 10 (0 = (132) Hz(0,2)-Ar - 19.567 0.043 (U’i) = (0’2) H2(0,2)-Ar - 19.474 0.040 ( V , i ) = (0,O) H2(0,2)-Ar - 19.590 0.034 k = O ~~ a All potentials taken from table 1 . illustrate the effect of diatom stretching dependence on the level widths for predissoci- ation by internal rotation. The first two rows of table 4 show that the lifetimes for internal-rotational pre- dissociation of complexes formed by Ar with H,(v = 0) will be almost three times longer than those for complexes formed from H,(v = 1).This suggests also that rotationally inelastic cross-sections for vibrationally excited H2 will be much larger than those for the ground-state species. Moreover, the difference between the second and third entries in table 4 shows that even the pure centrifugal stretching of the diatom has a significant effect on the predissociation widths. The potential used to obtain the fourth entry in this table corresponds to a fictitious rigid-rotor model with the diatom bond length fixed at its average value for ground- state H, (r-ro). While this potential does not correspond to any observable species, it is included here because these k = 0 functions have sometimes been used in cal- culations for this The substantial difference between the level widths for the third and fourth entries in table 4 shows that failure to perform proper vibrational averaging can introduce significant errors.It is clear therefore that predicted pre- dissociation level widths can have little physical significance unless the diatom stretch- ing dependence of the potential-energy surface is correctly accounted for and the vibrational averaging is performed properly. B. PHYSICAL PREDICTIONS AND 1MPLICATIONS REGARDING POTENTIALS The CC results for H2( 1,2)-Ar listed in tables 2 and 3 are essentially exact calcu- lations for the BC3(6,8) potential-energy surface of Carley and Le Roy,16 except that they neglect coupling to closed channels f o r j > 2. However, subsequent calculations showed that inclusion o f j = 4 channels shifts the levels downward by only ca.0.01 cm-’ and increases their widths by < 1 ”/. Inclusion o f j = 6 channels had no further significant effects. Thus, the constraint that j,,, = 2 does not significantly affect the accuracy of predicted H2(v,j = 2)-Ar resonance energies and widths. The main sources of inaccuracy in the physical predictions provided by the CC results of tables 2 and 3 are therefore deficiencies of the BC3(6,8) potential-energy surface of Carley and Le Roy.I6 The accuracy with which this surface reproduces the spectroscopic data and explains total differential and integral scattering cross- sections l6 implies that its (diatom stretching dependent) isotropic part is quite reliable. The accuracy of the anisotropy strength function at distances near and beyond the isotropic potential minimum is confirmed by its agreement with that determined by Zandee and Reuss 34 from their integral cross-section anisotropy measurements, but its inability to reproduce Buck’s inelastic differential cross-sections 35 suggests that its short-range behaviour is in error.This is a matter of concern here, since the short- range part of the anisotropy also determines the predicted level widths.354 PREDISSOCIATION OF VAN D E R WAALS MOLECULES In the work of Carley and Le Roy I5*l6 the exponent parameter of eqn (3) was assumed to have the same value for ;1 = 0 and 2 because the data could not determine p2 independently. More recently, Buck and Le Roy36 determined an improved estimate of the H,-Ar potential anisotropy which is in good accord with both the spectroscopic data l8 and inelastic differential cross-~ections.~~ This was obtained from fits to the spectroscopic data which held all of the isotropic (3.= 0) parameters constant at the values given in ref. (16) and fixed p2 = 3.90 A, a value suggested by the scattering data. The ratio of the widths associated with the first two entries in table 5 indicates that the predissociation level widths of this improved surface will be ca. 1/3 larger than those of the BC,(6,8) potential of Carley and Le Roy.16 Thus, the best available predictions of the predissociation level widths of H2( 1,2)-Ar are obtained by multiplying the CC results in table 3 by an approximate correction factor of 4/3. The only previously reported predictions of internal-rotational level widths for this system are values for the J = 0, 1 and 4 levels of H2(0,2)-Ar reported by Beswick and Requena.I2 Their results were obtained using the BFD and “ intermediate ” distortion approximations from a potential-energy surface based on those reported by Dunker and Gordon,37 and their predicted widths are roughly a factor of 3 too small.This discrepancy is believed to be largely due to the fact that the potential-energy surface used in ref. (12) is less realistic than the one used here. One of the most prominent features of the CC level widths shown in table 3 is the fact that the J = 0, 1 = 2 level is predicted to be more than a factor of two broader than any of the others. This prediction and most other features of the level width pattern are also expected to hold true for complexes formed from other inert-gas partners.The last two entries in table 5 show the energy and width of this J = 0 level calculated from Carley’s l5 BC3(6,8) potentials for H2(1,2)-Kr and H2( 1,2)-Xe. An interesting feature of these results is the prediction that level widths for internal- rotational predissociation decrease from H,-Ar to H2-Kr to H2-Xe. This implies that rotationalIy inelastic cross-sections will vary in this same order and hence that the system with the weakest potential anisotropy, as characterized by the anisotropy strength function depth E ~ , will have the largest inelastic transition probabilities ! This reflects the fact that level widths and rotational inelasticity are both largely determined by the strength of the anisotropy at distances shorter than the zero of the isotropic potential, and that relative to the position of these zeros the short-range repulsive waIls of V,(uJIR) are located at reIatively srnaIler distances for the heavier inert-gas partners.Thus, the anisotropy in the repulsive region is actually smaller for H,-Xe than H2-Ar, giving rise to the predicted trend in predissociation Ievel widths. The one experimental IeveI width obtained to date for the €3,-inert-gas complexes TABLE 5.-cc RESULTS FOR THE (n?/,j,J) = (0,2,2,0) LEVEL OF VARIOUS H2-INERT-GAS POTENTIAL-ENERGY SURFACES species EJcm - rlcm-1 potential H2( 1,2)-Ar - 20.71 8 0.147 Buck-modified f12 = 3.9OA-’ BC3(5? 8) H,( I ,2)-Ar - 20.727 0.110 (r7,i) = (L2) a H,(1,2tKr - 28.680 0.073 Carley’s I5 BC3(6,8) H2( 1,2)-Xe - 34.727 0.039 Carley’s * 5 BC3(6,8) ‘ From table 1 .R.J . LE ROY, G . C . COREY AND J . M . HUTSON 355 is for the n = 0, J = 0 level of H2(l,2)-Kr. McKellar l7 has reported its width to be 0.1 1 cm-’, which is distinctly larger than the calculated value of 0.073 cm-’ shown in table 5. In view of the above discussion of improvements to the H,-Ar potential- energy surface, it seems clear that this discrepancy should be attributed to deficiencies in the short-range behaviour of the assumed potential anisotropy for H,-Kr, since Carley’s l6 BC,(6,8) surfaces for H,-Kr and H,-Xe were also obtained subject to the arbitrary constraint that p, = Po. This provides the first direct confirmation that predissociation level widths do in fact contain information about the potential anisotropy which cannot be obtained from the level energies alone, and which may be competitive with inelastic cross-section measurements as a source of information about the potential anisotropy.(a) D. A. Micha, Chem. Phys. Lett., 1967, 1, 139; (6) D. A. Micha, Phys. Rev., 1967, 162, 88. (a) R. D. Levine, J. Chem. Phys., 1967, 46, 331 ; (b) R. D. Levine, J. Chem. Phys., 1968,49, 51. (a) R. D. Levine, B. R. Johnson, J. T. Muckerman and R. B. Bernstein, J. Chem. Phys., 1968, 49, 56; (6) J. T. Muckerman, J. Chem. Phys., 1969, 50, 627. J. T. Muckerman and R. B. Bernstein, J . Chem. Phys., 1970,52, 606. (a) R. E. Smalley, D. H. Levy and L. Wharton, J. Chem. Phys., 1976,64,3266; (b) M. S. Kim, R. E. Smalley, L. Wharton and D.H. Levy, J. Chem. Phys., 1976,65, 1216; (c) K. E. Johnson, L. Wharton and D. H. Levy, J. Chem. Phys., 1978, 69, 2719. R. E. Smalley, L. Wharton and D. H. Levy, J . Chem. Phys., 1977, 66, 2750. (a) C. J. Ashton and M. S . Child, Faraday Discuss. Chem. Soc., 1977,62, 307; (b) C. J. Ashton, D. Phil. Thesis (Oxford University, 1981). (a) J. A. Beswick and J. Jortner, Chem. Phys. Lett., 1977, 49, 13; (6) J. A. Beswick and J. Jortner, J. Chem. Phys., 1977, 68, 2277 and 2525; (c) J. A. Beswick and J. Jortner, J. Chem. Phys., 1978, 69, 512. lo (a) G. E. Ewing, J. Chem. Phys., 1980,72,2096; (6) G. E. Ewing, J. Chem. Phys., 1979,71, 3143. J. A. Beswick and J. Jortner, Ado. Chem. Phys., 1981, 47, 363. J. A. Beswick and A. Requena, J. Chem. Phys., 1980,72, 3018. l 3 J. A. Beswick and A. Requena, J. Chem. Phys., 1980,73,4347. l4 (a) R. J. Le Roy and J. Van Kranendonk, J . Chem. Phys., 1974, 61, 4750; (b) R. J. Le Roy, J. S. Carley and J. E. Grabenstetter, Faraday Discuss. Chem. SOC., 1977,62,169; (c) J. S. Carley, Faraday Discuss. Chem. SOC., 1977, 62, 303. (a) J. S. Carley, Ph.D. Thesis (University of Waterloo, 1978); (6) J. S. Carley and R. J. Le Roy, unpublished work, 1978. ’ T. E. Gough, R. E. Miller and G. Scoles, J. Chem. Phys., 1978,69, 1588. l6 R. J. Le Roy and J. S. Carley, Adu. Chem. Phys., 1980, 42, 353. l7 A, R. W. McKellar, Faraday Discuss. Chem. SOC., 1982, 73, 89. la (a) A. R. W. McKellar and H. L. Welsh, J. Chem. Phys., 1971,55, 595; (b) A. R. W. McKellar l9 G. Herzberg, Spectra of Diatomic Molecules (Van Nostrand, Princeton, 1950). 2o D. M. Bishop and S-K. Shih, J. Chem. Phys., 1976, 64, 162. ” U. Fano, Phys. Rev., 1961, 124, 1866. 23 A. U. Hazi, Phys. Rev. A, 1979, 19, 920. 24 A. M. Arthurs and A. Dalgarno, Proc. R. SOC. London, Ser. A , 1960, 256, 540. 25 W. N. Sams and D. J. Kouri, J . Chem. Phys., 1969, 51, 4815. 26 S. L. Holmgren, M. Waldman and W. Klemperer, J. Chem. Phys., 1977, 67, 4414. 27 J. M. Hutson and B. J. Howard, Mol. Phys., 1980, 41, 1123. 28 J. E. Grabenstetter and R. J. Le Roy, Chem. Phys., 1979, 42, 41. 29 M. S. Child, Molecular Collision Theory (Academic Press, New York, 1974). 30 M. V. Berry and K. E. Mount, Rep. Prog. Phys., 1972, 35, 315. 31 I. F. Kidd, G. G . Balint-Kurti and M. Shapiro, Faraday Discuss. Chem. SOC., 1981,71, 287. 32 S-I. Chu, J. Chem. Phys., 1980, 72, 4772. 33 S-I. Chu and K. K. Datta, J. Chem. Phys., 1982, 76, 5307. 34 L. Zandee and J. Reuss, Chem. Phys., 1977, 26, 345. 35 U. Buck, Faraday Discuss. Chem. Soc., 1982, 73, 185. 36 U. Buck and R. J. Le Roy, unpublished work, 1981. 37 A. M. Dunker and R. G. Gordon, J. Chem. Phys., 1978,68, 700. and H. L. Welsh, Can. J. Phys., 1972, 50, 1458. H. Feshbach, Ann. Phys., (a) 1958, 5, 357; (6) 1962, 19, 287.

ISSN:0301-7249    

DOI:10.1039/DC9827300339

出版商:RSC    

年代:1982

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Faraday Discuss. Chern. SOC., 1982, 73, 357-373 Photodissociation of Van der Waals Molecules Adiabatic Treatment BY N. HALRERSTADT AND J. A. BESWICK Laboratoire de Photophysique Molkculaire, Btttiment 2 13, Universitk de Paris-Sud, 91405 Orsay, France Received 1st February, 1982 In this paper an adiabatic distorted-wave treatment of vibrational predissociation in Van der Waals molecules is presented. The model considers the coupling between a vibration of a normal molecule and the Van der Waals stretch. In the zero-order approximation, the two motions are decoupled in a way analogous to the Born-Oppenheimer separation of electronic and nuclear motion. In contrast to the usual diabatic treatments (where the molecular vibration is considered to be independent of the Van der Waals bond length) the formulation presented on this paper explicitly takes into account the change of the molecular frequency and equilibrium internuclear distance as the Van der Waals bond stretches.The formalism is presented for harmonic as well as for an- harmonic descriptions of the asymptotic intramolecular vibrations. In both cases analytic expressions for the non-adiabatic coupling terms are obtained. The vibrational predissociation rates are cal- culated using the first-order distorted-wave approximation. A simple model leading to fully analytic results corresponds to a colinear X * * - BC molecule with linearization of the potential. It is shown that in this case the two treatments, adiabatic and diabatic, lead to exactly the same results. In all other cases an efficient procedure involving a fitting of thc interaction terms is suggested which also leads to analytic expressions.1. INTRODUCTION The experimental studies of the photodissociation of Van der Waals molecules formed in a supersonic expansion are providing for the first time detailed information on pure vibrational predissociation processes.1 This is of importance since a detailed knowledge of vibrational predissociation provides the key for the understanding of many molecular processes: vibrational relaxation at very low intramolecular vibrational energy redi~tribution,~*~ n~cleation,~ as well as the role of Van der Waals molecules in atmospherical and astrophysical phenomena.' In a series of experiment^,^ Levy et af. have studied the photodissociation of Van der Waals molecules in a supersonic freejet, induced by visible laser excitation. When complexes of I, bound to rare-gas atoms and molecular hydrogen were studied, it was found that the complex dissociated to electronically excited I, (B) with a lifetime short compared to the radiative lifetime of I2 ( B ) .By observing the dispersed fluorescence from I, (B) it was concluded that dissociation proceeds via vibrational predissociation, and the vibrational state distribution of the product I, (B) could be measured. that the vibrational predissociation rates in- creased almost quadratically as a function of vibrational excitation in the range u = 10-20. A strong propensity rule Av = - 1 was found for the vibrational distribution of the I, fragment. In other words, the first open channel (the one corresponding to the minimum relative kinetic energy of the fragments) was the preferred one.For example, when the ZI = 21 level of the complex was excited, only 3.5% of the I, ( B ) products were in the u' = 19 vibrational state. All these results were theoretically accounted for by the use of diabatic distorted- For HeI, complexes it was found3 58 ADIABATIC TREATMENT OF PHOTODISSOCIATION wave treatment of the coupling between the internal I, vibrational degree of freedom and the motion of the rare-gas atom with respect to the centre of mass of the I, molecule. More explicitly, if Y and R denote these two coordinates, then the zero- order wavefunction of the bound state of the system was written as a product wvl = ~ , ( r ) yL(R) where u is the vibrational quantum number of I, and I is a quantum number describing the motion of the Van der Waals stretch The wavefunction ~ , ( r ) was taken to correspond to a free I, (B) molecule while ql(R) described the motion of the rare-gas atom with respect to the centre of mass of I, frozen at its equilibrium geo- metry.This diabatic description of the zero-order wavefunctions of the complex was justified in view of the weak coupling between the two degrees of freedom as well as the large difference between the frequency of the I2 (B) vibration (127 cm-’) and the dissociation energy of the complex (ca. 15 cm-I). One would expect that the diabatic description will fail for more strongly coupled systems characterized by larger dissociation energies, as for example the I,Ar molecule (Dz 220 cm-I).In that case the motion of the I, molecule will be much more strongly perturbed by the presence of the rare-gas atom and an adiabatic description of the motion, similar to the treatment used in molecular c o l I i ~ i o n s , ~ ~ - ~ ~ should be more appropriate. This is obtained by considering the wavefunction yal = x,(Y; R) vvl(R) where now the wavefunction ~ , ( r ; R) describing the I, vibration depends para- metrically on the distance R of the rare-gas atom with respect to its centre of mass. In this paper we present such an adiabatic method to calculate vibrational pre- dissociation rates and final vibrational distributions of the fragments, using the first- order distorted-wave approximation and Golden-rule expressions.The formalism is presented for harmonic as well as for anharmonic descriptions of the molecular vibration. In the harmonic description, a model based on Morse potentials for the Van der Waals interaction and linearization is presented which leads to fully analytic expressions. We show that in this case the two treatments, adiabatic and diabatic, lead to exactly the same results. In all the other cases an efficient method involv- ing a simple fitting of some interactions terms is suggested. This is of importance to the order-of-magnitude calculations of vibrational predissociation rates and vibrational distribution in order to avoid the lengthy and costly close-coupling numerical methods. In section 2 we present the general method for the adiabatic distorted-wave description of vibrational predissociation of a Van der Waals complex formed by two normal chemically bounded systems held together by a Van der Waals interaction.The method applies therefore to a simple complex like X * - * I2 (X = rare-gas atom) but also to polyatomic systems: X - * - M (X = rare-gas atom, M = polyatomic molecule), or M * - * M’. Section 3 considers two types of intramolecular vibrations: the harmonic case and the anharmonic case described by a Morse potential. In section 4 we present the final expressions for the vibrational predissociation rates and final vibrational distributions. Finally, in section 5 we apply these general expressions to a simple case which gives fully analytical expressions and in section 6 we compare the adiabatic results for a model ArJ, molecule with those obtained in the diabatic approach,10 and also with those given by a direct close-coupling calculation.‘ 2.METHOD We consider a general system composed of two stable components M and M’ (in which at least one is a molecule) held together by a Van der Waals interaction, undergoing vibrational predissociation, i.e. (M * . * M’)? -+ M + M’ (1)N. HALBERSTADT AND J . A . BESWICK 3 59 where t denotes vibrational excitation. In eqn (1) the fragments M and M' can also be (and in general they will) vibrationally excited, the only requirement is the conserv- ation of the total energy initially in the bounded complex into vibrational energy of the fragments plus some relative kinetic energy E > 0.The assumptions made in writing eqn (1) are: (i) the complex has quasi-bound vibrationally excited states (M - - M')t which live longer-than several vibrational periods of the lowest frequency; (ii) the complex has been prepared initially in only one of these quasi-bound states by a short-time excitation process which it is not necessary to include in the calculation of the rates; (iii) the lifetime of the (M - - M')t is short compared to the radiative life- time. These assumptions allow us to describe the process through the use of pertur- bation theory and Golden-rule expressions. The initial state is described by the zero- order wavefunction (yi) and the final asymptotic state by Iyf). When energy- normalized wavefunctions are used for Iyf), the Fermi Golden rule will provide the approximate vibrational predissociation rate: where H is the total Hamiltonian of the system.partial rates while the total rate is given by When several final states lyf) are energetically accessible, eqn (2) provides the and the lifetime is z = ki-'. Iyf) will be given by The final distribution among the different final states Pf = k f e i / k i . ( 3 ) It is worthwhile to note at this point that in a time-independent experiment, in which the excitation pulse is much longer compared with the lifetime of the process (which is in fact the situation met in all the experiments in supersonic jets), eqn (2) and ( 3 ) are again approximately valid, the observables being in that case the linewidths Ti = tzki/2 (h.w.h.m.) and the final distributions Pf.The above assumptions are reasonably met in actual experiments on Van der Waals molecules formed in supersonic expansions.' What remains is to specify the initial and final wavefunctions (yi> and IVf>. This is the crucial point in all the treat- ments of radiationless transitions. The first quantum-mechanical theoretical treat- ment of vibrational predissociation has been provided by Rosen's paper l4 on unimolecular decomposition induced by collisions. The coordinates describ- ing initial and final states were inconveniently chosen, introducing spurious asymptotic couplings which made the results valid only for some special mass com- binations. Much later Coulson and Robert~on,'~ following a suggestion by Step- anov,16 studied vibrational predissociation in hydrogen-bonded systems using a diabatic basis set.Independently, Beswick and Jortner lo and Ewing 3 7 1 7 have studied vibrational predissociation in much weaker Van der Waals systems and they have also used the diabatic description. In those treatments, the wavefunctions I yi) and I yf) are written as direct products of a wavefunction for the vibrational motion of the components M and M , unperturbed by their mutual interactions, and a wavefunction describing their relative motion. In the most simplified versions, the last is calculated by freezing M and M' in their internal equilibrium geometries. It is clear that the diabatic description of the wavefunctions will fail whenever the intermolecular interaction is sufficiently strong to modify appreciably the wave- functions of M and M' while approaching each other.For the HeI,* case studied before, this is not too serious since the vibrational frequency of 1," (ca. 125 crn-') is360 ADIABATIC TREATMENT OF PHOTODISSOCIATION large compared with the dissociation energy of the complex (D z 15 cm-l), so that the coupling is small. This is not the case for much stronger complexes involving polyatomic molecules. In particular, the hydrogen-bonded systems could be a test case of the failure of the diabatic approximation. We present here a different, adiabatic,” -13 approach to vibrational predissociation which describes the change on the wavefunctions of M and M’ when their mutual distance varies. Let us denote by R this distance and by Y an ensemble of internal coordinates describing the intramolecular vibrations of M and M’.We will then write <r,RlWi> == ~u‘(r,R)qu‘l(R) (4) where x,,(r,R) is a vibrational wavefunction for the internal motion of M and M’ depending parametrically upon the distance R between the centres of mass of M and M’. The label u’ corresponds to an ensemble of vibrational quantum numbers necessary to describe this wavefunction. The wavefunction q,<,(R) describes the relative motion of the two partners M and M’ and in general will depend on the internal state u’. In addition, the label I denotes different bound vibrational levels of the Van der Waals stretch. The adiabatic wavefunction (4) is similar in spirit to the Born-Oppenheimer separation of electronic and nuclear motion. In our case, the fast motion is the internal vibrations of M and M’ while the slow one is the Van der Waals stretch vibration.At this point we notice that the wavefunction (4) depends only on the internal co- ordinates of the M and M’ molecules and on the stretch distance R between their centres of mass. Several other coordinates are in principle necessary to specify the bending and torsional relative motions of M and M’. The coupling of these degrees of freedom and the internal vibrations will result in rotational excitation of the fragments. In Van der Waals complexes it is expected that rotational excitation will be weak, and some experimental and theoretical evidence l9 on X * - - J2 complexes has been recently provided. The final wavefunction (r,RIryf) is written similarly to eqn (4) <r,RI Wf) = xu(Y,R)Pum (5) where E is now the relative kinetic energy of the fragments.write the total HamiItonian H as We have now to specify the wavefunctions x,(r,R) and qu{i)(R). For this, we h2 d2 2m dR2 H = HIM, + HM - - - + V(R,r) where HM and HM, are the Hamiltonians describing the free molecules M and M’, and V(R,r) is the Van der Waals interaction which vanishes as R goes to infinity. The term (-h2/2rn) d2/dR2 describes the relative kinetic energy of M and M’, with m being the reduced mass of the couple. We now define x,(r,R) to be the solution of the Schrodinger equation [ITJM + + v(R,r)lxu(r,R) = uu(R)xu(Y,R) (7) and qu{;}(R) the solution of: where ( I ), denotes integration only over the internal coordinates Y .N . HALBERSTADT A N D J .A . BESWICK 361 - _ - and A,, = 0. It is shown 2o that and 10’ , u‘ # u \ x v ’ / r which ensures the hermiticity of the right-hand side of eqn (1 1). Eqn (2) and (3) together with eqn (1 I)-( 13) provide all the ingredients necessary to calculate vibrational predissociation rates, linewidths and final vibrational populations. The actual implementation of the calculations necessitates the specification of the intermolecular interaction and knowledge of the spectroscopic constants of the M and M’ fragments. We are interested here in getting as many analytic expressions as possible which provide fast and efficient ways of calculating rates and probabilities. We therefore consider in the following section two cases of interest in this context. The first one corresponds to a harmonic description of the molecular vibrations of M and M’, together with an expansion of the intermolecular interaction in a truncated Taylor series on the displacements of r with respect to their equilibrium distances.This calculation is the same as the one presented by Eno and Baht-Kurti l 2 in their treatment of molecular collisions. In the second case we consider a Morse function description of the intramolecular vibrations, and we expand the intermolecular interaction in terms of exponentials of the displacements. 3 . INTRAMOLECULAR VIBRATIONS HARMONIC TREATMENT In the harmonic description of the intramolecular vibrations, we consider HM and HMr to be the Hamiltonians for a collection of harmonic oscillators corresponding to the normal modes of the M and M’ molecules, respectively.At the same level of approximation, we then expand the intermolecular interaction Y(r,R) in a Taylor series on the normal modes displacements (r - re), up to second order. This defines new frequencies and equilibrium distances which are now functions of R. For the simplest case, i.e. a diatomic molecule M with reduced mass ,u and frequency cc) and a rare-gas atom M’, we have362 ADIABATIC TREATMENT OF PHOTODISSOCIATION The new equilibrium distance is now The matrix elements Auu, have been worked out by Eno and Baht-Kurti l2 and can be obtained easily from eqn (12). They have the explicit form (for u' > v): and AUdu = -Auu,. From eqn (13) we get, for u' > u, Some discrepancies exist in the literature for the Buu, element^.'^^'^ and Eqn (18)-(20) give all the matrix elements necessary in the harmonic treatment of intramolecular vibrations in the case of an atom-diatom Van der Waals molecule.In more complex cases involving polyatomic molecules the harmonic expansion of V(r,R) will involve cross-terms between different normal modes. In all cases a linear transformation can be found which will decouple the problem in an ensemble of harmonic oscillators for which eqn (1 8)-(20) will apply.N . HALBERSTADT AND J. A . BESWICK 363 ANHARMONIC TREATMENT We start again with the simple case of a Van der Waals molecule formed by an In that case it is convenient to describe the intra- atom and a diatomic molecule. molecular vibration P of the diatomic molecule in terms of a Morse potential, i.e.h2 i12 2p ar2 . HM + HM. = - - - + do{exp[-2/3(r - re>] - 2exp[-p(r - re)]) (21) where p is the reduced mass of the diatomic, Y, the equilibrium interatomic distance, while do and 6 are related to the spectroscopic frequency o and anharmonicity wx by the relations w = 2do/tzK; OJX = w/2K (22) where K = (h/3)-i(2pdo)1/2. 423) In order to get analytic expressions for Auu, and B , , , we may expand V(R,r) in terms of exponentials, i.e. Y(R,r) = C,(R) - 2C,(R)expl--P(r - re>] + C2(R)exp[-228(p. - re)l. (24) This is similar to Pekeri’s treatment 21 of the centrifugal term in diatomic molecules. We get t i 2 a2 2 p ar2 HrYi + HM, + v ( R , ~ ) = - - - + c,(R) + dO(R)(exp!--2/l[r - ~ ~ ( ~ 1 1 with and We have then a new Morse potential with modified dissociation energy do and equilibrium distance Fe.Therefore where we have defined the modified parameter X(R) = [ 2p&(R>]””iri& In order to calculate the A,, elements using eqn (12), we need the auxiliary results364 ADIABATIC TREATMENT OF PHOTODISSOCIATION valid for uf 2 u. Substituting eqn (30), (31) and (28) in eqn (12) we get (for u’ > u) (2k - 2u - 1)(2X - 2u’ - l)d! n (2R - u - l)u! A,,# = ( u ’ - u I = I 1 d;io [+)(I -(2R - u - UI - l)(u’ - v) dn + %] 2R and as usual relation : -1 - A,,.. For B,, and B , , (v’ > v), we use eqn (13), with the (2K - 2u - 1) (2Z - 2v’ - 1)u’ ! + ( Gv (2Z- v - l ) u ! 1 1 = 1 with We notice at this point that although the sum over u” in eqn (13) has an infinite number of terms, A,,,, will decrease fast as v f f - v increases, so that only a few terms are necessary to get converged results.We conclude that even for the Morse oscillator wavefunctions we obtain rather simple analytic expressions for the coupling terms AuD, and Buv,. Tn the polyatomic case, anharrnonicity is usually treated in terms of linear com- binations of harmonic wavefunctions. We propose then to start using the method of section 2 to get the harmonic Auu.Ch”rl and BuUqchar1. Introducing then the anharmon- icity coupling terms, the vibrational wavefunctions will be written as linear combin- ations of harmonic wavefunctions with R-dependent coefficients, i.e. in vectorial form lxCanhl) = a ( R ) I X L ~ ~ ~ I > . (35) U lntroducing eqn (35) into the definitions of A,,, and Buup [eqn (12) and (9), respec- tively] we obtain andN.HALBERSTADT A N D J . A . BESWICK 365 4. VIBRATIONAL PREDISSOCIATION The last step of the calculations involves the evaluation of the coupling terms in eqn (1 I), where qUtz and q,, are solutions of the Schrodinger equation, eqn (8). In the general case, we propose a method similar in spirit to the one used by Eno and Baht-Kurti l2 for the collisional problem, which makes the whole calculation com- putationally fast. In order to do this, we would like to have analytic channel wave- functions and then matrix elements which can also be evaluated analytically. Since we know from earlier work lo that vibrational predissociation rates are sensitive mainly to the short-range part of the potential, and largely independent of the details of the interaction at large distances, we propose the following procedure: first we fit the potential terms in eqn (8) for the two channels u and u' involved in the cal- culation of eqn (1 l), by Morse functions with the same exponential factor, i.e.h2 W,(R) = Uv(R) - 2nzB,,,(R) = D,(exp[-2~@ - R,)] - 2exp[--(R - R,)]} (38) in the region R z I , , and similar expressions for u'. exponential constants, i.e. We now fit the A,,,, and B,,,, (u' # u ) in terms of exponentials with arbitrary A,,@) z 2 Ciexp(-BiR). (39) 1 The functional forms (38) and (39) are sufficiently flexible as to give a good representation of the potentials in the region of the well. The interest in using these forms is that all the calculations can be performed analytically.First, the derivative of a Morse wavefunction can be expressed in terms of other Morse wavefunctions. For example, the discrete wavefunction for a Morse potential with parameters Dv,, a and R,, is where (2& - 21 - 1) cc i ! r ( 2 ~ , , - i) c,.z = and WK,p is Whittaker's function.22 Now '' so that (414366 ADIABATIC TREATMENT OF PHOTODISSOCIATION while the energy-normalized continuum wavefunction is ~ ~ ~ E u ( Z y ) = C U E Z v - 1 ' 2 wKu,ie(zv> soIution of a Morse potential with parameters D,, Rv and the same CI, where I and with kinetic energy E and Z, = 2~,exp[ - E ( R - I?,)]. We therefore obtain the result that all the coupling matrix elements in eqn (11) reduce to a sum of terms involving integrals of the form W i t h The evaluation of these integrals is presented in the appendix.The final result is where 2f'1 (a, 6, c, y> is the confluent hypergeometric function 22723 Note that the integral l i n eqn (46) is real. Therefore, the imaginary part of the right-hand side of eqn (48) should vanish. This constitutes a further test of the convergence of the hypergeometric function. From our past experience, we may assert that 8-10 terms in the expansion (49) give 4-5 digits accuracy in this type of problem where p is of the order of 1. In most applications, we may further make the approximation D, z D,. and R, FZ &, which lets a = 1 , then we recover the well- known result for two identical Morse oscillators : Eqn (111, (38), (39) and (43)-(50) together with eqn (18)-(20) in the harmonic cases or (32)-(34) in the anharmonic case, constitute our final expressions for the matrix elements ( y i l H l ~ f > necessary to calculate the rate constants, eqn (2), and the final vibrational distribution, eqn ( 3 ) , of vibrational predissociation in Van der WaalsN. HALBERSTADT AND J .A . BESWICK 367 molecules. We present in the following section an application of the expressions to a particular simplified model. 5 . LINEARIZED POTENTIAL MODEL In this section we shall apply the general method presented in the above sections to a simple case which provides closed-form expressions for the rates. We consider one intramolecular coordinate Y coupled to the Van der Waals stretch coordinate R. This case may correspond either to the atom-diatomic-molecule Van der Waals complex or in the polyatomic case to the situation where only one normal mode is efficiently coupled to the Van der Waals bond motion.V(R,r) = D{exp{-2u[R - R - ~ ( r - re)]} -2exp{ - u[R - R - Y(Y - ~ e ) ] } } . (51) In the collinear model for X - - BC molecules, this particular form of the potential will correspond to an interaction between the neighbouring atoms X and B, and therefore y = m,/(m, + mc). We treat the vibration of BC in the harmonic approx- imation and we expand V(R,r) up to linear terms in (Y - re) only. From eqn (16) and (17) we then have 63 = o and In both cases we shall take 2uyD Fe = Ye - - ( y 2 - y ) P o 2 where we have defined y = exp[-u(R - R)]. (53) From eqn (18) and (19) we get (for v' > v ) and 3 +%(@ + + 2)) t2(2Y2 - A2& + 2.u' where +- 2a2yD pic02 a The adimensional parameter <, eqn (56), can also be written as 5 = Y- m (-)2 co, P a where co, is the frequency of the Van der Waals vibration This parameter will be usually smaller than one since w, < co.We shall neglect then in our calculations the terms involving t2 in the coupling terms, and the terms involving 5 in the diagonal terms.368 ADIABATIC TREATMENT OF PHOTODISSOCIATION Therefore we get B,, = 0 and U,(R) = hu(v + 3) + DO.)' - 2y). (58) Applying these results to eqn (8), we conclude that the wavefunctions qUsl and p,& will be the solutions of Morse potentials with the same parameters D, a and R. Furthermore from eqn (1 1) and using eqn (43) and (50), we get [02 + ( K - I - 1/2)2]2 where Inserting eqn (60) into eqn (59) and using the relation 22 (59) eqn (59) can be recast in the form lr(1/2 + K + i0)l [cos2(nK) + ~ i n h ~ ( ~ 0 ) ] ' / ~ ' X The result given by eqn (62) for the coupling matrix element responsible for predissociation is identical to the one obtained in the adiabatic treatment with linearization.'' Explicitly, in the diabatic treatment and linearized potential, the coupling is given by <YfIHlwi> = (wfI[V(R,r) - v ( ~ , r e ) ~ ~ v i ) = <n l ~ l r ~ 3V - re> 1 Wi/ \ (63) with where xi:,) (r) is the wavefunction corresponding to the " free " BC molecule, and pi;) is the solution of Introducing now the potential form given by eqn (51), and using the well-known matrix elements of (r - re) between harmonic oscillator wavefunctions, we obtainN.HALBERSTADT AND J . A . BESWICK 369 The integrals over the Morse wavefunctions are well known.lplo Their explicit forms are ($%lcxp[--24R - Rltv;) = (vzlexP[--@ - R)lIvJ (Ij2K)[(X - 1 -' li2)' + O2 +- 2 K ] . (67b) Substituting eqn (67) into eqn (66) and using definition (60) the result given in eqn (62) is recovered. The identity of these two results seems at first sight surprising, since the adiabatic and diabatic treatments are rather different in nature. In the adiabatic treatment the coupling responsible for the predissociation is the kinetic-energy operator (h2/2m) d2/dR2, while in the diabatic approach the coupling is provided by the potential difference [V(R,r) - Y(jP,rej]. However, note that the identity of the results in the two approaches is only true in the linearized approximation.We shall show that the expansion of the potential up to second order in the molecular disphcements (r - r,) results in differences between the adiabatic and diabatic treatments. 6 . APPLICATION AND mscussroN In this section we apply our general method of sections 2-4 in order to go beyond the Iinearized approximation for the potential. We have chosen a coIlinear model system Ari2. The choice of this system was dictated by two reasons: (a) X l 2 (X = H,, Ne, Ar) have been studied in the diabatic approximation lo and also by numerical close-coupling integration; (b) for the He and Ne compkxes the coupling is weak and the linearized approximation is good. Thus the results for the diabatic and adiabatic cases are identical for such systems (see section 5).The Ar12 molecule is a stronger-coupled Van der Waals system, so that we may expect differences between the two treatments. The parameters for the potential are those used in the close-coupling calcuIations: w = 128 crn-l; re = 3.016 A; r. = 2.02 A-l; D = 200 cm-l; a = 4.0 A. (68) We represent in fig. 1 the effective potentials W, [see eqn (1 5 ) and (38)] for u = 0 and 1. In fig. 2 and 3 we have shown the couplings Aol [eqn (18)] and B,, [eqn (19)], respectively. In table I we have collected the effective potentials W, (v = 0,l) as well as the couplings Aol and B,, for some selected values of R in the region of the potential well. The values in brackets are those corresponding to the linearization approximation presented in section 5.Two comments about the results of table 1 are now in order. First the differences in the effective potentials Wo and Wl with respect to the linearized approximation are rather small. On the other hand, the differences between linear- ized and non-linearized values for the couplings Aol and B,, are significant. The potentials Wo and W, can be accurateIy fitted by Morse functions with parameters [see eqn (38)] a =1.936 A-' Do = 196.2cm-' M == 1.936 A+' D, = 196.1 CIII-' Ro 2 5.52 A Rl = 5.52 A. (69) The couplings A,, and B,, were fitted in terms of exponentials [see eqn (39)3. We370 ADIABATIC TREATMENT OF PHOTODISSOCIATION 201 1oc d I ---- 8 G 0 -100 -2ou FIG. 1.-Effective potentials W, [eqn (38) and (191 used in the cakulations: (-) II = 0, ( - - - ) u = 1.TABLE l.-pOTENTIALS W,(R), U = 0, 1 AND COUPLINGS AolfR), &,(R) FOR MODEL COLLINEAR ArIz The definitions and parameters of the intermolecular potential are given in the text. The values in brackets are those obtained in the linearized approximation of section 5. ~~ 5.0 394.0 (382.0) 5.2 - 54.5 5.4 - f 87.0 (-56.8) (- 188.0) (- 194.0) 5.8 - 160.0 (- 160.0) 6.0 -121.0 (-121.0) 6.2 -87.0 (- 87.0) 5.6 - 194.0 412.0 (395.0) -48.5 (- 54.7) - 186.0 (- 188.0) - 193.6 (- 194.0) -160.0 (-160.0) - 121 .o (-12l.U) - 87.0 (-87.0) - 2.9 (-3.7) -1.3 (-1.5) - 0.52 (-0.54) -0.16 ( -0.16) -0.02 0.03 0.04 (0.04) (- 0.02) (0.03) 9.3 (16.6) 5.6 (7.05) 2.7 (2.9) 1.1 (1.1) 0.4 (0.4) 0.1 (0.1) -0.001 (- 0.001)N. HALBERSTADT A N D J . A . BESWICK c I I. I 5 6 RIA FIG.2.--Aol coupling [eqn (IS)] as a function of R. 37 1 have again found that a Morse functional form is satisfactory. are for Aol, and for Bol. Using these functions in our general expressions of section 4 we have calculated the predissociation half-width for a set of levels corresponding to v f = I and 1 = 2,3, . . ., 7, where v f is the vibrational quantum number associated with the T2 vibration and 1 the Morse vibrational quantum number associated with the Van der Waals stretch. Since the dissociation energy for ArI, is larger than one quantum of the I, molecule, there is no predissociation from v f = 1 and 1 = 0. In fact, even 1 =IT 1 lies below the dissociation threshold. We have then only levels from 1 = 2 and beyond. The results of the calculations (DWA) are presented in table 2, together with the close- coupling (CC) and the distorted-wave diabatic (DWD) results. The first remark concerning the results presented in table 2 is that the adiabatic treatment improves the results, bringing the calculated linewidths very close to the converged numerical results FO.This is gratifying and shows that the adiabatic approach may conveniently be used to accurately calculate predissociation linewidths. On the other hand, we notice that the difference between the DWA and DWD The parameters used cc = 1.45 A-' D = -0.1 cm-l D = -10.3 cm-l I? = 6.3 A cc = 4.1 A-' R = 4.9 A372 ADIABATIC TREATMENT OF PHOTODISSOCIATION I I I RIA FIG. 3.--B0, coupling [eqn (19)] as a function of R. results is not very large. bonded molecule. satisfactory for an order-of-magnitude estimation of the linewidths.This is surprising since the ArI, system is a fairly strongly In fact, even for such a system the diabatic approach is rather TABLE 2.-cOMPARISON BETWEEN CL9SE-COUPLING (cc) AND DISTORTED-WAVE [ADIABATIC (DWA) AND DIABATIC (DWD)] VIBRATIONAL PREDISSOCIATION HALF-WIDTHS FOR A MODEL COLLINEAR ArI, predissociated level v ' = 1 I = 0.13 0.14 0.17 0.17 0.18 0.22 0.19 0.21 0.25 0.19 0.21 0.25 0.17 0.20 0.22 0.12 0.17 0.18 CONCLUSIONS We have presented in this paper an adiabatic formulation for vibrational pre- Such a formulation provides an dissociation processes in Van der Waals molecules.N . HALBERSTADT A N D J . A . BESWICK 373 efficient and fast method of calculating vibrational predissociation linewidths (life- times) even in the case where the Van der Waals coupling is strong.We have shown that in the linearization approximation of the interaction potential the results of the adiabatic treatment are identical to those given by the diabatic approach. We have applied the general treatment (beyond linearization) to a model collinear ArI, which corresponds to a case of a fairly strongly coupled molecule. The results are in better agreement with the close-coupling numerical calculations than the ones obtained by the use of a diabatic approach. For the Ari, system however, the difference between the diabatic and the adiabatic results is not large. We conclude from such calculations that the adiabatic treatment will be particularly useful for much more strongly coupled complexes, such as hydrogen-bonded molecules.For those systems, however, the incorporation of rotational effects becomes The adiabatic treatment of vibrational predissociation, including rotation, is the natural extension of the present work. J. A. Beswick and J. Jortner, Intramolecular Dynamics of Van der Waals Molecules, in Photo- selective Chemistry, ed. J. Jortner, R. D. Levine and s. A. Rice (Wiley, New York, 1981). ' (a) E. E. Nikitin, Opt. Spectrosc., 1960,9,8; (b) J. Bellingsley and A. B. Callear, Trans. Faraday SOC., 1971,67, 257; (c) J. R. Airey and I. W. M. Smith, J. Chem. Phys., 1972,57, 1669; ( d ) P. Zittel and C . B. Moore, J. Chem. Phys., 1973, 59, 6636; (e) H. M. Audibert, C. Joffrin and J. Ducuing, Chem. Phys. Lett., 1974, 25, 158; ( f ) R. A. Lucht and T. A. Cool, J. Chem. Phys., 1975, 63, 3962. G. Ewing, Chem. Phys., 1978, 29, 253. (a) J. Tusa, M. Sulkes and S. A. Rice, J. Chem. Phys., 1979, 70, 3136; (6) G. McClelland, K. Saenger, J. J. Valentine and D. R. Herschbach, J. Phys. Chem., 1979, 83, 947; (c) T. D. Russell, B. M. Dekoven, J. A. Blazy and D. H. Levy, J. Chem. Phys., to be published. J. Tusa, M. Sulkes and S. A. Rice, J. Chem. Phys., 1980, 73, 5897. J. B. Hopkins, D. E. Powers and R. E. Smalley, J. Chem. Phys., 1981, 74, 745. G. E. Ewing, Annu. Rev. Phys. Chenz., 1976, 27, 553. See for example, J. M. Calo, Geophys. Res. Lett. (USA), 1980, 7, 289. For a review see, D. H. Levy, Van der Waals Molecules, in Photoselective Chemistry, ed. J. Jortner, R. D. Levine and S. A. Rice (Wiley, New York, 1981). lo (a) J. A. Beswick and J. Jortner, Chem. Phys. Lett., 1977, 49, 13; (6) J . Chem. Phys., 1978, 68, 2277; (c) J. A. Beswick, G. Delgado-Barrio and J. Jortner, J. Chem. Phys., 1979, 70, 3895. (a) E. Thiele and R. Katz, J. Chem. Phys., 1971, 55, 3195; (6) U. Halavee and M. Shapiro, Chem. Phys., 1977, 21, 105; (c) M. Baer, G. Drolskagen and J. P. Toennies, J . Chem. Phys., to be published. L. Eno and G. G. Balint-Kurti, J. Chem. Phys., 1979, 71, 1447. l3 R. J. Bieniek, J. Chem. Phys., 1980, 73, 851. l4 N. Rosen, J. Chem. Phys., 1933, 1, 319. ( a ) C. A. Coulson and G. N. Robertson, Proc. R. Soc. London, Ser. A, 1974,337, 167; (6) 1975, 342, 289; (c) G. Robertson, J . Chem. Soc., Faraduy Trans. 2, 1976, 72, 1153. l6 B. I. Stepanov, Nature (London), 1946, 157, 808. l7 (a) G. E. Ewing, J. Chem. Phys., 1979,74, 3143; (b) C. E. Ewing, J. Chem. Phys., 1980,71,2096. l 8 D. H. Levy, personal communication; see also ref. (9). l9 (a) J. A. Beswick and G. Delgado-Barrio, J . Cheni. Phys., 1980, 73, 3653. 'O F. T. Smith, Phys. Rev., 1969, 179, 1 1 1 . C. L. Pekeris, Phys. Rev., 1934, 45, 98. 22 M. Abramowitz and I. A. Stegun, Handbook of Muthenintical Functions (Dover, New York, 1965). 23 I. S. Gradshteyn and I. M. Ryshik, Table of Integrnls, Series and Products (Academic Press, New York, 1966).

ISSN:0301-7249    

DOI:10.1039/DC9827300357

出版商:RSC    

年代:1982

数据来源: RSC

摘要:

Furaduy Discuss. Chem. SOC., 1982, 73, 375-386 Dimer Spectroscopy BY Jo GERAEDTS, MARTIN WAAYER, STEPHEN STOLTE AND JORG REUSS Fysisch Laboratoriwm, Katholieke Universiteit, Toernooiveld, 6525 ED Nijmegen, The Netherlands Received 2nd December, 198 1 Measurements of Vibrational predissociation of SF6 clusters are discussed and compared with theoretical spectra. Attention is given to the line-shape and saturation behaviour as influenced by spatial averaging effects. Also isotopic dimers are considered. Hyperfine-structure measurements of H2-H2, H2-Ne, H,-Ar and H2-Kr dimers are dealt with, and information from these results concerning the intermolecular potentials is discussed. 1. INTRODUCTION Properties of vibrationally excited Van der Waals complexes will be discussed in section 2, whereas H, complexes will be considered in their ground state in section 3.Both investigations have in common only the method of production of clusters, i.e. by supersonic expansion in a molecular beam ; the method of probing their properties, however, is rather different in both cases. As described in section 2, dimers of SF6 are predissociated by laser vibrational excitation of their constituents, the v3 mode of SF, being conveniently accessible through the nearly coincident line spectrum emitted by a CO, laser. The predissociation spectrum is then observed as molecular-beam attenuation. In section 3 the magnetic-beam resonance (m.b.r.) method is employed to observe the hyperfine spectrum (h.f.s.) of H2-H,, H,-Ne, H,-Ar and H,-Kr; i.e. the change of magnetic deflection due to the h.f.s.transition renders these complexes observable. As far as we are aware the vibrational predissociation spectrum of (SF& is the first leading to sharp spectral features which permits a straightforward interpretation. In this the high symmetry of SF, and the threefold degeneracy of its v,-mode vibration along three orthogonal axes are important. Consequently the SF6 molecule behaves like a three-dimensional harmonic oscillator, to a sufficient approximation. The in- fluence of the possible rotation of the SF, molecules becomes negligible. The domin- ant interaction lifting the high degree of degeneracy of the singly excited SF6 clusters turns out to be the electric dipole interaction determined by the quantity pil (R-,), where pol stands for the dipole transition matrix element and R for the intermolecular distance between nearest-neighbour molecules in a cluster.On the other hand, the information from the ground state of H2 dimers resides in the shift that the h.f.s. lines experience due to the (angle-dependent) forces between the constituents. These forces slightly perturb, in a first approximation, the free rotation of the ortho-hydrogen (.j = 1) within the complex. Although admixture of higher.j states (e.g. j = 3) can still be neglected due to the large rotational constant of H2 (60 cm-I), the orientational coupling (determined by the quantum number mj) causes deviations from straightforward vector-model expectations. These deviations lead to h.f.s. shifts of ca. 1 kHz for (H,), to ca.100 kHz for H,-Kr. A theoretical model allows one then to translate these observed shifts into a single-potential para- The information obtained from the two experiments is very different.3 76 D 1 M ER SPEC TR 0s COPY meter for each system probing espucially the r/, (R)P, (2 i) contribution to the inter- molecular potential; R * i stands for t h u cosine of the angle hetween the moIeculnr axis, r , and the intermolecular distance vector, R. 2 . PREDISSOCIATION O F VIBRATIONALLY EXCITED SF, CLUSTERS 2.1. CALCULATIOI: O F TIIE S P E r T R U h l In fig. 1 an illustration by Lconardo da Vinui demonstrates the geometry of an octonier; each corner might be thoiight of 3s thc. position of ;in SF, molecule in the 1 XxKxl I1 I I , 1 FIG. 1.-Leonard0 da Vinci's illustration of an octomcr, from ref.(I). case of (SF,),. The special feature of this gcoinetry is that it can be stripped down one by one leading through hepiamer, hexamer P t c . , finalIy to the dimer and the monomer, each in its supposed cquil i briuni geometry.' We treat vibrationally excited (SF,),, clusters in the approximation that the 3n dcgenerate states with o n u v3 q u a n t u m ahsorbed are diagonalized, taking as off- diagonal coupling elements the terms derived from the electric dipole-dipole inter- action. ResponsihIe for the strong coupling is the dipole transitioii eIement of SF, for thc i q 3 mode, 0.39 D,* Each SF-', rnoleculc is ireated as a three-dimcnsional harmonic oscillator, neglecting Coriolis cffects. The positions of the SF6 molecules constituting the octoiner are given in table 1.The off-diagonal elements are shown in table 2, together with the energy eigenvalues and the corresponding transition strength. The proccdure is discirssed in ref. (3), here extended up to octomers. I n fig. 2 the results are displayed in a stick spectrum, where the height corresponds to the transition strength times the degeneracy. I t is * 1 I 1 2 3.3356 i< C m.J . GERAEDTS, M . WAAYER, S . STOLTE A N D J . REUSS 377 assumed that the nearest-neighbour distance remains unchanged for clusters growing from dimers to octomers. The absolute calibration is then taken from the precisely measured distance of the two sharp dimer lines. 31) ~ m - ' , ~ corresponding to ,L&(R-~) = 6.8 cm-' and (R-3)-1/3 = 4.8 A.TABLE 1 .-EQUILIBRIUM POSITIONS OF THE CLUSTER CONSTITUENTS, ACCORDING TO REF. (2), IN UNITS OF NEAREST-NEIGHBOUR DISTANCES 1 2 3 4 5 6 7 8 Y i 0 1 2 - - 1 2 - 0 0 0 5 6 - - 5 6 - In ref. (3) experiments were reported with fits up to the trimers, the latter especially contributing a significant absorption around 962 cm- '. Under conditions of heavy clustering some gross features were communicated in ref. (4) which are not contradic- tory to the stick spectrum (fig. 2). However, further and precise experimental confirmation is still needed. The above treatment resembles the theory of the Davydov effect [e.g. ref. (4)], where the vibration spectrum of solids with more than one molecule of the same sort per unit cell yields splittings owing to electric-dipole forces similar to those found by us.2.2. THE EXPERIMENT To measure the predissociation of clusters by vibrational excitation a molecular beam (containing clusters at concentrations determined by the choice of the source conditions) is crossed by a line-tunable CO, laser of moderate power. After excit- ation the clusters leave the collimated beam owing to the recoil upon dissociation. Thus a beam attenuation is measured which reflects the absorption spectrum of the clusters. Beam detection takes place using an efficient electron-bombardment ionizer, followed by a magnetic mass spectrometer. As violent fragmentation of the cluster ions takes place in the ionizer we have generally measured the attenuation spectra on mass SF,+. Quantitative conclusions about this fragmentation and details of the ex- perimental arrangement are discussed in ref.(3). So far we have not been successful in utilizing the selective properties of the mass spectrometer to disentangle the congested spectrum in the case of heavy clustering. Normally, the beam is ionized by 100 eV electrons. Reducing the electron energy to 25 eV was of no avail; the monomer fragmentation appeared markedly diminished whereas the clusters unvaryingly kept their propensity to show up at the mass of the SF,+ ion. 2.3. POWER DEPENDENCE A N D LINE-SHAPE In ref. (3) we have described a simple model, which, owing to dissociation, yields [I - exp(--r,t)] for the relative attenuation of dimers in the beam, where r2 de- scribes the rate of excitation of dimers to a predissociative state and t the time of interaction.As common practice we had written [see eqn (4.3) of ref. (3)]378 6 - 4 - 2 - 6 - 4 - 2 - 6 - 4 - 2 - 6 - 4 - 2 - 6 - 4 - 5 5 2 - 6 - 4 - 2 - 6 - 4 - DIMER SPECTROSCOPY 1 I I 1 I I I I - - monomer - - 1 dimer I - - - trimer - - - tetramer - - - pentamer I I I - - - hexarner I I I II 1 - 1 , - - heptamer octomer I t 1 I I t I I I - 4 - 3 -2 -1 0 1 2 3 4 A FIG. 2.-Stick spectrum of (SF,),,, n 1 1 , . . . .,8. The height of each bar corresponds to the Note how the The three largest transition strength times degeneracy. high symmetry of the tetramer and the octomer is reflected by high degeneracy. The energy shift A is given units of pi1(R-3>. clusters contribute lines near A = 0.J . GERAEDTS, M . WAAYER, S .STOLTE AND J . REUSS 379 The fluency is indicated by F ; r2,ml g2,m and S2,m describe the line-width, the degener- acy and the line-strength of the mth transition of the dimer, m2,,, being the correspond- ing transition frequency at the centre of the assumed Lorentz line profile. In the following, we wish to take into account the possibility of orientational hole- burning effects, which in eqn (I) are neglected, introducing the factor 1 / 3 on the right- hand side, i.e. by taking the usual spatial average over all orientations of the dimer axis. As long as saturation effects are weak, this is a permissible procedure; however, as soon as the fluency exceeds a few tenths of 1 J m-2 these orientational effects show up clearly for (SF,), predissociation, as already indicated in ref.(4), fig. 3, where the asymptotic beam attenuation for large laser power was shown to no longer foIlow an exponential law. The blue dimer peak around the P(8) CU2 laser line corresponds to an excitation of the two oscillating dipole moments parallel to each other and perpendicular to the dimer axis. If the end-over-end rotation of the dimer is characterized by the quantum numbers L and ML, ML being defined along the direction 2 of the polarization of the C02 laser, one finds that the blue peak corresponds to a transition with IALI = 0,l. In this case, the factor 3 in r,t has to be replaced by $(l + cos20) and to be spatially averaged over a11 directions, in the exponent of [l - exp(--r,t)]. Here, 0 stands for the angle between L and 2. The red peak around the P(30)C02 laser line corresponds to an excitation of the two oscillating dipole moments parallel to each other and parallel to the dimer axis.Thus one has L --f L 5 1 transitions with the factor 4 replaced by 3 sin20. Again one has to average over all directions with equal weight, sin2 8 occurring in the exponent of [I - exp(- r 2 t ) ] . It is evident that for weak fluency with El - exp(- r2t)] z r2t this spatial averag- ing yieIds the factor 1/3, in a11 cases. In general, however, I - exp(- v,t) becomes, for the red line For the blue line one finds ~ ' ( 3 r z r 1 4 ) exp( - 3v2tj4) exp( - X2)dX2 spatial averaging 1 1/(3r.d/4) i - exp(-r,r) ~ - 1 - 0 exp(- 3r,t/4). 2 d ( 3 f - 2 1/41 (for strong fluency) --+ 1 - Here, r2t is defined as in eqn (I).The asymptotic formulae show that the blue line reaches its saturation value significantly faster than the red one, in disagreement with fig. 3 of ref. (4). The line-shape, too, is affected by this spatial averaging, for high Auencies. Whereas the far wings of the Lorentzian remain unchanged the line-width may appear wider than corresponds to a power-broadened pure Lorentzian. However, whereas * To be consistent with the use of c.g.s. units in the rest of the paper, put E,, = 1147~.TABLE 2. -ENERGY MATRIX, EIGENVALUES AND TRANSITION STRENGTHS OF v,-EXCITED (SF,),, n = 1, . . ., 8 upper right) is symmetric and has to be complemented by values reflected with respect to the diagonal; the diagonal elements are all equal (948 cm-1). near-diagonal n = n' being zero.For the operator the dipole-diple interaction is chosen, [(ui . pj) - 3(pi. Rij)(pj . Rtj)]/R;j. The eigenvalues E (lower left) were obtained by units of pil( R-3), as are the matrix elements. The corresponding transition strengths transition strength. The indication for an unperturbed degenerate state (e.g. 7z) is a shorthand (meaning that of all 3n possible va-excitations the 7th molecule is excited Here, poi corresponds to the transition matrix elements for the v3-mode (0.39 D). 2x 2y 22 3x 3y 32 4x 4y 42 5x 5y 5z 6x 6y 62 7x 7 y 72 8x -1.85 -1.85 -1.85 0.50 0.50 2.39 2.39 2.39 5.00 0 2.23 2.23 2.23 0 0 1.77 1.77 1.77 0 -5432 - 5 -344 -5 3 2 -42 42 - 1 4 8 2 -543 -242 2 4 3 4 3 3 36 36 18 36 - - 3 4 4 1 -543 -13 -546 9 36 0 0 1 0 0 1 0 0 3.\/32 0 - 6 36 12 -148 14/16 -242 -546 1 - 4 2 0 0 1 0 0 1 - 4 2 0 - 1 d 2 0 - 1 - 3 O - - - 6 18 9 - - 9 9 3 -144 2 2 3 -144 -142 -13 7412 -749 --7 - 4 d 3 3 .3 1 / 3 9 8 4 8 4 3 2 7 --- 3 2 3 6 3 2 9 9 1 0 0 - - E -144 -! 142 -1d4 _1 -142 7-\/12 J- 1 s 4 2 2 -246 -3 1 1 2 2 2 0 9 T -- 9 4 3 3 3 6 3 4 6 3 4 6 -142 --142 -749 * d2.8 -2116 2/3 2 6 2 6 2 6 9 9 9 9 16 0 - 2 0 - strength 0 0 1 1 4 2 142 - 1 - - 1 E 3 12/4 1 s 3 1 4 4 -142 -13 -743 -749 4 3 3 43 -7 4 3 2 z --- 3 2 36 -- 12 2 8- 4 3 16 - 9 - -1.00 -1.00 1.43 1.43 2.00 3.50 I 144 -142 1 2 6 3 2 6 2 0 0 0.83 0.83 3.00 0 I z2 - 1 - 142 -.- - strength -142 -7412 1 - 1 4 3 -- - - 4 6 3 4 6 142 ~ -749 -143 1 2 6 9 -- 6 I - - - 3 4- 42 16 -343 8 d 8 -3 16 - -443 9 8 4 2 9 - 3* 0 6 0 - -6d32 5./9 6 2 0 1 0 5.\/9 2 0 7 9 - 11 36 - 2543 36 18 - -542 - 2543 36 - - 13 12 18 - 5L6 -542 11 - - - 18 36 546 -2543 ---- 18 36 7 -542 9 - - - 18GERAEDTS, M .W A A Y E R , S. STOLTE AND J . REUSS3 82 DKMER SPECTROSCOPY one observes pronounced effects for the above-discussed power dependence, the changes of the line-shape amount to at most 2% for the red line and 10% for the blue line ; this can be seen from table 3. In fig. 3 experimental results are compared with a fit with and without proper spatial averaging, for fluencies differing by about a factor 10. TABLE 3.-CHANGE OF ABSORPTION FOR PROPER SPATIAL AVERAGING The first column displays different values for the transition probability, r2t. For the red transition of (SF,), vred = 11 - h-1 exp ( - - P ) / ~ ~ I .*eexp ( . ~ 2 > 1 / [ 1 - exp ( - r Z t ) l is shown; here, b stands for (3r2t/2)+.in the last column. The equivalent ratio for the blue line is given red line, blue line, r2 t vred Vbluc 0.1 1 .oo 1 .oo 0.5 0.96 0.99 1 .o 0.93 0.99 1.5 0.92 0.98 2 .o 0.9 I 0.99 10.0 0.97 1 .oo experimental error:+ I v1crn-l FIG. 3.-Beam attenuation plotted against laser frequency. The full circles and curves with narrow peaks correspond to 1 W laser power, i.e. 1.3 J m-' fluency. The open circles and wider curves correspond to 10 W, i.e. 13 J m-2 fluency, The calculated curves are fit with the same parameters (r =: 1.6 cm-l, ,u&(R-~> = 6.75 cm-', unshifted frequency 948.5 cm-I)), applying eqn (3) and (4); broken lines are calculated using eqn (l), i.e. without proper spatial averaging, The two strong- fluency peak heights correspond to a beam attenuation of O.S%.2.4. PURE AND M I X E D DIMERS The vj vibration frequency of SF6 is especially sensitive to isotopic substitution. For 34SF6 (4% natural abundance) the excitation occurs 16 cm-l red-shifted withJ . GERAEDTS, M . WAAYER, S . STOLTE AND J . REUSS 383 respect to the 32SF, molecule (ca. 948 cm-l). Dimer spectra are obtained by diagonalizing 2 x 2 sub-matrices (see the general scheme in table 2). The results of this diagonalization are given in table 4, including the corresponding eigenfunctions. For pure dimers all diagonal elements are identical, for the mixed ones the difference of 16 cm-I leads to the significantly altered eigenfunctions and eigenvalues of the lower part of table 4.The symmetry breaking renders all four eigenvalues observable, in TABLE 4. -EIGENVALUES, TRANSITION STRENGTHS AND EIGENSTATES OF (32SF6)2 AND 32SF6*34SF6, CALCULATED WITH ( R - 3 ) = 6.8 Cm-' eigenvalues/ cm- g strength eigenfunction - 24.7 1 1.85 0.49 lzl+ ) +0.87 lz; ) - 19.6 2 0.36 -0.941x,+) +0.341x,f) or x+y +2.4 2 I .64 0.341x,f) +0.941x,f) or x+y 1-7.5 1 0.14 -0.871~,+) +0.49]~,+) principle; up to now we have searched and found only the line shifted to 921.5 cm-', slightly different from the theoretical prediction but this difference still being within the experimental ~ncertainty.~ 3. T H E HYPERFINE STRUCTURE OF MAGNETIC TRANSITIONS OF H2-H,, H2-Ne, H,-Ar AND H,-Kr 3.1. THE EXPERIMENT The experimental apparatus for m. b.r. measurements has been described in detail in ref.(6) and (7). It consists of a beam machine with two deflection fields each 40 cm long, a field-free transition region where transitions are induced by means of Ramsey's double-coil technique,8 a sensitive detection system which permits the measurement of the dimer intensities on the masses H3+ for (H2)2, HNe+ for H,Ne, HAr' for H2Ar and HKr+ for H2Kr, and a nozzle-skimmer system to produce dimers by supersonic expansion, in sufficient concentrations. It is a convenient feature of the systems studied here that their transitions can be well predicted so that lengthy scanning of wide spectral ranges can be avoided before finding the first inkling of the presence of dimers. In any event, to resolve lines from a large noisy background many hours of signal averaging were required utilizing a completely automated set-up.The quantum numbers L and ML describe the end-over-end rotation of the com- plexes, the L = 0 states being predominantly occupied as we achieved rather low internal temperatures (T = 3 & 2 K) during the expansion. States with L = 0,1,2 were investigated. 3.2. OUTLINE OF THEORY dependent forces within a H,-X dimer are rather weak. The theoretical analysis is facilitated by a number of facts. First, the angle- Although we adhered to384 DIMER SPECTROSCOPY exact methods of diagonalizing the complete Hamiltonian [ref. (6) and (7)] pertur- bative approaches have served to guide us and to give a good insight into the physics involved. Secondly, as the rotational constant of the H2 molecule is large (60 cm-') the admixture of higher j states becomes negligible, i.e.the H, molecule rotates freely inside the complex. Thirdly, H2 being homonuclear there only occur even Legendre polynomials in the expansion of the intermolecular potential V : v = V,(R) + V,(R)P,(L' * A) + V4(R)P4(? * i) + . . . . Furthermore, the V@) terms even can be omitted in our case, as we have seen that we can restrict ourselves to t h e j = 1 state of ortho-hydrogen, which does not couple with the fourth-order Legendre polynomial. Roses hardly bloom without thorns; a number of complicating factors must, therefore, be mentioned. As we are considering hyperfine states we have to cope with the nuclear spin I of H,. For the coupling scheme we have chosen [(jL)Jl]f; with j = I = 1. Further, the continuum inffuences the h.f.s.transitions once we have made our choice for this coupling scheme; e.g. L = 2 states are predissociating for ortho-hydrogen-para-hydrogen dimers, (H2),, but they are still the main terms yielding off-diagonal elements in the diagonalization procedure. Consequently, one has to incorporate the continuum states in the analysis. In the following we restrict ourselves to h.f.s. states which, apart from some ad- mixture, belong to L = 0; the general analysis is extensively treated in ref. (7). The chosen coupling scheme (without off-diagonal elements) leads to a transition [ ( j = 1, L = 0) J = I , I = 11 F = 0 + F' = 1 with a frequency at 546.437 kHz, unshifted with respect to the corresponding H2 h.f.s. transition and independent of the dimer partner X.In table 5 the line positions are given for the various clusters as calculated For four systems, results from theoretical or empirical potentials [ref. (9)-( 12)J are compared with experimental ones [ref. (7)]. In the last column, the transition frequencies are given (to be compared with the unshifted value of 546.437 kHz). In the next to last column, corres- TABLE 5.-RESULTS OF Hz DIMER H.F.S. MEASUREMENTS ponding values of the dimensionless experimental parameter (see text) are given. ~~ transition/kHz system ref. I V2,/(& - &)I L = O , J = l,F=O-+F'=l H2-Ne 10 0.041 9(29) 1 1 0.1663(8) 12 0.1696(8) (expt.1 7 0. I892(78) 546.36 545.12 545.07 544.74( 1 5) H2-Ar 10 0.97 79( 4) 11 0.9491 (4) (expt.1 7 1.0537 497.12 499.84 489.94( 15) H2-Kr 10 1.73 18(7) 11 1.3472(7) (expt.1 7 I .622( 15) 421.86 460.24 432.42 transition/kHz system ref.[XI V Z O / ( G - mI:d' L = 0, J = 0, F = O+F' = 1 V' H2-H2 9 0.174(2) (expt. 1 7 0.23(1) 545.07 544.05(10)J . GERAEDTS, M . W A A Y E R , S . STOLTE A N D J . REUSS 385 using recent potentials for (H2)2,9 for H,-Ne and for H,-Ar, H,-KT.~O*~~ The shift varies from ca. 1.5 kHz for (H,), to ca. I 15 kHz for H,-Kr. In all cases, the V2(R) term of the intermolecular potential is mainly responsible for this shift: having chosen to analyse here only o-H,-p-H,, even the normally dominant quadrupole- quadrupole interaction between hydrogen molecules is eliminated. The shift in the h.f.s. transition frequency reflects the value of the parameter 1 V2,/(E2 - Eo)( (see table 5). It turns out that (E2 - E,) is astonishingly system- independent, corresponding to ca.96 & 4 GHz for H,-Ne, H,-Ar and H2-Kr. Here, EL designates the energy of bound dimer states with L = 0 and 2, calculated from the isotropic potential V,(R) and corresponding to zero Van der Waals stretching excitation. Consequently, the large variation of this shift is predominantly due to the different values assumed by lV2,1. It does not mean, however, that the isotropic potential has no influence, as V,, corresponds to the matrix element of V,(R) sand- wiched between the Van der Waals stretching vibration functions (RIL,v) with v = 0 and L = 0 and 2. Classically the V2 term is probed between the turning points of the vibrational eigenfunctions, the precise position of the turning points being deter- mined by the isotropic term Vo(R).[In reality the probing region extends well beyond these turning points, as is discussed in ref. (7).] In the case of (H,),, the L = 2 state is not bound, and therefore one has to deal with continuum states. Nevertheless, the determining parameter still permits one to recognize its counterpart for bound L = 2 states, only that here a number of states (obtained with box-normalization) ' contribute additively. This parameter is nearly independent of where precisely the properly chosen box-normaliz- ation is introduced; we used R,,, = 32 A. 3.3. RESULTS In table 5 experimental results are displayed yielding the corresponding molecular parameter of the first column. For H,-Ne, excellent agreement was found with the empirical potential obtained by the Gottingen group of Buck and coworkers l 3 from elastic and inelastic differential scattering measurements, simultaneously taking into account total cross-section measurements with oriented H, molec~le.'~ For evidence of a small blister term in V2(R) in the neighbourhood of the potential well see ref.(7). For H,-Ar and H,-Kr, the agreement with existing empirical and semi-empirical potentials in general is less satisfying; however, it is not disquieting, in view of one's still rather crude knowledge of these systems. Thepi2ce de rksistance seems to be the system (H2)2, where very precise experimental and empirical potentials e x i ~ t , ' ~ ~ ' ' as well as an excellent and highly sophisticated ab initio calculation by Meyer.' The discrepancy arises near the zero crossing of Vo(R), where experimental data point to a 0.1 A shift to shorter distances with respect to the ab initio results (see fig.4).l53l6 Based upon the dimer-spectroscopic results we had to add a sizeable blister term to V2(R) [admittedly in an entirely ad hoc manner, see ref. (7)] to find agreement with the experimental molecular parameter of table 3 without impairing the agreement with other experimental findings. In particular, the inelastic differential cross-section measurements are expected to go along with the potential of fig. 4, as care was taken to conserve V2(R)/ V,(R) in the repulsive region, a requirement dictated by recent results of Buck.16DIMER SPECTROSCOPY RIA FIG. 4.--lnterrnolecular potential for ortho-H2-para Hf, i.e. the isotropic Vo(R) term and the V2(R) term. The broken lines correspond to ab inifiu caIcuIations, the full curves represent empirical adjustments of these ab iniriu results. This work is part of the research program of the “ Stichting voor Fundamenteel Onderzoek der Materie (F.U.M.) ” and has been made possible by financial support from the “ Nederlandse Stichting voor Zuiver-Wetenschappelij k Onderzoek ( Z . W.O.).” Leonard0 da Vinci, in De diflinaproportione by L, Pacioli (MiIan, 1496). J. Geraedts, S. Stolte and J. Reuss, 2. Phys., in press. J. Geraedts, S. Setiadi, S. Stolte and J. Reuss, Clzern. Phys. Lett., 1981, 78, 277. D. P. Craig, in Physics and Chemistry of rhe Organic SoIidSrate, ed. D. Fox, M. M. Labes and A. Weissberger (J. Wjley, New York, I963), vol. 1. J. Verberne and J. Reuss, Chem. Phys., 1980, 50, 137; 1981,57, 189. M. Waayer, M. Jacobs and J. Reuss, Chem. Phys., 1981, 63, 247; 257; 263. N. F, Ramsey, Mo/eculav Beams (Oxford University Press, 1956). W. Meyer and J. Schafer, personal communication. Carley, Adv. Chem. Phys., 1980, 42, 353. ’ A. Bonissent and B. Mutaftschiev, J. Chem. Phys., 1973, 58, 3727. lo J. S. Carley. Thesis (University of Waterloo, Waterloo, Ontario, 1978); R. J. Le Roy and J. S. l1 K. T. Tang and J. P. Toennies, J, Chern. Phys., 1978, 68, 5501 ; 1981, 74, 1148. l2 J. Andres, U. Buck, F. Huisken, J. Schleusener and F. Torello, J. Chern. Phys., 1980,73,5620. l3 L. Zandee and J. Reuss, Chem. Phys., 1977, 26, 327; 1977, 26, 345, l4 U. Buck, F. Huiskens, J. Scheusener and J. Schafer, J. Chem. Phys., 1981, 74, 535. l5 I. F. Silvera and V. V. Goldmann, J. Chem. Phys., 1978, 69, 4205. l6 U. Buck, personal communication.

ISSN:0301-7249    

DOI:10.1039/DC9827300375

出版商:RSC    

年代:1982

数据来源: RSC

摘要:

Faraduy Discuss. Chem. Soc., 1982, 73, 387-397 Vibrational Predissociation Spectra and Dynamics of SmalI Molecular Clusters of H20 and HF BY MATTHEW F. VERNON," JAMES M. LISY,? DOUGLAS J. KRAJNOVICH," ANDRZEJ TRAMER, $HU€-SING KWUK,S Y. RUN SHES 7 AND YUAN T. LEE * Materials and Molecular Research Division, Lawrence Berkeley Laboratory, University of California, Berkeley, California 94720, U.S.A. Receitvd 9th December, 1381 Experimental results are presented for the vibrational predissociation spectra in the frequency range 3000-4000 cm-I for the species (HF), and (H,O),, n = 2-6, using moIecuIar-beam techniques and a tunable infrared laser. The observed spectra show a dramatic change between the dimer and larger clusters which is thought to be a result of the cyclic structure of the trimer and larger clusters.The spectra are compared with calculated harmonic force constants of available intermolecular potentials to understand how these small, gas-phase dusters relate to the liquid and solid phases of HF and H,O. Additionally, the angular distributions of the predissociation products show that little energy appears as translational motion of the fragment molecules. This conclusion is consistent with recent theoretical rnodeIs of the predissociation process. An upper limit of ca. 2 p s is observed for the lifetime of the vibrationally excited clusters. Van der Waals (VDW) molecules have long been suggested as a well defined, accessible model for studying the spectroscopic and dynamicaI properties of condensed phases. Hydrogen-bonded liquids, in particular, have received much attention because of the noticeable spectroscopic changes which occur in the Iiquid and solid phases.' Despite the extensive study, little is known about the detailed electronic restructuring (potential-energy surface) which is responsible for the large red shifts and intensity changes associated with hydrogen bonding.' Until the physical mechanisms for these changes are known, it will be difficult to parameterize analytical model potentials in an accurate and economical way.Tndeed, most potential models proposed to date consist of simpIe atom-atom interactions which rarely predict more than one property correctly over a substantial range of conditions.2 The experimental results presented below were obtained in an attempt to understand the rate at which the bulk-phase spectroscopic behaviour is approached for these two molecules.* Also associated with the Department of Chemistry, University of California, Berkeley, Cali- t Permanent address: School of Chemical Sciences, University of Illinois, 505 South Mathews 1 Permanent address : Laboratory of Molecular Photophysics, CNRS, Universite Paris-Sud, 5 Permanent address: Department of Electrical Engineering, State University of New York, 'r Also associated with the Department of Physics, University of California, Berkeley, California, fornia, U.S.A. Avenue, Urbana, Illinois 61801, U.S.A. 91405 Orsay, France. Buffalo, New York 14226, USA. U S A .388 DYNAMICS OF SMALL MOLECULAR CLUSTERS EXPERIMENTAL The (HzO), and (HF),, clusters are formed in an adiabatic expansion using neat or rare- gas mixtures of HF and HIO.The infrared radiation is obtained from a Nd : YAG pumped LiNb03 optical parametric oscillator (OPO) based on the L-shaped cavity design of B ~ e r . ~ For the frequency range 3000-4000 cm-’, the pulse energy is 1-4 mS with a pump energy fluence of 1 J cm’z, and the linewidth (f.w.h.m.) varies from 3 to 4 cm-I. The predissociation of the clusters is detected in two different configurations using a mass spectrometer. PERP END I C U LAR LASER- MOL E C U LA R - B EA M ARRANGEMENT Fig. 1 shows an in-plane view of this experimental configuration. The vibrational pre- dissociation is observed by monitoring the appearance of the predissociation fragments. FIG. 1 .--In-plane view of perpendicular laser-molecular-beam apparatus.Labelled components are: (1) 0.007 in, diameter quartz nozzle heated to 125 C, (2) first skimmer, (3) second skimmer, (4) third skimmer, ( 5 ) power meter, (6) germanium filter, (7) ionizer assembly, (8) quadrupoIe mass spectro- meter. 0 measures the angle of rotation of the detector from the molecular beam. The electron bombardment ionizer-mass spectrometer assembly rotates on a 20 cm radius circle about the intersection point of the laser and molecular beams. The defining apertures in the mass spectiometer limit the detector’s line of sight to a 3 mm x 3 rnm region at the intersection point. Clusters that predissociate before travelling beyond this viewing region (nominal residence time is 2 ps) and whose fragments recoil into the detector are detected as an increase in the ion signal at the masses characteristic of the cluster predissociation fragments.The predissociation signals are collected by a muItichanne1 scaler triggered by the laser pulse and accumulated for 2000-10000 laser pulses. The laser power is monitored by a power meter placed after the intersection region. The power dependence of the signal is measured to ensure that the predissociation yield is linear with photon number. The angular distri- bution of the predissociation fragments is used to determine the translational energy distribu- tion of the product molecule^.^M. F. V E R N O N ~ ~ al. 389 C O A X I A L LASER-MOLECULAR-BEAM ARRANGEMENT A cross-sectional view of this apparatus is shown in fig.2 . Here, the laser and molecular beams are superimposed while the mass spectrometer monitors the molecular beam. Vib- rational predissociation is observed as depletion in the signal of the parent cluster mass. FIG. 2.-Side view of the coaxial laser-molecular-beam apparatus. Labelled components are: (1) BaFz entrance window for the OPO beam, (2) quadrupole mass spectrometer, (3) ionizer assembly, (4) final molecular-beam-defining aperture, ( 5 ) second skimmer, (6) first skimmer, (7) nozzle. Compared with the perpendicular arrangement, this method has thirty times the interaction volume between the laser and molecular beams, and detects all predissociation events, in contrast to the small fraction sampled by the perpendicular arrangement. The increase in the detectable signal (ca.lo6) permits the use of 1 .O-0.1 % seeded beam ratios (HF or H,O in a He or Ar carrier) and faster data acquisition. However, it cannot supply angular distribution information. RESULTS Vibrational predissociation spectra of (H20), and (HF), are displayed in fig. 3-6. The angular dependence of the (H20),, predissociation fragments is shown in fig. 7. Similar results were obtained for (HF),t. The narrow angular range of the products (0 < 10') indicates that little of the excess energy appears as product translation. The steep monotonic decrease in the angular data with increasing laboratory angle can only be fitted by a product translational energy distribution which peaks at or near zero degrees. The excess energy thus remaining after the cluster dissociates must be in the internal degrees of freedom of the fragments.For energetic reasons * the dominant predissociation channel for both (HF), and (H20),, is thought to be (H20)n + Iiv --+ (H20)n--1+ H20 (HF), + hv --f (HF)n-, + HF (1) (2) since this is consistent with the observed independence of the shape of the angular distribution with OPO frequency. The electron bombardment ionization cam- * Calculations of the dissociation energy, D,, of water clusters using a variety of intermolecular Dissociation into non-monomeric potentials indicate that this process [eqn (l)] is always allowed. products is not allowed over the complete frequency range probed for these same potentials.390 D Y N A M I C S OF SMALL MOLECULAR CLUSTERS 1.c 0 * 5 0 .0 1 . a 7 0 . 0 C M ._ ul 1 . 0 0.5 0 . o 1 . o 0.5 . (el 0 .o 3000 3200 3400 36003800 3000 3200 3400 3600 3800 f'requency/cm - FIG. 3.-Water-cluster and condensed-phase spectra. Panels (a)-(e) are spectra observed in the present work. (a) (H20)3, (b) (H20j4, (c.) (H20j5, (4 ( H z ~ ) ~ , ( P ) (HzO)~, ( f ) NZ matrix, ( g ) liquid, ( h ) solid. plicates the assignment of the signal observed at il particular mass peak to a specific cluster. The dominant ionization channel is most probably (H20),, +-- e- -+ (H,O),-,H+ -1 OH $- 2e- (3) (HF), - I - e- 3 (NF),-,H-+ + F -1 2e-. (4) However, the channels (H20), 4- e- -+ (H,O),,-,-H + 1.- . . . (HF),, + e- + (HF),l-,n- Ht + . , .M . F . V E R N O N ~ ~ al. 39 1 with rn > 1 are also observed to occur. Fig. 3 (a) and (e) show two frequency scans at the H30+ mass under two different source conditions.The changes observed for the two different cluster distributions are a result of the larger cluster predissociation fragments from process (1) ionizing by channel (5) with m > 1. In cases where 3 500 3600 3700 frequency /cm - FIG. 4.-Vibrational predissociation spectra of (H20)2 and (H20)3 obtained by the coaxial method. several clusters absorb at the same frequency, varying the source pressure and seeding ratios and recording the spectra at many masses can distinguish the bands belonging to different polymers. DISCUSSION From the data displayed in fig. 3-6 it is obvious that the dimer species (H,O), and (HF), are substantially different compared with the larger clusters. Molecular-beam electric resonance spectroscopy has established the ground vibrational state structures of (H20), and (HF)2.6 The three (HF), bands are assigned to the HF stretch of the hydrogen-bonded proton (3720 cm-'), the HF stretch of the " free " proton (3878 cm-l), and a combination band involving an intra- and inter-molecular vibration (3970 cm-l).The (H,O), bands are similarly assigned to free (3715, 3625 cm-') and hydrogen-bonded (3596, 3524 cm-') protons. The bands at 3715 and 3500 cm-' are also seen in the spectra of larger water clusters. Their presence at the dimer parent mass could be due to the larger neutral cluster ionizing via eqn (5). More detailed studies will be necessary to measure unambiguously the complete (H20), spectra in this region. The (HF), and (H20),*, n = 3-6, clusters exhibit a strong, broad, red-shifted band structure which is assigned to the hydrogen-bonded protons.The lack of any absorption peaks above 3500 cm-l for the (HF),, n = 3-6, clusters indicates the absence of a terminal - - H-F or - * * F-H molecule. Consequently, these larger clusters must be cyclic. Similarly, the absence of a band at 3596 cm-I for the larger392 10 a 6 4 m 2 A Y .- 5 .E 0 c 10 8 6 4 2 0 DYNAMICS OF SMALL MOLECULAR CLUSTERS 3100 3200 3300 3400 3500 water clusters is consistent with a cyclic geometry for these clusters. The sharp, narrow peak at 37 I5 cm-' is assigned to the " free " hydrogens which point away from the ring. Typical minimum energy water polymer structures predicted by available intermolecular potentials are shown in fig.8 and agree with this assignment of the observed spectra. Molecular-beam deflection experiments for the larger (HF), and (H20),t clusters indicate small to undetectable dipole moments for these species, as wouId be expected for cyclic structures. The band structure of the hydrogen-bonded absorption seems to be best described as a progression of combination bands. The equation 4- t d n t e r ? m = 0,lJ (7) V ~ b s - Vintra where vintra is the fundamental frequency of the hydrogen-bonded intramolecular H-F(H0) stretching motion, Per is a frequency of the intermolecular F * - - HFM . F. VERNON et al. 393 H 10- I 8- -I I-Av tr 4 - 2 - 0 I 3700 3800 3900 .4( frequcncy/cni - 1 O l 1 30 FIG. 6.--Vibrational predissociation spectrum of (HF)2 corrected and normalized for the photon number, with the structure of (HF)2 as determined by tnolecular*-beatn electric resonance spectroscopy.10 9 8 7 6 5 4 3 s 2 -% E: on .& vl .N, 10 2 9 5 6 5 5 4 3 2 5 ; 1 \ \ I 0 1 3 4 5 6 7 8 9 labor at ory angle /" FIG. 7.--Laboratory angular distributions for the detected mass: 0, H,O' ; I-, (HzO)ZH+ ; u, ( H ~ w ; a, ( H ~ ~ ) ~ H + .394 DYNAMICS OF SMALL MOLECULAR CLUSTERS FIG. 8.-Minimum energy cluster geometries for the Watts potential energy function (a,b,d,e,f), for the polarization model trimer (c), and for the configuration characterizing a transition state for the tetramer aissociating into a trimer and a monomer (g). (0 - HO) hydrogen bond, and vobs is the frequency of the observed bands, is con- sistent with the more advanced treatments of hydrogen bonding.For the (HF)n series, the decrease of vintra and increase of vinter with increasing cluster size is in accordance with progressively weaker intramolecular bonds and stronger inter- molecular bonds in the larger clusters. The question arises as to why the hydrogen-bonded protons of the cyclic structure are so red-shifted compared with the dimer species. Since the dimer absorptions are characteristic of the pair-interaction potential, it might be a consequence of the changeM. F . VERNON etal. 395 in the reduced mass of the proton motions in the cyclic as opposed to the open dimer configuration. To investigate this possibility, the harmonic vibrational frequencies at the minimum energy configurations of the (H20)n polymers were calculated using 3 widely used potentials.? -9 The harmonic frequencies of the polymer vibrations were multiplied by a constant chosen to bring the calculated monomer vibrational frequencies into close agreement with the known vibrational frequencies of H,O.This was meant to serve as a crude correction factor for anharmonicity. The best agreement in all cases was from the Watts potential,' which uses the known gas-phase H,O anharmonic vibrational potential to describe the intramolecular forces. The polarization * and central force9 models of Stillinger do not predict the appropriate tight grouping of the free OH peaks, and also indicate blue shifts relative to the monomer. These discrepancies are expected since no particular attention was paid to predicting the high-frequency proton vibrations.The Watts potential predicts the observed dimer spectra quite accurately given the approximations of the normal-mode treatment and the type of data used to derive the intramolecular potential, namely second virial coefficients and solid-state nearest-neighbour distances. When this pair potential is applied to larger clusters, the calculated spectra show two band groups : the free and hydrogen-bonded protons. The calculated frequencies of the larger clusters are more commensurate with the observed and calculated dimer bands than the red shifts observed in the larger clusters. We conclude that the pair potential cannot account for the red shifts, which must then be a result of non-additive interactions.When an OH group is involved in both donating and accepting a hydrogen atom for hydrogen bonding, the electronic structure of that OH bond is significantly perturbed beyond that accountabIe for by pair interactions. An interesting implication of our assignments is how they suggest that past explanations for the infrared structure of liquid water are in error." The basic con- clusion of these previous studies was to assign each observed (or deduced) band to a different local environment. The present work, along with theoretical treatments by Rice and Sceats,l' Marechal and o t h e r ~ , ' ~ suggests that the broad-band structure is intrinsic to the hydrogen bond. Recent studies by Byer et aI.l4 on the depolariza- tion CARS spectra of liquid water have shown similar structure in the wavelength region of this study.Two aspects of the predissociation dynamics were also observed in our experi- ments, an upper limit to the vibrational predissociation lifetime of the cluster, and the partitioning of energy between translational and internal degrees of freedom of the predissociation products. The lifetime was determined by our observations of pre- dissociation (in collision-free conditions) in the perpendicular beam arrangement, and the agreement obtained between spectra measured by the two experimental configurations. Although the information from the product angular distribution is limited, it is possible to understand why little energy appears as translation and why the shape of the angular distributions are independent of the photon energy for the range probed (9- 10.5 kcal).Calculations were performed of the locally adiabatic predissociation (LDP) paths using the Watts potential for the dirner, trimer and tetrarner, as shown in fig. 9. The dimer path is straightforward, and simiIar to a diatomic molecuIe dissociation. The trimer path indicates a two step process. First, the separation of two oxygen atoms from ca. 2.75-5.50 A breaks the cyclic structure to form an open chain, then a terminal monomer molecule dissociates with a curve similar to the dimer. The tetramer indicates more exotic behaviour depending on the order with which the water mole- cules are separated. This is a result of numerous local minima on the potential396 D Y N A M I C S OF SMALL MOLECULAR CLUSTERS 0 .0 -5 -1 0 3 2 2 -15 t! C -20 -25 -30 1 1 1 1 1 1 1 1 1 1 , 2 4 6 8 10 12 14 0-0 separation/A FIG. 9.-Locally adiabatic dissociation energy curves for (A) (H20)2-+2H20, (B) (H20)3+(H20)2 + H20, (C) (H20)4+(H,0)3 + H 2 0 when adjacent hydrogen-bonded waters are separated, (D) (HzO)4t (H20), 4- H20 when it is formed from a monomer adding to a cyclic trimer, (E) (H20)r-+(H20)3 + H20 when opposing non-hydrogen-bonded water molecules of the cyclic tetramer are separated. surface. (The LDP were generated by small sequential displacements of ca. 0.2 8, of the oxygen-oxygen distance, so the curves do not represent global minima.) These paths indicate that the geometry of the cluster when it dissociates is far removed from the equilibrium configuration of the separated products.Many low- frequency vibrational motions will be excited in the cluster fragments left in these extended orientations. This would explain why little energy appears in translation, and why the product energy distribution should not depend dramatically on which motions are excited by the photon. CONCLUSIONS A preliminary study of the fundamental hydrogen-atom vibrations in small clusters of HF and H,O has indicated that the key to understanding the origins of the spectral changes induced by hydrogen bonding is in the differences between the dimer and larger clusters. Accurate quantum-mechanical calculations of the electronic structure should be possible for these systems. These new results will then enable more accurate analytical potential functions suitable for modelling solution behaviour.The predissociation lifetime (c: 2 j i s ) and energy partitioning among the product degrees of freedom were also obtained. The large geometry changes that the trimer and larger clusters must undergo during dissociation appear to explain the small amount of energy released in translation and the insensitivity of this product energy to the photon energy in these experiments, Future experiments on isotopically substituted clusters should further the under- standing of the dynamic coupling present in these clusters. A study of the behaviourM . F . V E R N O N ~ ~ ~ ~ . 397 of large clusters should also indicate whether it is possible to increase the cluster’s heat capacity t o the range where the predissociation can be measured in real time, if indeed the energy is substantially redistributed within the cluster before dissociation.This work was supported by the Assistant Secretary for Nuclear Energy, Ofice of Advanced Systems and Nuclear Projects, Advanced Isotope Separation Division, U .S. Department of Energy under contract no. W-7405-Eng-48. For a recent review, see The Hydrogen Bond, ed. P. Schuster, G. ZundeI and C. Sandorfy (North Holland, Amsterdam, 1976). 1. R. McDonald and M. Klein, Faraday Discuss. Chem. Soc., 1978, 66, 48. S. Rrosrian and R. Byer, fEEE J. Quantum Electron., 1979, QE-15, 415. M. F. Vernon, J. M. Lisy, H. S. Kwok, D. J. Krajnovich, A. Tramer, Y . R. Shen and Y . T. Lee, J. Phys. Chem., 1981, 85, 3327. (a) T. Dyke, K. Mack and J. S. Muenter, J. Chenr. Phys., 1977, 66, 498; (b) T. Dyke and J. S. Muenter, d. Chem. PJiyhy., 1974, 60, 2929; ( c ) T. Dyke and J. S. Muenter, J. Chern. Phyx., 1972, 57, 501 1 ; (d) J. A. Odutola and T. Dyke, J. Chem. Phys., 1980,72, 5062. ’ T. Dyke, B. J. Howard and W. Klemperer, J. Chem. Phys., 1972, 56, 2442. R. 0. Watts, Chem. Phys., 1977, 26, 367. a (a) H. Lernberg and F. Stillinger, J. Chem. Phys., 1975,62, 1877; (b) F. Stillinger and C. David, J . C’hem. Phys., 19711, 68, 666. (a) F. Stillinger and C. David, J . Chem. Phys., 1978, 69, 1473; (6) F. Stillinger and C. David, J . Chem. Php., 1980, 73, 3384. M. Sceats and S. Rice, J. Chem. Phys., 1980, 72, 3236. (a) G . C. Hafacker, Y . Marechal and M. A. Ratner, ref. (I), chap. 6; (b) Y . Marechal and A. Witkowski, J. Chem. Phys., 1968, 48, 3697. lo D. Hadzi and S. Bratos, ref. (11, chap. 12, p. 575. lJ M. Wojcik, Mol. Php., 197X, 36, 1757. l4 N. Koroteev, M. Endemann and R. Byer, Phys. Rec. Left., 1979, 43, 398.

ISSN:0301-7249    

DOI:10.1039/DC9827300387

出版商:RSC    

年代:1982

数据来源: RSC

摘要:

GENERAL DTSCUSSlUN Dr. J. N. L. Connor (UPziuersity of' Marichester) said: Holmer and Certain have reported calculations of complex energy pole positions for the scattering matrix of a square-well potential. At Manchester, we have also been interested in methods for calculating the positions and residues of poles of the scattering matrix for interatomic potentials. As well as poles in the complex energy plane we have aIso investigated poles in the complex angular-momentum plane (Regge poles). We have used accurate quantum and uniform semiclassical methods to calculate complex energies for a Lennard-Jones (12,6) potential, an exponential potential and a cubic anharmonic oscillator.2 The quantum calculations employed the complex rotation method3 The uniform semiclassicd quantization formula we used was actually derived about I0 years but has eluded a numerical investigation until We have found that the uniform semiclassical results agree well with the quantum ones.z*s In addition, we have compared the resonance energies and widths obtained from the compIex energy definition of a resonance with those given by the time delay definition.6 Our investigation of the positions and residues of Regge poIes for Lennard-Jones potentials started somewhat earlier.' Quantum and uniform semiclassical techniques were again employed.We used the complex coordinate method of Sukumar and Bardsley ' for the quantum calculations. An advantage of the semiclassical approach i s that essentially the same theory applies to the complex angular-momentum eigen- values as to the complex energy eigenvalues [for reviews of the formalism, see ref.(9) and (lo)]. When the imaginary parts of these eigenvalues are large, the semi- classical theory simplifies to a simple Bohr-Sommerfeld quantization forrn~la.~*~' As for the complex energies, we have obtained good agreement between the serni- classical and quantum Regge-pole result^.^ An example of the usefulness of Regge poles is that they provide a simple interpretation of diffraction scattering as an inter- ference between a surface wave that propagates around the core of the potential and st directly reflected wave." Experimental examples of diffraction scattering in molecular beam experiments have been reported at this rneeting.l3*l4 B. K. Holmer and P. R. Certain, F ~ V O ~ Q Y Discuss. Chem.Suc., 1982, 73, 3 11. J. N. L. Connor and A. D. Smith, Chem. Phys. Leu., 1982, 88, 539; J. Chem. Phys., 1983, in press, lnnt. J . Quanfum. Chem., 1978, 14, no, 4. J . N. L. Connor, Mol. Phys., 1973,25, 1469. H. J . Korsch, H. Laurent and R. Mohlenkamp, h f d Phys., 1981,43,1441; J . Phys. B, f982,15, 1. J . N. L. Connor, W. Jakubetz and C . V. Sukumar, J . Phys. B, 1976,9, 1783; J. N. L. Connor, D. C. Mackay and C . V. Sukumar, J . Phys. B, 1979, 12, L515; J . N. L. Connor, W. Jakubetz, D. C. Mackay and C . V. Sukumar, J . Phys. B, 1980, 13, 1823. C . V Sukumar and J. N. Bardsley, J . Phys. B, 1975, 8, 568. J. N. L. Connor, Chem. SOC. Rev., 1976, 5, 125. Child (Reidel, Dordrecht, 1980), pp. 45-1 07. ' J. N. L. Connor and A. D. Smith, Mol.Phys., 1981,43, 397; Mol. Phys., 1982, 45, 149. lo J. N. L. Connor, in Semiclassical Methods in Molccrilar Scattering und Spectroscopy, ed. M. S. " J. B. 13elos and C . E. Carlson, Phys. Rev. A , 1975, 11, 210. l2 J. N. L. Connor, D. Farrelly and D. C . Mackay, J . Chern. Phys., 1981, 74, 3278. l4 M. Faubel, K. H. Kohl, J. P. Toennies, K. T. Tang and Y . Y. Yung, Furadcry Discuss. Chern. U. Buck, Faraday Discuss. Chem. Soc., 1982, 73, 187. Suc., 1982, 73, 205.400 GENERAL DISCUSSION Miss N. Halberstadt (Unitiersity of Paris, Orsuy) said: In his paper1 Prof. Ewing describes the dissociation channels of a Van der Wads molecule undergoing vibrational predissociation. We have recently performed2*3 a series of experiments on the vib- rational predissociation of the Van der Waals complex M - * X, where M is a poly- atomic moleculc (glyoxal: CHO-CHO) and X a diatomic moleculc (H, or D,).This case involves in principle two of the channels mentioned by Prof'. Ewing,' namely : M* * . X+M 1 X -k vibration translation M* - * * X+M + X* + AE,,, vibration vibration but in our experiments the second channel, involving excitation of the H, or D, molecule, is energetically closed. On the other hand, dealing with complexes of a polyatomic molecule introduces new channels (as mentioned by Prof. Ewing). The vibration of a polyatomic molecule is described by several normal modes, so that vibrational predissociation can be produced by the transfer of vibrational energy from one mode vi initially excited to another one vj, provided that h(vi - vj) :> D (the dissociation energy of the complex).The energy difference h(v, - v J ) is transformed into dissociation energy of the complex and relative kinetic energy of the fragments. We shall write this process as MT - * X+MT + X + AEv,-V,,T where AEviav,,, is the energy left over in translation motion and will be called E in the following. This process provides interesting insights into the more general problem of vibrational energy redistribution. We have performed a series of experiments on glyoxal-H2 and glyoxal-D, corn- plexes, 2a,b selectively excited to different vibrational levels of the first electronic excited ' A , state. Under excitation of the 8' level of complexed glyoxal (noted g1 to differentiate it from the 8' level of uncomplexed glyoxal, which has a vibrational energy excess of 735 crn-l) vibrational predissociation populates only two among the five energetically accessible states of the glyoxal fragment: 0" and 7' (Evih -2 0 and 233 cm-l, respectively).Both complexes only differ by the branching ratio R = z & ~ / Z ~ I \ ~ I yielding R(H2) > 9 and R(D2) = 2.3. As underlined by Ewing,' it follows from application of the Golden Rule that the predissociation rate z;!,f is given by where wi describes the initial (bound) state M* * - X, and t,uf the final state of the fragments M -1 X, u,, being their final relative velocity. In a diabatic description of the process, when the intermolecular potential is approximated by a Morse function: V(R) --- D{exp[-2a(R - R,)] -- 2 cxp[-a(R -- Re)]} (1) Ewing,l and also Beswick and J ~ r t n e r , ~ have shown that the predissociation rate T& is proportional to exp(-x/h) where A : h/d(2rn~) is the De Broglie wavelength governing the oscillations of the continuum wavefunction.Tntroducing Ewing's parameter q,,, -- 27~/;1a, which he interprets as the number of De Broglie waves in a distance 27t/a(range of the Morse potential) it can be seen that z& is proportional to e-1/2y,. qm is obviously higher for D, than for EI, because the difference between the reduced masses. Thcrcfore T ~ I \ ~ I ( H ~ ) r T{IL~~(D~)GENERAL DISCUSSION 40 I and Moreover, the mass in qm is multiplied by the kinetic energy, so that the effect of the mass on 7;Lf is larger for the $ ' L O channel than for the 8l-7' channel because the kinetic energy involved is larger.T~L+~C(H,) > TC~!+,~(DZ). Therefore T C I ! ! ~ 1( H2) > T ~ I > T ~ ( Dz) but Z&,O (HI) & sK~!+~~(DZ). Introducing Y = R(H,),/R(D,), where we concluded that ~gL+,o-(Hz) T.F~~L+T~(DZ) ~p,i!+,o '( D') ~$j!+7 1 (H,) Y = - > 1. From the experimental results, Ye,, > 3. We have performed more complete calculations The polyatomic rnoIecule was described as a set of harmonic oscillators (its normal modes) and this model shows that only the branching ratios should be affected !y changing the mass of the using a diabatic description of the system. 1 0 1 50 100 150 D,!cm-' FIG. I .-Distorted-wave diabatic results for vibrational predissociation of the 8' vibrationd level of glyoxal-H, and glyoxal-D, complexes.R = T ~ I ! + ~ ~ / & + ~ I is the branching ratio between the two dissociation pathways 8'+0" and 8l-7'. The figure is a plot of Y = R H ~ : ! R D ~ as a function of the potentiaI parameters rx and D [see eqn (I)]. Values of 2 as follows; (a) 1.2, (b) 1.6, (c) 2, (d) 2.4 A-I. partner of the polyatomic. In fig. 1 we present Y as a function of D (dissociation energy) for several values of x . < 4), Y is significantly higher than 1 , and depends strongly on SL but weakly on the dissociation energy of the complex. For reasonabIe values of SL (1.5 <402 GENERAL DISCUSSION These results show that usual models describing the vibrational predissociation of complexes of diatornics can be usefully extended to polyatomics. A knowIedge of the interaction potential is obviously necessary for a complete theoretical description but essential features of such simple models can already be tested by simple experi- ments such as those presented here.Moreover, they may provide useful information on the parameters of the interaction potential. Finally, it should also be stressed that similar isotope effects should be present for other initial states in glyoxal as well as in other molecular systems such as 12-H2 (D2).5 G. Ewing, Faraday Discuss. Chem. SOC., 1982, 73, 325. (a) N. Halberstadt and B. Soep, Chem. Phys. L e f f . , 1982, 87, 109; (b) J. A. Beswick, N. Halber- stadt, C. Jouvet and B. Suep, Laser Chemistry, to be published. N. Halberstadt, Thesis (UniversitC Paris XI, Orsay, 1982); J. A. Beswick, N. Halberstadt and B.Soep, to be published. J. A. Beswick and J. Jortner in Photosefecliue Chemistry, Adu. Chem. Phys., 1981, 47, 363. J. E. Kenny, T. D. Russel and D. H. Levy, J. Chem. Phys., 1980,73, 3607. Prof. G . E. Ewing (Indiana Unitwrsity) said: LeRoy, Beswick and Child have noted, on conclusion of extensive calculations, that internal-rotational predissociation of excited Van der Waals molecules is much more efficient than vibrational predis- sociation. There is a simple expIanation of their results which also lends itself to the identification of efficient relaxation channels of other excited Van der Waals molecules. I offer here this explanation in terms of an extension of my propensity rules.’ These propensity rules, stated in a different Ianguage than that given previousIy,‘ are a consequence of %s rductance of a molecule to change quantum numbers during a radiationleg(;_ :ransition between non-crossing potential surfaces.In the further deve!Spment of these propensity rules we must keep track of all the quantum numbers for each predissociation channel. That channel with the smallest total change in quantum numbers will be the most efficient for the predissociation process. The translational quantum number change is obtained from examining fig. 2 of ref. (I). Here A-B* - *C begins with no nodes in the translational wavefunction R,(r) and ends with the large number of nodes of R,(r) for A-B + C , The trans- lational quantum number for R,,,(r), a near-plane wave, is roughly the number of nodes in the region of the Morse well.(It is to be recalled that quantum numbers merely count nodes in wavefunctions.) It is the large change in nodes (i.e. large change in translationaI quantum number) which accounts for the inefficient reIaxation in the example of fig. 2. A measure of the change in translational quantum number can be read off from the abscissa of fig. 3 of ref. (I). This leads to an approximate estimate of the change in translational quantum number: The predissociation lifetime depends exponentially on this quantum number change as fig. 3 shows. Rotational-quantum-number change, Anr, also effects the relaxation rate ex- ponentially. This has been demonstrated both through experiments and by the01-y.~ We then write where AJ measures the change in internal rotational state of A-B within A-B - - * C and the rotational state of A-B as the fragment of the predissociation process.The final quantum-number change identifies vibrational states of motions against chemical bonds. We writeGENERAL DISCUSSION 403 where vi labels the sum of the vibrational quantum numbers of A--B* - * * C and uf labels the quantum states of the fragments A-B + C*. Again the rate for relaxation is known by experiment5 to depend exponentially on these changes in quantum numbers. We write then that the total change in quantum numbers, An,, for predissociation is given by AfiT = Anl + Anr +An,. (41 Eqn (4) then keeps track of the total number of wavefunction nodes that change during the relaxation process. For the internal- rotational predissociat ion or rotati onal-translation (R-T) channel, the translational energy of the fragments is AE z 350 cm-', received from the J = 2 state of H2 after breaking the weak Van der Waals bond.Using a z 2 x 10s cm-I (a typical range parameter)l and the appropriate reduced mass, ,uv, we have by eqn (1) Arzt w 2. Since the H2 fragment is produced in the J = 0 state we have Anr z 2. Finally, since the u = 1 level of H, has not changed we have An, = 0. The total quantum change for the R-T channeI is thus AnT = 4. For the vibrational-translation (V-T) pre- dissociation channel, AE x 4150 cm-' is released into translational motion as I! = 1 of H2 in the complex is relaxed to u = 0 in the fragment, i.e. An, z I . Using this translational energy into eqn ( I ) we find Ant z 6. No rotational energy has been exchanged so An, = 0.The net quantum change for vibrational predissociation is thus An, z 7. The large number of nodes that must change for the V-T channel is consistent with the inefficient relaxation process. In contrast the relatively more efficient R-T channel is understood by the modest total quantum change required. Our qualitative conclusions are thus consistent with the numerical results of extensive calculations.' The results for H, (2: = I ? J = 2) - * * Ar predissociation are summarized in table 1 . Let us consider the example of H, ( v = I, J == 2) - - * Ar. TABLE 1 .-QUANTUM-NUMBER CHANGE FOR PREDISSOCIATION OF EXCITED VAN DER WAALS MOLECULES molecule An L Anr A H , . An channel H,(u= 1 , J = 2 ) . . . A r 2 HCI ( u = 1 , J = 3) - .. Ar 6 15 :O ZO 13 Z O 6 NlO(001) * * . NZO(000) 2 0 0 13 3 0 31 0 0 4 1 7 1 16 1 14 0 3 1 14 1 32 2 8 R-T V-T V-T V-R R-T V-T V-R v-v We also present the example of HCI ( P = 1, J = 3 ) - - - Ar predissociation. The large reduced mass, ply, for this complex accounts for the large value of AnT by the V-T channel in which HCl (11 = 0, J = 3) is the predissociation fragment. Production of HCI (tl = 0, f = 16) by the V-R channel causes a smaller value of An, even though Anr z 13 since essentialIy all kinetic energy appears in rotation. Relaxation by V-R is then mure efficient than by V-T, as Ashton and Child have shown. However, it is internabrotational predissociation (the R-T channel), producing HC1 ( u = I , J = 0) and Ar with little translational energy, which will dominate the relaxation process.404 GENERAL DISCUSSION As a final example we recast the problem of NzO(OO1) - - - N20(000) relaxation con- sidered previously ' in terms of the propensity rules. The high vibrational frequency of N,O(OOI) and large p, result in a large AilT by the V-T channel.Likewise the large moment of inertia of N20 requires a large Anr NY IAJ1 so that the V-R channel is also closed. Clearly the V-V channel will dominate the reIaxation process where the fragments produced are in the NzO(OOO) + N,0(100) vibrational state. Unfortunately it is difficult to offer these propensity rules in a more quantitative form, although it has been tried.' The uncertainty in range parameter, a, places significant errors in Ant of eqn (1). The anisotropic nature of the intermolecular potential will influence the quantitative relationship between An, and AJ in eqn (2).Finally the efficiency of the coupling of vibrations through the van der Waals bond will alter the effective value of Anv in eqn (3). Nevertheless the bookkeeping on the total number of nodes that the initial-state wavefunction (for A-€3 r * .C*> and final- state wavefunction (for A-B + C) must change is a convenient way to assess the relative importance of relaxation channels of excited Van der Waals molecules. ' G. E. Ewing, Faraday Discuss. Chem. Soc., 1982, 73, 325. ' G. E. Ewing, in Proc. 15th Jerusalem Symposium in Quantum Chemistry and Biochemistry, ed. B. Pullman (Reidel, Dordrecht, 1982). F. LeGay, in Chemical and Biological Applications of Lasers, ed.C. B. Moore (Academic Press, New York, 2nd edn, 1977). K. F, Freed and H. Metiu, Chenr. Phys. Lett., 1977, 48, 262. Y . T. Yardley, IntroductiotI lo Molecular Energy Tram-er (Academic Press, New York, 1980). R. J. LeRoy, G. C. Corey and J. M. Hutson, Faraday Discuss. Chem. Soc., 1982, 73, 339. ' ( 0 ) C. J. Ashton and M. S. Child, Faraday Discuss. Chem. Soc., 1977,62,307; (b) C. J. Ashton, D . Phil. Thesis (Oxford University, 1981). Dr. G. G. Balint-Kurti and Mr. I. F. Kidd (Uiriuersify of Bristol) said : As previously mentioned in an earlier comment, we have been performing exact quantum-mechanical calculations of the photofragmentation cross-sections for the Ar * - - H, Van der Waals complex. We have used the same potential-energy surface as used by Le Roy, Corey and Hutson together with the dipole moment function of Dunker and Gordon.' We have calculated the absorption cross-sections and fragmentation patterns for many lines which are both rotationaIly and vibrationally predissociated.Of particular interest is the line arising from the vibrational excitation process: Ar - H2(y~)]*+ Ar + The intermediate metastable excited complex in this process can only break up through vibrational predissociation, as there is no rotational predissociation pathway which can lead to break up available. As expected the line-width for this process is very narrow. In fig. 2 we show our calculated total photofragmentation cross-section for. the process. The line is centred around a photon energy of 4161.2214 cm-' and has a half-width of 2.5 x lo-* cm-'.s. This result may be compared with a value of 3.4 x CM-' calculated by Beswick and Requ ena.2 This corresponds to a lifetime of 2 x 'A. M. Dunker and R. G . Gordon, J. Chem. Phys., 1978,68, 700. J. A. Beswick and A. Requena, J. Chem. Phys., 1980, 73,4347. Dr. J. M. Hutson (Utiirevsity of Waterloo) said : When performing close-coupling calculations, including all relevant open and closed channels, it makes no difference whether body-fixed or space-fixed basis functions are used; either choice forms a complete set and gives accurate results. Our comments on the applicability of body- fixed and space-fixed functions referred to approximate caIculations in which onIy one closed channel was retained, and under these circumstances the results of body-fixedGENERAL DISCUSSION 405 - 4 - 3 - 2 -1 0 1 2 3 4 5 photon energy/lO-* cm-' FIG.2.TotaI photofragmentation cross-section for the process Ar-H2(J = 0, u = 0 , j = O ) h v _ [Ar-H,(J = 1, u = 1,j = O>]* -Ar + H,. and space-fixed decoupling approximations are not equivalent. The criterion for a single-channel approximation to hold is that the (neglected) off-diagonal matrix elements between closed channels should be small compared to the energy separation between the channel potentials, since in first-order perturbation theory the correction to the angular wavefunction is given by the ratio of these quantities. In Ar-H,, our fig. 1 and 2 demonstrate that the space-fixed decoupling scheme is more appropriate than the body-fixed one, and the numerical results of our tables 2 and 3 confirm this.It is important to realise that the competition between space-fixed and body-fixed coupling schemes depends on the system concerned. Body-fixed quantum numbers are appropriate when the rotational energy of the complex is small compared to the anisotropy of the potential, and space-fixed quantum numbers are appropriate when the reverse is true. For Ar-HC1, which is considerably more anisotropic than Ar-H,, we have observed a changeover from body-fixed to space-fixed coupling with increasing diatom rotational quantum number in close-coupling calcdations of rotational predissociation.' For total angular momentum J = 1, we have found that states up to . j = 3 are well described by body-fixed quantum numbers, but for j 2 4 a space-fixed coupling scheme is more appropriate, reflecting the increasing rotational406 GENERAL DISCUSSION 0.18 0.15 0.12 energy of the complex.Such changes in coupling scheme may be expected to be a general phenomenon in Van der Waals molecules, and have important consequences for the development of approximate schemes for calculating Van der Waals energy levels. C . J. Ashton, M. S. Child and J. M. Hutson, J. Cliem. Phys., to be published. - - - Dr. J. A. Beswick (University ofParis, Orsay) and Dr. G. Delgado-Barrio, Dr. P, Villareal and Dr. P. Marcca (Madrid) said: We would like to report some new theo- retical results on the vibrational predissociation of the He - - * I, Van der Waals molecule, which may contribute to the discussion on the dynamics of vibrational pre- dissociation as well as on the role of rotation.Recently, W. Sharfin et al.’ have conducted experiments on Hcl, near the dissoci- ation limit of I@ 312). They have shown that the vibrational predissociation rates exhibit a different and very interesting behaviour as a function of the vibrational excitation of the TI subunit, as compared with the results of Sharfin et al. for low vibrational excitation.2 Sharfin et a1.2 found that the rates follow an almost quadratic 3 I E --? 0.09. L 0.06 0.03 * 0 0 / ‘A A 0 h 0 0 I I I I I 10 15 20 25 30 35 V(I2) FIG. 3.-Vibrational predissociation linewidths for the He12(B) molecule as a function of the stretching quantum number u(I2) corresponding to the vibration of the I2 bond. Comparison of quasiclassical values (0) with the T-shape quantum-mechanical results (- - -) and the experimental data (A).law Bv2 + Cu3(C < B) with u being the vibrational quantum number of I,. This behaviour has been reproduced theoretically by the use of an approximate quantum- mechanical treatment,3 as well as by a quasi-classical calc~lation.~ The potential- energy surface used was a sum of pairwise atom-atom potentials. Tn fig. 3 we have represented the results of those calculations, together with the experimental results of Smalley et a1.’ The overall agreement is good and the quasi-classical results re- produce the behavrour of the rates very well, especially for the larger values of v .GENERAL DISCUSSION 407 The recent experiments of Sharfin ef aZ.‘ show that the function Bu2 f Cu3 fits the data well for levels up to u z 45, but deviates for higher L’.In particular, they have determined a further enhancement of the rates peaking at L’ ==: 57 followed by a pro- nounced decrease. They have interpreted this effect as the gradual closing of the Av = -1 channel. Tn order to check this interpretation we have recently performed quasiclassical calculations for higher u levels, using the same potential surface as the one determined previ~usly.~ In brief, the classical Hamilton equations have been solved in three dimensions with a standard integrator. The integration step was 0.4 x s. The initial conditions were randomly chosen using a non-rotating Morse distribution for the I, stretching vibration and an initial energy corresponding to the zero-point level of the Van der Waals bond.The con- dition for a trajectory to be considered as predissociated was taken to be that the intermolecular distance He - - I2 being larger than 12.5 A. This also defines the dissociation time. Representing the number of trajectories as a function of the dissociation time gives an exponential curve from which the lifetime z, and hence the linewidths r, can be extracted. The results of our calculations are represented in fig. 4. The first important conclusion is that the calculations indeed reproduces the The method has been presented elsewhere.4 0.6 l,ol A A A A A I f I I I 45 50 55 60 4 1 2 ) FIG. 4.-Same as fig. 3 for the quasiclassical values (a) and the experimental data (A) near the dissociation limit.experimental behaviour for large 2;’ levels. This result is encouraging since it should be stressed that the calculations were performed with exactly the same potential as the one used to fit the experimental points for low L’ levels. In addition, we have aIso determined vibrational distributions of the I, fragments by the use of a box-quantization procedure. The resuIts are presented in table 2. We note that the distributions show very neatly the graduaI closing of the A2;. = - 1 channel and therefore they support the interpretation of Sharfin et aZ.l regarding the behaviour of the linewidths in the region of u = 56 to v = 63, Incidentally, for u = 60, we find that the Au = -2 channel is already more than a factor of two larger than the Au = - I , and this is in good agreement with the experimenta1 findings of Blazy et al.5408 GENERAL DISCUSSION TABLE 2.-PROBABILITIES FOR THE DIFFERENT CHANNELS U’ AS A FUNCTION OF STRETCHING QUANTUM NUMBER U CORRESPONDING TO THE VIBRATION OF THE 12 BOND 21’ U u - 1 v - 2 0 - 3 v - 4 50 74 24 2 54 63 30 7 56 52 36 9 3 59 33 44 16 7 60 20 52 21 7 63 - 47 33 20 Another result of our calculations is that the amount of energy going to rotation We then conclude that of J2 is a very small fraction of the excess available energy.He * * J, predissociation corresponds to a nearly pure V-T process. W. Sharfin, P. Kroger and S. C. Wallace, Chem. Phys. Lett., 1982, 85, 81. W. Sharfin, K. E. Johnson, L. Wharton and D. H. Levy, J. Gem. Phys., 1979,71,1283. J. A. Beswick and G. Delgado-Barrio, J.Chem. Phys., 1980,73, 3653. G. Delgado-Barrio, P. Villarreal and P. Mareca, unpublished results. J. A. Blazy, B. M. DeKoven, T. D. Russell and D. H. Levy, J . Chem. Phys., 1980,72,2439. Dr. I. W. M. Smith (University of Cambridge) said: I should like to draw attention to one important aspect of the Van der Waals or hydrogen-bonding interaction which has been almost totally ignored in our discussions so far. I refer to the effect of weak or medium-strength attractive forces on molecular collision dynamics. In addressing this question, it may be useful to define two general limiting cases. The first extreme is reached when the magnitude of the attraction, the depth of the intermolecular potential well, is much greater than the collision energy. This is the situation in collisions between free radicals at low and moderate temperatures. As long as the radicals are not both atomic, collision complexes form and the association- dissociation kinetics can be treated with a fair measure of success by statistically based theories.’ In the other extreme, the collision energy is much greater than the well depth and collisions are direct.The energy associated with relative motion, including that released as a result of any small attractive interactions, cannot be distributed sufficiently or quickly enough into other degrees of freedom to prevent “ immediate ” redissociation. I think that it is important to establish the factors which determine where the boundary lies between these two distinct types of be- haviour-and how sharp that boundary is.My own concern with this problem results from an interest in vibrational energy transfer, which appears to be facilitated by hydrogen-bonded interactions. Con- sequently, we have suggested that such processes [for example, the relaxation of HF(u = 1) by HCN] might be viewed in terms of the following elementary steps: k + HCN + HFt T, (HCN * * - HFt)t k_ (HCN * - - HF)tt+HCN + HF. Here, HFt is the initially excited HF molecule, (HCN - - - HFt)t is a hydrogen- bonded complex which retains the initial excitation in the HF stretching mode, and (HCN - * - HF)ff is a complex in which the energy initially in the HF vibration has been redistributed. This species will rapidly dissociate, so the unimolecular process with rate constant k is one of vibrational predissociation, which has been discussed k-GENERAL DISCUSSION 409 here at considerable length.The questions which 1 wish to direct attention on are (i) whether (or when) one can calculate a steady-state concentration of the complexes (HCN - - - HFt)-/ by statistical methods, and (ii) should all internal states in this adduct be included in the stale c o u n t which is performed in a cdculation of this type ? Answers to these questions are also rcquired for treating the kinetics of formation of Van der Waals or hydrogen-bonded species-and ultimately of higher clusters. Here the mechanism may be written for example as k 4- k- HCN + H F Z ( H C N - - - HF)t 'M(M? HCN a - - H F where HCN - * * HF is the collisionally stabilised adduct. Once again the correct treatment of the equilibrium between separated monomers and collision complexes i s vital.M. J. Howard and 1. W. M. Smith, Progr-. Recic. Kinet., 1982, in press. ' G. S. Arnold, R. P. Fernando and I. W. M. Smith, J , C'hem, Phys., 1980,73, 2773. Dr. C. Jouvet (Uniuersity of Paris, Orsay) said: The main features of the inter- molecular potential surfaces may be deduced either from ineIastic scattering (vibrational and rotational collisional relaxation) or from vibrational predissociat ion of Van der Waals complexes. In the first case we obtain information about the repulsive part of the potential corresponding to translational energies ET E 300 cm-l, while in the latter one the shape of the potcntial-energy surface in the minimum region plays an essential role.I would like to report new experiments filling the gap between collisional relaxation and predissociation: low-energy relaxation (ET M I to 100 cm-') involving meta- stable levels of Van der Waals compIexes. For non-central colIisions with light col- lision partners (e.g. He), the effective intermolecular potential contains a significant centrifugal barrier. Tunnelling through this barrier results in formation of long-lived molecular complexes (orbiting resonances) which determine the mechanism of relaxation. Such low-energy collisions take place during the supersonic expansion and their energy and rate depend in the well known manner on the distance from the nozzle (expressed usually in terms of x / D , where D is the nozzle diameter). In all experi- ments described here, the molecule seeded with helium-free jet is excited with a tunabIe dye laser at a variable, well defined distance from the nozzle and fluorescence (energy, time, time and energy resolved) is recorded.The main feature of low-energy collisions is a strong enhancement of cross-sec- tions as compared to room-temperature experiments for all kinds of relaxation pro- cesses. (i) Electronic relaxation (collision-induced intersystem crossing) for the lowest excited state (lAJ of giyoxal by helium is more efficient by one order of mag- nitude in the supersonic expansion than in the room-temperature gas.' (ii) a similar increase of cross-sections fur rotational relaxation was observed in the case of glyoxal- ('A,+)--heliurn (400 A2 compared to 40 A2) and I,(B 0;) 3-helium collisions; (iii) the enhancement of cross-sections for vibrational relaxation at low collision energies has been observed for a number of molecules such as Ilr4 aniline toluene, fluombenzene ' and glyoxal.' The relaxation paths are highly selective (a small number of ener- getically accessi bIe states is efficiently popuIated), the propensity rufes being more or less similar to the room-temperature collisions and in the vibrational predissociation of the corresponding Van der Waals complexes.(iv) For aniIine and glyoxal some propensity rules have been observed on the energy range of collisions efficient for41 0 GENERAL DISCUSSION depopulating the initially excited level. As an example, the 8l level (Ev z 735 cm-') of ' A , glyoxal is efficiently relaxed in collisions with an average energy of 1 cm-', while for the 2l level of the same molecule (Ev 1400 cm-l), a collision energy of 10 cm-' is necessary for efficient r e l a ~ a t i o n .~ * ~ If the high efficiency of low-energy collisions may be considered as a general rule, some peculiar exceptions have been observed. Low-energy collisions are not efficient for vibrational relaxation of low vibronic levels (E, > 4000 cm-l) of benzene in the lBZu state.6 The basic theory of low-energy collisions has been developed but more extended experimental studies and more quantitative data are necessary for a better under- standing of their mechanism. ' C. Jouvet and B. Soep, J. Chem. Phys., 1980,73, 4127. B. Soep, to be published. J. Tusa, M. Sulkes and S.A. Rice, Proc. Nut1 Acad. Sci. USA, 1980,77, 2367. S. A. Rice, Phofoselective Chemistry, Ado. Chem. Phys., 1981, 47, 237 and references cited therein. J. Tusa, M. Sulkes, S. A. Rice and C. Jouvet, J. Chem. Phys., in press. C. Jouvet, M. Sulkes and S. A. Rice, Chem. Phys. Lett., 1981, 84, 241. V. Sethuraman and S. A. Rice, J. Chem. Phys., submitted for publication. ' C . Jouvet, M. Sulkes and S. A. Rice, J. Chem. Phys., in press. Dr. J . Schaefer (Max-Planck-Institut f u r Physik und Astrophysik, Garching) said : My first comment refers to the interaction potential of the H,-N2 system as discussed by Reuss and coworkers, and by Buck, in a previous paper of the present meeting. It is worth mentioning that the empirical improvements of the most recent ab initio potential of Meyer et al.' (see fig.4 of this paper) yield relatively small correc- tions of differential and integral, elastic and inelastic cross-sections, of the order of ca. 10% or less.2 In quite a number of tests of this ab initio potential done by Kohler and my~elf,~ effective (temperature dependent) cross-sections have been calculated giving accurate agreement with experimental results obtained by the Molecular Physics Group in Leiden, within experimental error bars of 10%. It has also been shown in this work that at least some of the effective cross-sections are very sensitive to changes of the anisotropy of the potential. However, we may expect that the correction of the lead- ing non-spherical potential term by an additional blister, as introduced by Reuss and coworkers, does not yield significant corrections to the results obtained.The two empirical fits of the isotropic potential term mentioned in the same paper (one has been obtained by Silvera and Goldman from equilibrium properties of solid hydrogen, the second one by Buck et a1.2b from a crossed-molecular-beam experi- ment) differ very little and can be considered identical. Instead of assuming now that the final correction of the isotropic potential had been obtained from the results of two completely different experiments, we should realize that both experiments had been sensitive with respect to about the same range of the potential. In fact, both experiments did not sensitively probe the well depth and the main part of the attractive region. We should also reflect on the quantum-chemical consequences of a potential shift as implemented in the two models.On the larger distances side of the well depth the shift gives a small positive correction which does not make sense as far as the ab initio energy points have been obtained in a variational or equivalent procedure. There- fore the unchanged potential points of this side have been kept in Dr. Buck's model. On the smaller distance side of the well depth the result of the shift is a large negative correction switched on in a relatively small range. The dispersion attraction energyGENERAL DISCUSSION 41 t needs roughIy a 30% increasing correction due to the shift of 0.1 A, within a distance of 0.4 A. We thereforc should still think about a better correction of the ab iplifio potential being acceptable from the point of view of quantum chemistry.Very recently, Watts tested both the ab initiu potential and the Silvera model by calculating the second virial coefficient, among other things. This is the most sensitive test of the attractive region. The finding was that the ab initio potential gives values that are consistently higher than the experimental data whereas the results of the Silvera model are consistentiy too low (see fig. 5). Agreement with experiment has i A I I I I 200 4 00 TIK FIG. 5.-Sccond virial coefficient for hydrogen: (--I experimental data; (. * .) ab initio potential; x , rescaled ab inicio potential; A , SiIvera-Goldman potential. been reached by dividing all distances by 1.02 and multiplying the spherical term by I .I , giving a new empirical model V;;pirical(&) ~ 1 1 v ah i n i r l a uoo (1.02 m. The empirical well depth of this model is 10% larger, and the value of & is between those of the ab initlo potential and the Silvera and Buck models. In addition, we should apply this scaling to the total interaction potential which (at least to a good approximation) conserves V,,,(.R>/ Vooo(R) in the repulsive region, as required in Buck’s paper at this meeting. Now the question arises whether one should use this new potential model as long as there is no significant disagreement found with any other measurement. I t seems412 GENERAL DISCUSSION interesting to me to show that a negative answer to this question might be opportune for those who work theoretically in the field of dimer spectroscopy.The bound-state energies and eigenfunctions of molecule dimers can be calculated most conveniently by using an approximate two-body interaction procedure, which is formally similar to the two-body scattering formalism. When applying this procedure for the H2-H2 system, the bound-state energies derived from the ab initio potential are found to be in accurate agreement with spectroscopical results published by Watanabe et al.7 The explanation is that one has beautifully compensated the short- comings of the ab initio potential with a small deviation of the true zero-vibrational energy of the four-atom system caused by the approximate two-body formalism. In concluding this discussion of the " true " interaction potential of the H2-H, system, I would like to note that (perhaps with the exception of the second virial co- efficient) the ab initio potential is useful if we accept errors of the order of 10%.Theoretical problems become enormous if one wants to do better. My second comment refers to a new theoretical attempt in the field of dimer spec- troscopy, the results of which I would like to report here briefly because of their great importance in the far-infrared astronomy, and because there is almost no experimental confirmation to be expected in the near future. This attempt has been started with a new advance in quantum chemistry, when Meyer calculated the induced dipole moment of H2-H,, depending upon the orientations of the two molecules and the centre-of-mass distance.This information allowed one to achieve the entirely ab initio calculations of collision-induced dipole radiation of normal hydrogen gas, in a temperature range up to 50 K, and in a frequency range up to 200 cm-' (rotationally inelastic collisions have been neglected). I can give only a brief description of the results obtained, in terms of line spectra and intensities, and of the first application. Fig. 6 shows the calculated emission intensities plotted as a function of frequency in the very low-temperature range. rn N vjcm - FIG. 6.-Theoretical emission intensities of the collision-induced dipole radiation of normal hydrogen gas plotted against frequency. The inset shows the measured (corrected) spectrum of Gush and two thermal Planck curves representing about the upper (2.9 K) and lower (2.7 K) limits of the predicted cosmic background radiation.The quantity W on the ordinate axis is the probability for the spontaneous emission of a photon in the frequency range v + dv per second and cm3, in normal hydrogen gas of number density n H z . The line spectrum caused by continuum-bound transitions displays the orbiting resonance spectrum of the dimers. The longer living quasi-GENERAL DISCUSSION 41 3 bound states give the narrow lines on the left side. At the very low temperatures shown here the continuum-continuum radiation is almost negligible. The inset shows the first measured spectrum which can be nicely explained by hydrogen emission, between ca. 10 and 25 cm-', emitted from a gas cloud of ca. 6 K.Gush has observed this spectrum in a rocket-borne measurement above 150 km alti- tude when he intended to measure the Wien tail of the cosmic background radiation. For about one third of the total observation time the galactic plane was in the field of view of the instrument, which explains the abundance of molecular hydrogen. It will be very interesting to search for this emission in regions of the sky where large amounts of molecular hydrogen can be expected. Observations in the laboratory should also be attempted to confirm the theoretical spectra. (a) W. Meyer and J. Schaefer, to be published; (b) J. Schaefer and B. Liu, to be published. (a) L. Monchick and J. Schaefer, J . Chem. Phys., 1980, 73, 6153; (6) U. Buck, F. Huisken, J. Schleusener and J. Schaefer, J.Chem. Phys., 1981, 74, 535. W. E. Kohler and J. Schaefer, to be published. I. F. Silvera and V. V. Goldman, J . Chem. Phys., 1978, 69, 4209. R. 0. Watts and J. Schaefer, to be published. J. Schaefer and W. Meyer, to be published. A. Watanabe and H. L. Welsh, Phys. Rev. Lett., 1964, 13, 810; A. R. W. McKellar and H. L. Welsh, Can. J. Phys., 1974, 52, 1082. * H. P. Gush, Phys. Rev. Lett., 1981, 47, 745 Dr. P. G. Burton (University of Wollongong) said : The two parallel series of com- putations that have recently been completed at Wollongong on the H,-H, potential surface which I have referred to in a number of comments earlier in this meeting are characterized by the same large basis set (78 functions for the supermolecule, which can account for 96.3% of E,,,, at R = a), but different treatments of the influence of electron correlation on the supermolecule total energies.When we examine how the PNOCI (correlating almost one pair of electrons at a time) and CEPA2-PNO (allowing for independent correlation simultaneously of all pairs-non-variational but TABLE 3 .-INFLUENCE OF ELECTRON CORRELATION TREATMENT ON ISOTROPIC AND ANISOTROPIC COMPONENTS OF THE H2-H2 INTERACTION POTENTIAL. DIFFERENCES IN MICROHARTREE (pH), FOR A RANGE OF INTERMOLECULAR SEPARATIONS (ao) R A Vooo(PNOC1-CEPA2PNO) A V202(PNOCI-CEPA2PNO) 3 .O 4.0 5 .O 6.0 7.0 8.0 579.4 250.6 101.8 40.18 15.87 6.62 3.39 5.85 3.34 1.69 0.72 0.27 systematic and size consistent) results differ in various components of the interaction potential, we find that the extent of treatment of correlation in the two approaches influences the isotropic V,,, component very strongly, but has a much reduced effect on the anisotropic components of the potential.Table 3 illustrates numerically that the V,,, term is 1-2 orders of magnitude more sensitive to the details of the electron correlation treatment than the leading anisotropy VzOz = V022, as given by our new calculations. In the light of this, we might expect that the leading anisotropy Vzoz = Vozz that414 GENERAL DISCUSSION we have already computed for Hz-H2, taken as an average of the PNOCI and CEPA2- PNO results, will be even closer to the converged, infinite basis, complete CI result at each R than is the case for the much more correlation-dependent Yo,, term of H,-H,. Table 4 illustrates the leading anisotropy thus computed, expressed as a V2 term to be consistent with Reuss’s paper (this entails multiplying V,,, by a factor of 5 ) in the same units as he has used.The uncertainty represents the distance in energy TABLE 4.-LEADING ANISOTROPY OF THE HI--H2 INTERACTION TOGETHER WITH THEORETICALLY ESTIMATED UNCERTAINTIES, EXPRESSED AS A LEGENDRE POLYNOMIAL EXPANSION ( v 2 == 5 x vz,,) 1.58 2.12 2.38 2.64 2.91 3.17 3.44 3.70 4.23 4.76 57 947 + 56 9093 96 3401 * 76 1039 57 143.3 40.2 - 144.1 5 27.7 - 196.4 j~ 18.7 -160.2 * 11.9 -30.1 & 4.5 -34.5 rt 1.4 terms from the auerage V, values to the upper (PNOCI) or lower (CEPA2-PNO) energy level at each R. Notice that the uncertainty estimated from the PNOCI and CEPA2 comparison increases as R decreases, but decreases again at the smallest R considered.We interpret this in terms of the increasing dominance of effects accessible at the SCF level in the computation of the anisotropy and the fact that the correlation contributions themselves become more isotropic as the molecular wavefunctions increasingly over- lap. When these data are compared with the ad hoc adjustments to the Schaefer-Meyer potential shown in fig. 4 of Reuss’s paper, one sees a similarity in the R, values for V2 and that the deeper V, required by Reuss is certainly in the direction that our calcula- tions indicate. To this extent we can see a theoretical basis for the adjustments made to fit the dimer hyperfine spectra. However, as we saw earlier for the V,,, term in a comment on paper by Buck, this more attractive V2 that we find compared to Schaefer and Meyer does not suddenly become different from their V, at ca.3.8 A as Reuss’s fig. 4 indicates, but is more attractive over the whole well of the V2. A second differ- ence bctween our V , and Reuss’s is that our repulsive wall of Vz does not remain shifted in from, and parallel to the Schaefer-Meyer V,, as R decreases indefinitely, but our V2 crosses the latest Schaefer-Meyer V, to become more repulsive than for R < 2.4A. It is interesting to note, looking back at Buck’s paper in this Discussion, that the total elastic cross-section data of Buck et al. for H,-D, were taken at 89.1 meV, which has a classical turning point at slightly smaller R values on either our PNOCI or CEPA2 V,,,. At such R values, our V2 would be even greater than the value of Schaefer and Meyer, but as discussed above, our V2 shortly crosses below their V2 as R increases. Accordingly we cannot support an inward shift being applied to the whole of V , (or V,,, = V,,, of course) as Buck advocated, but feel instead that any modifications to the Schaefer-Meyer Vz must be made selectively.It is true to say,GENERAL DISCUSSION 41 5 1.2 however, that in the vicinity of the R, of the isotropic potential, the magnitude of the inward shift of V, advocated by Buck and also by Reuss is reasonable on the basis of our own theoretical determination of V2. At this stage we feel that a somewhat larger basis than we have already used is required before definitive results for particularly the V,,, term (on which all else depends in accounting for experimental data) will be available from theory.We have underway a new series of calculations based on 52 functions per H2 [cf. 39 per H2 in ref. (l)] (which can account for 98% of the correlation energy of each H2) to this end. P. G. Burton and U. E. Senff, J . Chern. Phys., 1982, 76, 6073. - - Prof. T. E. Gough, Dr. M. Keil, Mr. D. G. Knight and Prof. G. Scoles (University of Waterloo) said: An experiment similar to that of Geraedts et ai.'S2 was performed at the University of Waterloo for the purpose of determining photodissociation spectra of the species present in beams of SF, seeded in Ar. The final goal of the experiment is the determination of the angular distribution of the fragments produced in the infrared photolysis of molecular clusters.The major difference between our work and that of Geraedts et al.'p2 is the use as detector, instead of a mass spectrometer, of a liquid-helium-cooled bolometer which records the loss of signal when a species is photolysed and its fragments leave the main beam path. The 934.0 Fig. 7 shows three distinct features of the predissociation spectrum. C02 laser frequency/cm-' FIG. 7.-Dissociation spectrum of 1 % SF6 in Ar, with gas pressure at 30 p.s.i. and a 17 pm molecular beam nozzle at room temperature. Laser power was adjusted to 7.5 W for each of the 001+100 10pm C 0 2 laser lines. All intensities are normalized to that of the P(18) laser line. Crosses denote data points, and circled crosses are laser lines where saturation studies were carried out.The arrow indicates the absorption of SF6 in the gas phase. The continuous line was drawn by eye. and 954.6 cm-I peaks are assigned to the parallel and perpendicular transitions of the SF, dimer respectively, as reported by Geraedts et The central peak we assign to the SF,-Ar dimer; since the dipole-dipole resonance force is now absent the peak frequency (946.4 cm-') is very close to that of the SF, monomer. The experimental linewidths are 3.3 cm-' for the SF, dimer peaks, and 1.3 cm-I for SF; Ar, with an uncertainty of k0.3 cm-I.41 6 GENERAL DISCUSSION Saturation curves were taken on the 001 +I00 P(30), P(8) and P(18) CO, lines, and Each saturation curve was fitted to an equation in the form the data shown in fig. 8.I = Io[l - exp(--P)] ( 1 ) where 1, is thc asymptotic signal for the curve, CI is a constant and P is the laser power in W. This equation is equivalent to the simple expression for dimer dissociation discussed in ref, (2). * P(18) * + + P(8) m ' P(30) 1 I 0 3 6 9 12 15 laser power/W FIG. 8.-Relative saturation data using the same gas and molecular-beam conditions as in fig. 7, along with the best-fit curvcs to each set of saturation data. Laser lines used are shown, where the 001+100 CO, P(18), P(8) and P(30) lines represent the saturation of the SF,-Ar, (SF6), perpendicular, and (SF& parallel transitions, respectively. The most striking result of the saturation study is thc fact that the asymptote for the (SF,), parallel transition does not equal that for the perpendicular transition, although the same species is being photolysed in both cases.This indicates that the (SF,), spectra are heterogeneously broadened, which casts doubt on all theories relying exclusively on homogeneous broadening. Two possible canses of hetero- geneous broadening are the presence of other species such as (SF,)2-(Ar)n (n > 1) which absorb at P(8) and P(30), or rotational structure in the perpendicular and parallel transitions of (SF,),. Computer simulation of the rovibrational transitions of (SF,),, assuming a 4.8 8, separation between the sulphur atoms, yields a rotational temperature of 25 K to fit the 3.3 cm-l linewidth observed. The presence of higher polymers with our source conditions may be less likely, since no gross changes in the spectrum occur at pressures less than 30 p.s.i. J .Geraedts, S. Stolte and J. Reuss, 2. Phys., 1982, A304, 167. J. Geraedts, S. Setiadi, S. Stoltc and J. Reuss, Chem. Phys. Left., 1981, 78, 277. ' J . Geraedts, M. Waayer, S. Stolte and J. Reuss, Farduy Disc~uss. Chem. SOC., 1982, 73, 375. Mr. J. Geraedts (Universiry of Nljmegen) said: Using the N,O laser lines available around 955 cm-' the spectral resolution of the blue peak of the dimer-attenuation spectrum has been increased by more than II factor 2. As shown in fig. 9, a separateGENERAL DISCUSSION 41 7 peak at 959.5 cm-' was found, shifted ru. 4.2 0.7 cm-' to the blue side of the prin- cipal attenuation peak at 955.3 cm-I, This new structure is interpreted as a simultaneous excitation of the Van der Waals stretch vibration and the u3 mode oscillating perpendicular to the dimer axis.The small value of 4 cm-l for the Van der Wads stretch corresponds to a broad potential well around 5.8 A. Subsequently, G. Luijks of OUF laboratory has found this dimur-stretch frequency in the Raman spectrum of ground state (SF,),, resolving further details. - VI w .& E error 1 0 t 0 uicm-' Fw. 9.--Dimer attenuation spectrum. The improved resolution of the bfue peak is due to the use of a N20 laser (e, NzO laser lines; 0, C02 laser lines). A 5 :< SF6-95 % He mixturc was employed. The source conditions were 7; = 233 K, po --x 1000 Torr and a 30 pni nozzle diameter. The laser fluence was ca. 2 J m-I. Signal attenuation was observed on mass SF'+, In fig. 9 the strength of the peak at 959.5 cm-' is in agreement with a straight- forward estimate, using the electrical dipole-dipole interaction as coupling agent between the uj vibration and the Van der Wads stretch.The principal peak at 955.3 cm-' i s asymmetric; compatible with a symmetric Lorentzian would be a second peak at 953 cm-', with a maximum height of 30%, originating from a ground-state dimer with the Van der Waals stretch in its first excited state. The laser excitation would lead then t o a single excitation of the u3 mode (perpendicular to the dimer axis), without change of the Van der Waals stretching vibration. The observed shift corresponds, in this tentative interpretation, to a change of the value (47~t;,,)-~ p i l ( R - 3 : , for the excited Van der Wads stretching vi- bration.In fig. lO(a) the attenuation spectrum is presented for 0.57; SF6 in Ar, as detected on mass Ar-SF,+. The monomer peak appears to be shifted by ca. 7.5 cm-I to the red, in consequence of the presence of the Ar atoms in the neutraf parent-duster (Ar),-SF,. Under essentially the same conditions, the attenuation spectrum has been recorded as observed on mass SF,+-SF,, see fig. 10(h). The dimer spectrum appears to be red- shifted by cu. 4 cm-', and only slightly changed. The presence of the Ar atoms in the parent cluster, (SF6)2-Ar,i, increases 1 he ohservcd linewidth. Prof. J. Jortner (Tel-Auiv University) said: It would be extremely interesting to extend the studies of the infrared spectra of hydrogen-bonded and other molecular clusters to the overtone region.The infrared spectra of molecular crystals in the over- tone region exhibit, in addition to the v : 2 overtone, a cooperative vibrational41 8 GENERAL DISCUSSION excitation involving a simultaneous excitation of two u = 1 vibrations on a pair of interacting molecules.1 A theoretical treatment of cooperative vibrational excitations was provided by Jortner and Rice in terms of a local site approximation, which rests on the notion of a Fermi resonance between the molecular overtone 120) and a doubly excited 11 I ) , whose energies are split in zero-order due to the anharmonicity defect. The intermolecular coupling, essentially originating from short-range repulsive intcr- actions, scrambles the zero-order states and redistributes the intensity of the 120) overtone among the two transitions.The local-site approximation is adequate for a I I I 1 I I 1 L 930 940 950 960 v/cm - ’ FIG. lO.--(u) Monomer-attenuation spectrum for SF, imbedded in Ar. The sotirce conditions wcre To = 233 K and pu = 2250 Torr. A 0.5 ”/, SF6-99.5 % Ar mixture was employed. The laser fluence was about J m-’. (b) Attenuation spectrum for dimers imbedded in Ar. A0.5”/, SF,-99.5?4 Ar mixture was employed. The attenuation was Signal attenuation was observed on mass SF,+-Ar. The source conditions were To 7 233 K andpo = 2250Torr. The laser fluence was 6 J m-2. measured on mass SF,+-SF,. dimer and for small clusters. For larger clusters the effects of vibrational exciton band structure can be incorporated in a complete manner, using a band model.Such a band model starts from a manifold of zero-order two-particle excitations, which form a two-vibron continuum, while the intramolecular anharmonicity induces a diagonal local perturbation resulting in the branching of a two-particle bound state (the molecular overtone) from the continuum. This situation can be accounted for by the formalism of two-particle excitations in solids [see for example ref. (3)]. TheGENERAL DISCUSSION 419 observation of the energetics, intensities and absorption lineshapes of cooperative vibrational excitations in clusters will be of considerable interest for the elucidation of vibrational exciton band structure in finite systems. A. Ron and D. F. Hornig, J . Chetn. Phys., 1963, 30, 1 1 29. J. Jortner and X. Rice, J. Chern.Phys., 1966,44, 3364. J. Klafter and J. Jortner, Chern. Phys., 1980, 47, 25. Prof. A. W. Castleman Jr, Dr. B. D. Kay, Dr. F. J. Schelling and Dr. R. Sievert (University of Colorado) (communicated) : Studies of neutral clusters formed in a supersonic expansion yield a wealth of information on structures and formation dyn- amics of weakly bound Van der Waals clusters and on the continuous course of change from isolated molecules to a degree of aggregation analogous to that existing in the condensed phase. Examination of the intensity distribution of clusters of up to 40 H20 molecules provide evidences for the formation of particularly stable cyclic struc- tures of these complexes. Electrostatic focusing experiments in quadrupole and hexapole fields of clusters up to 9 waters also complement the results of Vernon et al.in establishing the existence of these structures. Finally, isotopic studies of H20 clusters provide several details of the dynamics of formation of these species. In our experiments, neutral clusters formed in a supersonic jet are studied by analysing intensity distributions mass-spectrometrically following electron-impact ionization. A number of interesting features in the distribution for water clusters were classified as being one of two types: (1) those which are independent of stag- nation conditions but have some dependence on the ionization energy, and (2) other relatively weak features which depend strongly on stagnation conditions and are independent of the energy of the ionizing electrons.' Features of the first type are seen as breaks in the observed distribution corresponding to water clusters containing 22,27,29 and 3 1 molecules.These effects are observed under all expansion conditions and may be related to the particular structural stability of these clusters. A careful study showed that these features are obtained under all electron energies above threshold, but diminished in magnitude within the first few volts of threshold (ca. 13 eV). In agreement with other work,l these intense peaks correspond to known unit-cell clathrate structures in the condensed phase. Upon ionization, the 22-mer yields, for example, a clathrate composed of 12 5-member rings formed from 20 water molecules, having one molecule trapped inside, and with the extra mobile proton residing on the surface.The neutral 27-mer produces a known slightly distorted clathrate consisting of 2 planar 6-member rings and 12 5-member rings formed by 24 molecules on a case surface, the case trapping one or two molecules inside.' Similar structural considerations apply to the 29- and 31-mer clusters.' A mechanism of rapid reorientation of the water molecules and the stability of these ionic species formed from the original neutral cluster can be explained by the 3-coordinate hydrogen bonding assisted by a mobile proton.' In other experiments, such as with methanol, smooth intensity distributions are observed ; an equivalent 3-coordinate hydrogen- bonded network required for this mechanism is not possible in systems of protonated alcohol clusters. Examination of the intensity distribution for smaller clusters, namely, those con- taining '-lp to 20 water molecules, also revealed several reproducible structures.These are of the second type, in that their intensity was influenced by stagnation con- ditions, with decreasing intensity for conditions producing higher supersaturations on expansion. Discontinuities in the derivative of the cluster distribution were seen for clusters of 6, 8, 10 and 12 molecules and were found to be totally independent of the energy of the ionizing electrons.420 GENERAL DISCUSSlON Tn the electrostatic refocusing experiments conducted on neutral water clusters, net dipole moments are indicated for the monomer and dimer only, while larger clusters (up to the 9-mer studied) were found to be non-polar.z This result suggests a cyclic structure for the intermediate size chsters, and in this respect supports the spectral data of Vernon ef al.More recently, we have examined the focusing properties of acetic acid clusters. In these experiments, significant dipole moments were measured for the monomer, trimer and pentamer, using second-order Stark focusing; the dimer and tetramer were non-polar. This evidence suggests the non-cyclic nature of the clusters containing an odd number of moIecules, in systems for which there is a predisposition for the form- ation of a strongly bound head-to-tail dimer. In other work, studies of isotopicalIy mixed water clusters have provided informa- tion related to the formation and growth of these species. Early work with pure H20 and NH3 systems showed that virtualIy identical cluster distributions are obtained under widely different expansion conditions, providing the absolute dimer concentration in the source vapour was maintained at the same fixed value.Over the same range of temperature and pressure, very different cluster distributions were obtained under con- ditions for which the calculated dimer concentration varied. This strongly suggests that growth in expansion nozzles proceeds from pre-existing dimers in the source, rather than upon dimers formed during the expansion. Further cluster growth then appears to occur uia interaction of monomers with dimers already present. This is in general agreement with recent isotopic experiments performed in our laboratory (de- scribed below) which indicate that clusters are formed in the beam vibrationalIy ex- cited, only undergoing subsequent growth following cooling collisions. The lifetime of new dimers formed in the expansion must be so short that appreciable collisional stabilization cannot occur, whereas the increased number of degrees of freedom in the larger clusters enables them to survive long enough for this stabilization to occur. Evidence that the clusters are vibrationally hot when formed in the expansion was established in a series of experiments involving water and its deuterated analogue. Under well defined isotopic ratios and source stagnation conditions, intensity distri- butions observed folIowing ionization in the mass spectrometer showed a cumulative isotopic enrichment. Instead of the expected binomial distribution of isotopic peaks, masslcharge ratio FIG. 1 1 ,--Intensity plotted as a function of the mass,change ratio for water clusters.GENERAL DISCUSSION 42 1 a pronounced shift (see fig. 11) to the heavier deuterated species was observed. For clusters of up to 13 molecules the effect was found to be cumulative and did not have any apparent trcnd with increasing cluster size. This result can be explained using a statistical (R.R.K.M.) model of unimolecular decomposition for vibrationally excited clusters.2 Using a simple model for the transition state, an isotopic enrichment factor, p, for a unimolecular decomposition-controlled process can be calculated from 112 p = (:I exp[--Afi2.,./kT]. In this equation, is defined as the zero-point energy difference between the tran- sition state for the elimination of either an H 2 0 or a D,O molecule. In agreement with theoretical prediction, an enrichment factor of 1.26 & 0.02, independent of the initial isotopic ratio and stagnation conditions, was found experimentally. More complete verification of the production of vibrationally excited clusters was obtained by conducting experiments in seeded expansions. In agreement with theory, under these conditions of lower effective temperature, a larger enrichment factor of 1.35 was found. V. Herrnann, B. D. Kay and A, W. Castleman Jr, Chem. Phys., submitted for publication, and references therein. B. D. Kay, Ph. D , Thesis (University of Colorado, 1982), and references therein.

ISSN:0301-7249    

DOI:10.1039/DC9827300399

出版商:RSC    

年代:1982

数据来源: RSC

摘要:

GENERAL DISCUSSION 42 1 Closing Remarks BY A. D. BUCKINGHAM We have had a splendid Discussion on a topic of considerable general interest. Chemists have often been tempted to postulate " complex formation " to cover ignorance or inability to reconcile experiment and simple theory. The study of the weakly bound species that we call Van der Waals molecules clarifies this notion of a " complex." At this meeting we have seen and heard that many different techniques can provide information on the structure and properties of these " complexes " in the gas phase. What is a Van der Waals molecule? Between sessions I have been asking experts for their views, but I have to report that my consultations have not brought out an agreed definition. There is a feeling that the concept may be temperature-dependent, and that at high enough temperatures molecules that we normally regard as " good ", such as H, and N,, might qualify.There is general agreement that Van der Waals molecules are weakly bound and have large-amplitude vibrations. But what do we mean by " weakly bound " ? Do we mean that the depth of the potential well is small compared with that in H,, or in (H,O),, or relative to kT? My own suggestion is that a Van der Waals molecule is one whose dissociation energy is less than ca. 40 kJ mol-' (10 kcal mol-1 or 3000 cm-l). So some hydrogen-bonded complexes are on the borderline between Van der Waals molecules and " good '' molecules. An im- portant feature of a Van der Waals molecule is the large-amplitude vibrational motion, and this means that the structural and electronic properties (" bond " lengths and422 GENERAL DISCUSSION angles, dipole moments, polarizabilities, etc.) may differ widely in different vibrational states, and perhaps even in different rotational states.And in a gas of Van der Waals molecules at equilibrium at room temperature, many different quantum states will be populated. This definition of a Van der Waals molecule is not temperature-dependent, but those who think that H2 would be a Van der Waals molecule in an environment at T E 5 x lo4 K may be able to reconcile their view with the definition by noting that the dissociation energies of highly excited vibrational states of H, are much less than 40 kJ mol-l. If we accept this definition then we should have to rule that Dr. J.Tennyson’s poster entitled “Ab initio structure and dynamics of a Van der Waals molecule: potassium cyanide (KCN) ” does not qualify! It is not for me to attempt to summarise the many contributions to this Discussion -the authors have done that for themselves, and the varied and lengthy discussions have brought out numerous points of interest. 1 was particularly impressed by the beautiful results reported by the spectroscopists; their progress will obviously en- hance our knowledge of intermolecular potential-energy surfaces. Atld the study of molecular clusters is very interesting, particularly in bridging the gap between the gaseous and condensed phases. We heard much on predissociation and on calcula- tions of rates of dissociation, and those contributions are specially topical because of their relevance to unimolecular reaction rate theory, To conclude, 1 should like to suggest that, in spite of what we have heard in this Discussion, there is still hope that we can achieve a qualitative understanding of the structures of Van der Waals molecules through the simple notions of long-range forces and molecular shape.Prof. Klemperer reported that his observation of a structure of the type shown in (I) for the formaldehyde-hydrogen-fluoride Van der Waals molecule meant that dipolar forces could not be responsible. I agree that dipole-dipole forces do not describe the attraction at these short distances, but can we rule out the electro- static interaction? It would be interesting to examine the actual electrostic energy in this case. But whether or not it favours the observed structure, I believe that useful predictions can be made on the basis of electrostatic forces coupled with the shape of the interacting molecules, as represented, for example, by atomic Van der Waals radii.After Klemperer et al. had reported some years ago that the benzene and hexa- fluorobenzene dirners (C6H6), and (C6F& were dipolar, I suggested that the dimer of I ,3,5-C6F3H, might be non-polar, with a face-to-face structure staggered by 60” so that the H atoms are near the F atoms of the neighbour. Such a structure is favoured by the electrostatic and dispersion forces, because of the closeness of the approach. This particular dimer has, I understand, been found by Klemperer and his group to be non-dipolar. It will be interesting to discover if the crystal structure of 1,3,5-tri- fluorobenzene differs radically from those of benzene and hexafluorobenzene. Similarly, the dimer of s-triazine, (s-C,N,H,),, might be non-dipolar. So, following Prof. Ewing, I suggest that there are ‘‘ propensity rules ” for under- standing and predicting the structures of Van der Waals molecules. The ‘‘ rules ” are based only on considerations of the long-range forces associated with the properties of the isolated molecules, including their shapes, and they will be valid if the effects ofGENERAL DISCUSSION 423 electron exchange can be neglected except in so far as they determine the Van der Waals radii of the atoms. Finally, it is abundantly clear that there is no shortage of interesting research to be done by both experimentalists and theorists on Van der Waals molecules. W. Klemperer et al., J. Chem. Phys., 1975, 63, 1419; 1979, 70, 4940. A. D. Buckingham, personal communication to Prof. Klemperer, March 1978.

ISSN:0301-7249    

DOI:10.1039/DC9827300421

出版商:RSC    

年代:1982

数据来源: RSC

摘要:

LIST OF POSTERS 1. Ab initio structure and dynamics of a Van der Waals moIecule: Potassium cyanideJKCN J. Tennyson Uniuersity of Nijmegen, The Netherlands 2. Laser-induced fluorescence from the Van der WaaIs molecule LiAr J. P. Toennies, I . Biittner and E. Hulpke Max-PIanck-Institilt fiir Srroniungsforschung, Gottingen, West Germany 0, Echt, M. Knapp, E. RecknageI, A. Reyes Flotte and K. Sattier Uniuersify of Konstanz, West Germany 4. Calculation of the far-infrared vibration-rotation spectrum of Ar-HCl J, A. Vliegenhart Universily of Urrecht, The Netherlands 5 . State-seIect ive pho topredissocat i o n of gly oxal-rare-gas complexes B. Soep and N. Halberstadt Uniceusity of Paris South, France 6. Structure of rare-gas oxides and fluorides by scattering experiments with magnetically seIected atoms V.Aquilanti, E. Luzzatti, F. Pirani and G. G. Volpi Uniuersity of Perugia, Italy P. A. Freedman and A. M. Grifiths Unirersity CoUege of Wales, Aberysrwyth M. Rigby, S. H. Ling and R. R. Miledi, Queen Elizabeth College, London G. C . Maitland, W. A. Wakeham and V. Vesovic Imperial College, London from scattering experiments R. Diiren Max-Planck-lnsritut fir Stromungsforschung, Gottingen, Wesf Germany potentials Ph. Brechignac Uniuersity of Paris South, France dynamics R. S . Berry and G. S. Ezra Unitlersity of Chicago, U.S.A. 13. Calculation of birner bound states using the R-matrix method G. Danby and D. R. Flower Uniuersity of Durham 14. Dynamics, energetics and structure of weakly bound molecules--eIucidating transitions from the gaseous to the condensed phase A. W. Castleman, B. D. Kay, F. J. Schelling and R. Sievert Unicersity of Colorado, U S A . 3. Mass spectrometry of Van der Waals clusters 7. Van der Wads complexes with aromatic mdecules 8. Towards an intermolecular potential for N2 9. The effects of potential anisotropy on the transport coefficients of dilute polyatomic gases 10. Ground-state and excited-state potentials of alkali-metal-rare-gas molecules determined 1 1. Inelastic cross-sections from infrared laser experiments and anisotropic intermolecular 12. A model of the transition between molecule-like and weakly coupled '' atomic

ISSN:0301-7249    

DOI:10.1039/DC9827300425

出版商:RSC    

年代:1982

数据来源: RSC

摘要:

INDEX OF NAMES* AItman, R., 116 Amirav, A., 153, 183 Baiocchi, F. A . , 126 Balint-Kurti, G. G., 133, 404 Barker, J. A., 235, 305 Barton, A. E., 45 Battaglia, F., 257, 307 Beenakker, J . J. M., 277, 295 Berkovitch-Yellin, Z., 153 Beswick, J . A., 357, 406 Brechignac, Ph., 115, 288, 303 Brobjer, .I. T., 123, 128 Brumbaugh, D. V., 137 Buck, U., 187, 277, 279, 281, 308 Buckingham, A. D., 116, 129, 421 Burton, P. G., 109, 111: 181, 275, 413 Candori, R., 289 Casavecchia, P., 257, 307 Castleman, A. W., 124, 297, 419 Certain, P. R., 311 Connor, J. N. L., 399 Corey, G. C., 339 Cruickshank, A. J. B., 127, 305 DeLeon, R. L., 63 Delgado-Barrio, G , , 406 Douketis, C., 286 Eley, C. D., 292 Even, U., 153, 175, 381, 183 Ewing, G. E., 122, 306, 325, 402 Faubel, M., 205, 277, 282, 284 Fuchs, R.R., 285 Geraedts, J., 375, 41 6 Gianturco, F. A., 177, 257, 282, 289, 307 Gough, T. E., 182, 415 Halberstadt, N., 357, 400 Hayes, I. C., 19 Haynam, C. A., 137 Holmer, B. K., 311 Howard, B. J., 45, 121, 122, 129, 308 Hutson, J. M., 114, 132, 276, 308, 339, 404 Jortner, J., 109, 114,153, 173, 175, 180, 181, 183, Jouvet, C., 409 Kay, B. D., 124, 297, 419 Keil, M., 182, 286, 41 5 Kidd, I. F., 133, 404 Klemperer, W., 115, 116, 126 Knight, D. G., 182, 415 Kohl, K. H., 205 Krajnovich, D. J., 387 La!, M., 295 Lee, Y. T., 387 417 Kwok, H-S., 387 Legon, A, C., 71, 127, 128 Le Roy, R. J . , 121, 122, 280, 339 Leutwyler, S., 153 Levy, D. H., 137, 173 Ling, S. H., 117 Lisy, J. M., 122, 174, 387 Liu, W-K., 241, 302 McCourt, F. R. W., 241, 280, 294, 301 McKellar, A. R.W., 89 Magnasco, V., 109 Maitland, G. C., 278, 290, 301 Mareca, P., 406 Marshall, M., 116 -Meyer, H., 281 Miledi, R. R., 117 Millen, D. J., 71, 127, 123 Mills, P. D., 129 Muenter, J. S., 63, 124 Murrell, J. N., 130 Ondrechen, M. J., 153 Pirani, F., 257, 289, 307 Pople, J. A,, 7, 109 Price, S. L., 296 Reuss, J., 375 Rigby, M., 117 Schaefer, J . , 410 Schelling, F. J., 124, 297, 419 Schramm, B., 283, 294 Scoles, G., 182, 286, 415 Shen, Y. R., 387 Sievert, R., 124, 297, 419 Smith, E. B., 221,292,295, 296,297,298,305 Smith, I. W. M., 408 Soep, B., 179 Stolte, S., 375 Stone, A. J., 19, 111, 113, 114 Tang, K. T., 205 Tennyson, J., 118, 121 Tjndell, A. R., 221, 292, 295, 297, 305 Toennies, J. P., 205 Trarner, A., 123, 174, 387 van der Avoird, A., 33, 114, 1 15, 1 I8 Vecchiocattivj, F., 257, 289, 307 Vernon, M. F., 387 Vesovic, V., 278, 290 Villareal, P., 406 Waayer, M., 375 Wakeham, W. A., 278, 290 Welis, B. H., 306 Western, C. M., 129 Yokozeki, A , , 63 Yung, Y . Y., 205 Zhu, Z. H., 130 * The page numbers in heavy type indicate papers submitted for discussion.

ISSN:0301-7249    

DOI:10.1039/DC9827300427

出版商:RSC    

年代:1982

数据来源: RSC

摘要:

THE Date 1907 1907 1910 1911 1912 1913 1913 1913 1914 1914 1915 1916 1916 1917 1917 1917 1918 191 8 1918 1918 1919 1919 I920 1920 I920 1920 1921 1921 1921 1921 1922 1922 1923 1923 1923 1923 1923 1924 1924 1924 1924 1924 1925 1925 I926 1926 1927 1927 1927 1928 1929 I929 1929 1930 1931 1932 1932 GENERAL DISCUSSIONS OF FARADAY SOCIETY/FARADAY DISCUSSIONS OF THE CHEMICAL SOCIETY Subject Osmotic Pressure Hydrates in Solution The Constitution of Water High Temperature Work Magnetic Properties of Alloys Colloids and their Viscosity The Corrosion of Iron and Steel The Passivity of Metals Optical Rotatory Power The Hardening of Metals The Transformation of Pure Iron Methods and Appliances for the Attainment of High Temperatures in a Laboratory Refractory Materials Training and Work of the Chemical Engineer Osmotic Pressure Pyrometers and Pyrometry The Setting of Cements and Plasters Electrical Furnaces Co-ordination of Scientific Publication The Occlusion of Gases by Metals The Present Position of the Theory of Ionization The Examination of Materials by X-Rays The Microscope: Its Design, Construction and Applications Basic Slags: Their Production and Utilization in Agriculture Physics and Chemistry of Colloids Electrodeposition and Electroplating Capillarity The Failure of Metals under Internal and Prolonged Stress Physico-Chemical Problems Relating to the Soil Catalysis with special reference to Newer Theories of Chemical Action Some Properties of Powders with special reference to Grading by Elutria- The Generation and Utilization of Cold Alloys Resistant to Corrosion The Physical Chemistry of the Photographic Process The Electronic Theory of Valency Electrode Reactions and Equilibria Atmospheric Corrosion.First Report Investigation on Oppau Ammonium Sulphate-Nitrate Fluxes and Slags in Metal Melting and Working Physical and Physico-Chemical Problems relating to Textile Fibres The Physical Chemistry of Igneous Rock Formation Base Exchange in Soils The Physical Chemistry of Steel-Making Processes Photochemical Reactions in Liquids and Gases Explosive Reactions in Gaseous Media Physical Phenomena at Interfaces, with special reference to Molecular Atmospheric Corrosion. Second Report The Theory of Strong Electrolytes Cohesion and Related Problems Homogeneous Catalysis Crystal Structure and Chemical Constitution Atmospheric Corrosion of Metals.Third Report Molecular Spectra and Molecular Structure Colloid Science Applied to Biology Photochemical Processes The Adsorption of Gases by Solids The Colloid Aspect of Textile Materials tion Orientation Volume Trans. 3* 3* 6* 7* 8* 9* 9* 9* 10* 11 12* 12* 13* 13* 13* 14* 14* 14" 14* 15* 15* 16' 16* 16* 16* 17* 17* 17* 17* 18* 18 19* 19 19* 19 19* 20* 20* 20* 20* 20* 21 * 21 22 22 23 * 23* 24 24 25 * 25 * 26* 26 21 28 29 10:430 FARADAY DISCUSSIONS OF THE CHEMICAL SOCIETY Dare Subject 1933 1933 Free Radicals 1934 Dipole Moments 1934 Colloidal Electrolytes 1935 The Structure of Metallic Coatings, Films and Surfaces 1935 The Phenomena of Polymerization and Condensation 1936 Disperse Systems in Gases: Dust, Smoke and Fog 1936 Structure and Molecular Forces in (a) Pure Liquids, and (b) Solutions 1937 The Properties and Functions of Membranes, Natural and Artificial 1937 Reaction Kinetics 1938 Chemical Reactions Involving Solids 1938 Luminescence 1939 Hydrocarbon Chemistry 1939 The Electrical Double Layer (owing to the outbreak of war the meeting 1940 The Hydrogen Bond 1941 The Oil-Water Interface 1941 The Mechanism and Chemical Kinetics of Organic Reactions in Liquid Systems 1942 The Structure and Reactions of Rubber 1943 Modes of Drug Action 1944 Molecular Weight and Molecular Weight Distribution in High Polymers (Joint Meeting with the Plastics Group, Society of Chemical Industry) 1945 The Application of Infra-red Spectra to Chemical Problems 1945 Oxidation 1946 Dielectrics 1946 Swelling and Shrinking 1947 Electrode Processes 1947 The Labile Molecule 1947 Surface Chemistry (Jointly with the SociktC de Chimie Physique at Bordeaux) Published by Butterworths Scientific Publications, Ltd 1947 Colloidal Electrolytes and Solutions 1948 The Interaction of Water and Porous Materials 1948 The Physical Chemistry of Process Metallurgy 1949 Crystal Growth 1949 Lipo-proteins 1949 Chromatographic Analysis 1950 Heterogeneous Catalysis 1950 Physico-chemical Properties and Behaviour of Nuclear Acids 1950 Spectroscopy and Molecular Structure and Optical Methods of Investi- 1950 Electrical Double Layer 195 1 Hydrocarbons 1951 The Size and Shape Factor in Colloidal Systems 1952 Radiation Chemistry 1952 The Physical Chemistry of Proteins 1952 The Reactivity of Free Radicals 1953 The Equilibrium Properties of Solutions on Non-electrolytes 1953 The Physical Chemistry of Dyeing and Tanning 1954 The Study of Fast Rcactions 1954 Coagulation and Flocculation 1955 Microwave and Radio-frequency Spectroscopy 1955 Physical Chemistry of Enzymes 1956 Membrane Phenomena 1956 Physical Chemistry of Processes at High Pressures 1957 Molecular Mechanism of Rate Processes in Solids 1958 Interactions in Ionic Solutions 1957 Configurations and Interactions of Macromolecules and Liquid Crystals 1958 Ions of the Transition Elements 1959 Energy Transfer with special reference to Biological Systems 1959 Crystal Imperfections and the Chemical Reactivity of Solids 1960 Oxidation-Reduction Reactions in Ionizing Solvents 1960 The Physical Chemistry of Aerosols 1961 Radiation Effects in Inorganic Solids 1961 The Structure and Properties of Ionic Melts 1962 Inelastic Collisions of Atoms and Simple Molecules 1962 High Resolution Nuclear Magnetic Resonance 1963 The Structure of Electronically Excited Species in the Gas Phase 1963 Fundamental Processes in Radiation Chemistry Liquid Crystals and Anisotropic Melts was abandoned, but the papers were printed in the Transactions) gating Cell Structure Volume 29 * 30 30 31 * 31 * 32 * 32* 33* 33* 34" 34* 35* 35* 35* 36* 37 * 37* 38 39 40: 41 42 * 42 A 42 B Disc.1* 2 Trans. 43* Disc. 3 4* 5 6 7 8* Trans. 46* Disc. 9 Trans. 47 Disc. 10 1 1 12* 13 14 15 16* 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 * 34 35 36Dare 1964 1964 I955 1965 I966 1966 1967 1967 1968 1968 1969 1969 1970 1970 1971 1971 1972 I972 1973 1973 1974 1974 1975 1975 1975 1977 1977 1977 I978 1978 1979 1979 1980 1980 1981 1982 i9ai FARADAY DISCUSSIONS OF THE CHEMICAL SOCIETY Subject Chemical Reactions in the Atmosphere Dislocations in Solids The Kinetics of Proton Transfer Processes Intermolecular Forces The Role of the Adsorbed State in Heterogeneous Catalysis Colloid Stability in Aqueous and Non-aqueous Media The Structure and Properties of Liquids Molecular Dynamics of the Chemical Reactions of Gases Electrode Reactions of Organic Compounds Homogeneous Catalysis with Special Reference to Hydrogenation and Bonding in Metallo-organic Compounds Motions in Molecular Crystals Polymer Solutions The Vitreous State Electrical Conduction in Organic Solids Surface Chemistry of Oxides Reactions of Small MolecuIes in Excited States The Photoelectron Spectroscopy of MoIecules Molecular Beam Scattering Intermediates in Electrochemical Reactions Gels and Gelling Processes Photo-effects in Adsorbed Species Physical Adsorption in Condensed Phases Electron Spectroscopy of Solids and Surfaces Potential Energy Surfaces Radiation Effects in Liquids and Solids Ion-Ion and Zon-Solven t Interactions Colloid Stability Structure and Motion in Molecular Liquids Kinetics of State Selected Species Organization of Macromolecules jn the Condensed Phase Phase Transitions in MoIecular Solids Photoelect rochernis try High ResoIution Spectroscopy Selectivity in Heterogeneous Catalysis Van der Waals Molecules Oxidation Precipitation 43 1 Volume 37 38 39 40 41 42 43 44 4.5 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 * 66 67 68 69 70 71 72 73 * Not available; for current informrrfion on prices, etc., of auaiiuble uofolumes, please confact fhe Morkering Oficer, Royal Society of Chemistry, Bidingtoan House, London WI V OBN stating whether or nof yuu are a member of the Society.

ISSN:0301-7249    

DOI:10.1039/DC9827300429

出版商:RSC    

年代:1982

数据来源: RSC