J. CHEM. SOC. FARADAY TRANS., 1994, 90(21), 3229-3236 Interaction Potentials and Fragmentation Dynamics of the Ne-Br, Complex in the Ground and Electronically Excited States Alexei A. Buchachenko, Alexei Yu. Baisogolov and Nikolai F. Stepanov Department of Chemistry, Moscow State University, Moscow I 19899,Russia Quantum calculations have been performed on the structure and vibrational predissociation dynamics of Ne. -.Br, van der Waals complexes in the ground and electronically excited states associated with the X'Z:,+(O:) and B 311,(0:)terms of Br, using simple empirical potential-energy surfaces. A reasonable approximation has been found for interactions in the ground state and the best potential for the excited state is chosen from three trial ones. Some results of classical dynamics calculations are also presented and discussed. The structure and dynamics of weakly bound systems (van der Waals molecules, hydrogen-bonded complexes, clusters, etc.)have attracted increasing attention by theoreticians over the last two decades (e.g.ref. 1 and 2). Developments in high- resolution spectroscopic techniques have made accurate data available on the bound and metastable states of the com- plexes and have allowed one to probe the intermolecular potential in the vicinity of an attractive well, which is hardly accessible in collisional measurements. Theory provides the essential step in the inversion procedure for converting the observed spectroscopic data into the interaction potential parameters. The ground-state potential-energy surfaces (PES) are usually determined from reliable microwave or IR spectra,' whereas the information on bound states of electronically excited complexes is much poorer as a rule.In this case one can gain additional information by analysing the dynamics of various decay (predissociation) processes. The rare-gas atom-excited halogen molecule van der Waals (vdW) complexes are excellent examples for studying the vibrational predissociation (VP), or decay, of a metastable state uia vibrational energy transfer, Rg. .-Hal,(u, n,, nb)+Rg + Hal,(o', j') (1) where u and u' designate the initial and final vibrational states of the halogen fragment, n, and nb are the (approximate) initial quantum numbers of the vdW stretching and bending modes, and j' is the diatom rotational momentum.Among the Rg. ..Hal, (B 'nu)complexes, the chlorine complexes were the subject of detailed high-resolution fluorescence studies4 and accurate quantum calculations5 based on realis-tic empirical PESS.~ Extensive experimental investigations, including time-resolved measurements, were carried out for Rg--.I,(B) systems as well.7 These complexes traditionally serve as a test for various approximate theoretical approaches.' Less theoretical attention has been paid to bromine com- plexes despite the substantial body of experimental informa- tion Vibrational predissociation widths of the Ne. --Br2 (B 'nu)complex for a wide range of initial vibra- tional excitations (u = 11-30) were estimated from low-resolution spectra by Janda and co-workers.'' Subsequent measurements with higher resolution performed by the same group allowed one to establish the T-shaped equilibrium con- figuration, determine the vibrationally averaged geometric parameters of this complex" and improve the data on decay widths for several values of u which fall within the range" of 10-20. Limited information on the product vibrational and rotational state distributions was obtained in dispersed fluo- rescence studies.' 2,1 ' In addition, direct lifetime measure- ments were carried out for the ground-state Ne. -.Br, (X'El, u = 1) ~omplex.'~ Theoretical studies of Ne...Br, are exhausted by the implementation of Ewing's momentum gap law" and our classical and quantum calculations within a restricted two- dimensional (2D) T-shaped model.'' These approaches are useful for qualitative analysis of VP dynamics, but they do not provide a quantitative test of the PES involved. This is the subject of the three-dimensional (3D) study of the structure, spectra and fragmentation dynamics of the Ne. * .Br, complex discussed here. The important point of the present investigations is the simultaneous consideration of ground and electronically excited states of this system, which allows us to treat the measured frequencies of the BtX transition directly. Together with the calculations on lifetimes and VP product state distributions, this analysis provides an almost complete spectroscopic characterization of the PESs.The main theoretical approach implemented here is the decoupled Fermi golden rule approximation. At the same time, we also involve in the discussion some results of quasi-classical trajectory (QCT) calculations on Ne. -.Br2(B) VP dynamics. Computational Procedures The total Hamiltonian of the triatomic complex is Jacobi coordinates has the form5 p2 p2 j2H = -+ -+ -+ -l2 + U(r,R, 6) (1)2m 2p 2mr2 2pR2 where r is the Br, internuclear distance, R is the Ne...Br, centre of mass separation, and 6 is the angle between the r and R vectors, p and P are the classical linear momenta con- jugated to r and R, respectively, or their quantum counter- parts, j and I are the angular momentum variables (or corresponding quantum mechanical operators) representing the rotational momentum of the diatom and the orbital momentum of the Ne atom; m and p are the reduced masses, U(r,R, 6) denotes the PES.QuasiclassicalTrajectory Method The procedure used here is similar to that applied by Wozny and Gray for the Ne. -C12(b) fragmentation dynamics.' In brief, initial conditions are selected quasiclassically using the action-angle form of the separable Hamiltonian function HO= h,(p, r) + h,(~,R) + h~, (11)e) evaluated by the Taylor expansion of the PES nearby equi- librium point r = F, R = 2, 8 = n/2. For each initial excita- tion of the diatom u specified as v + 1/2 action of the h, oscillator, the ground state of the vdW subsystem (n,= 0, nb = 0) is considered by setting the corresponding actions to 1/2.The Hamiltonian equation of motion with the total Hamil- tonian function [eqn. (I)] for zero total angular momentum, are integrated up to a preselected time limit, and the trajec- tories reaching some large critical interfragment separation R are regarded as dissociative. An exponential fit to the survival probability obtained from an ensemble of lo00 trajectories defines the lifetime of a complex. Internal state distributions of the Br, fragment are computed uia the standard histogram method by making assignments of the final classical action of the h, oscillator and angular momentum j to their nearest quasiclassical (half-integer and even) values.' Fermi Golden Rule Approximation In quantum formalism, the vibrational predissociation dynamics may be treated within the Fermi golden rule (FGR) approximation' which implies a splitting of the total Hamil- tonian into two terms, H=H,+V (111) where intra- and inter-molecular motions of a complex are decoupled in the unperturbed part H,, i.e.H, = h,, + hvdW, by picking up all coupling terms in the perturbation operator V. Hence, to the zeroth order, the metastable states of H cor-respond to the bound of H, , and their decay widths associated with process (1)are evalu- ated in first order: where Yu,j, is the continuum eigenstate of the total Hamilto- nian H with the eigenvalue exactly equal to Eungnb. However, we introduce an additional simplification by assuming sepa- rability of intra- and inter-molecular motions for the final wavefunction (or, in other words, we neglect the coupling between the different scattering channels u').Hence, Yu#j, is replaced in eqn. (V) by Y:rj(,the unperturbed continuum solu- tion for H, at the same energy. We therefore omit the null superscript at all wavefunction designations and, by dealing only with the ground vdW level n, = 0, nb = 0, we drop these indices too. The computational scheme implemented here for FGR integrals relies upon further approximations which resemble J. CHEM. SOC. FARADAY TRANS., 1994, VOL.90 where the radial function $(') depends parametrically on 8. For the initial discrete state it results in a series of 1D radial Schrodinger equations at fixed 8 values whose eigenvalues, n,, form the effective angular potential which determines F(") for the nb bending level. For the continuum states eqn.(VIII) is equivalent to the well known rotational infinite order sudden approximation,22 which substitutes angular momen- tum operators j2 and P by the effective expectation values iG+ 1) and (r+ l), respectively. In contrast to the inelastic scattering, the 'half-collision-' problem allows one to make a consistent choices of 3 and 1 parameters (treating the former as a true angular momentum quantum number rather than a measure of average rotational energy, i.e. for the case of the zero total angular momentum considered here, 1=j.Among them the most accurate is 1 =j'.,' The function F in eqn.(VIII) is simply the spherical function with angular momen- tum j' and zero body-fixed frame projection. Provided that only potential terms are responsible for the interaction between intra- and inter-molecular degrees of freedom, the expression for the partial width, eqn. (V), acquires the final form where E is the translational energy release which is equal to the vibrational energy loss of the diatomic fragment, E, -E,, minus the final rotational energy, BUtj'(j'+ l), and the initial vdW vibrational energy, EZw,defined by eqn. (VIII). The total predissociation width is evaluated by summing partial ones for all significant v' andj' channels. The above procedure which we term the vibrationally dia- batic decoupling rotational infinite order approximation (VDD/RIOSA) has been shown to provide a sufficiently accu- rate description of the direct (one-quantum) VP processes of the triatomic vdW complexes.2' However, it is necessary to indicate its most vulnerable place : vibrational decoupling in the continuum wavefunction which leads to the single-channel scattering picture disabling the account for asymp- totic state interactions.Strictly speaking, this approximation is valid only if the decay via the one-quantum, Au = u' -u = -1, vibrational channel strongly dominates over the consecutive transfer of several vibrational quanta. A second less important inaccuracy is connected with the fixed- angle approximation for the bound-state problem which introduces an error in the resonance position which, in turn, implicitly affects the resonance width.Potential-energy Surfaces The PES U(r,R, 0) consists of two terms describing intra- and inter-molecular interactions, the early approach of Delgado-Barrio and co-w~rkers,'~~~~ but formulated at a more quantitative level." Namely, intra- and inter-molecular variables are decoupled diabati~ally,~ so that the 'slow' intermolecular motion is described by the Hamiltonian averaged over 'fast ' vibrations of the diatom, aw = (XU(d I H I XUW> (VI) where xuis the eigenfunction of the hmoloperator correspond- ing to an isolated non-rotated diatomic fragment. The eigen- value problem for the 2D h3* operator e) = ~g~p)(~, (VII)h%w p(~, e) is eliminated by the diabatic decoupling approximation :19v20 +(u)(R,e) z $(u)(R; (VIII)e)F(V)(e) Potential curves of isolated Br, molecules constituting the intramolecular part of the global PES Umolare represented by the RKR turning points for both ground and excited states.23 Being suitable for quantum calculations, this choice is not optimal for classical dynamics.For this reason, in QCT calculations on the excited-state complex we use the Morse function with dissociation energy D, = 3788 cm-',inverse-length parameter a, = 2.045 A-', and the equilibrium dis- tance f = 2.667 A, which provides a good fit to the RKR potential for u = 10-30. Test quantum calculations with this Morse curve do not exhibit considerable deviation from those carried out with the RKR potential.J. CHEM. SOC. FARADAY TRANS., 1994, VOL. 90 The intermolecular potential, UvdW, is taken as a sum of pairwise Morse potentials UVdW(r,R, 0) = U,(9,) + U,(9,) (XI) U,(@) = D{exp[ -2a(9 -@)I-2 exp[ -a(B -@)I> (XI0 Here 9,and 9,denote the two Ne-Br distances, related to Jacobi coordinates by 9:,,= R2 + r2/4 & rR cos 0 (XIII) and the bar indicates an equilibrium separation. This simple form of UvdWwas found to provide quite a reasonable description of the VP dynamics of rare gas-halogen complex- es in many previous applications (see, e.g. ref. 17, 20 and 24). The separable Hamiltonian, Ho, appropriate for quasi- classical quantization consists of two Morse oscillators, h, and h,.Parameters of the former obviously coincide with those of the isolated Br, oscillator, whereas those of the latter are related to the constants of pairwise interaction by the fol- lowing formulae:25 DR = 20; aR = aR/@; R2 = B2-r2/4 The angular term h, is approximated by the harmonic oscil- lator Hamilt onian ' where Parameters of the interaction potential, eqn. (XII), are deter- mined using the Ho Hamiltonian which allows us to express observables Do and R,, the bond energy and vibrationally averaged intermolecular separation, via the pairwise potential parameters analytically. Assuming that the range parameter o! is 1.72 A-' (the value for the NeKr interaction26) we iter- atively adjust D and I? to reproduce Do = 70.5 cm-' and R, = 3.67 A determined experimentally for the ground state."*'2 The final set of parameters of the Ne-Br,(X) inter-action are given in Table 1.Three different PESs were involved in the study of the elec- tronically excited complex. The first, PES I, was obtained by a similar 'inversion' procedure, but neglecting the bending contribution he to the energy.I6 Since it provides a good description of the Ne. -.Br ,(B) fragmentation dynamics within the restricted 2D model,I6 this surface may be con- sidered to be a reasonable initial guess for 3D studies. The second, PES 11,is constructed on the basis of the 3D QCT simulations discussed below. Finally, the third set of param- eters PES I11 have been suggested by Walter and Stephenson for an Ne- * -IBr(A) complex.27 Actually, this potential pri- marily reflects the interactions in rare-gas dimers.Table 1 Parameters and constants of the Ne-Br potentials 323 1 -1 00 4.1 4.6 5.1 5.6 6.1 6.6 7.1 RIA loo! . *.. -1 00 InI I I1 I I I, I I I! I 14 r 1 I I1 111 I1 I I I I I 4 1 t I I1 I rnl I ' 11 I "~r'-' '"'1 " 2.7 3.2 3.7 4.2 4.7 5.2 5.7 RIA Fig. 1 Sections of Ne. .-Br, interaction potentials at (a)r = i,8 = 0 a)and (b) r = i,8 = 42. (. *. Ground-state PES; (---) PES I; (-- - -) PES II;(-) PES 111. Parameters of all these PESs are listed in Table 1. Also given are crude estimations of Do and R, constants derived for the Ho Hamiltonian. Fig. 1 shows the sections of inter- action potentials at r =? for linear and T-shaped configu- rations of the complex.Results Ground-state Complex The energy of the ground rovibrational state of the Ne. -.Br, complex is found to be -71.25 cm-' [relative to the Nee --Br,(X) dissociation limit] and the vibrationally aver-aged distance, R,, is 3.64 A. These figures agree with the experimental estimations described above (Do = 70.5 f2.0 cm-' and R, = 3.67 & 0.01 A) as should be expected since the latter have been used to parametrize the PES. An independent test of the interaction potential quality is provided by VDD/RIOSA calculations of the vibrational pre- dissociation lifetime for the first metastable state Ne. --Br,(X, ground state 46.0 1.72 3.7 92.0 3.52 70.2 3.68 38.52 1.72 3.58 77.04 3.32 56.84 3.50 43.0 1.75 3.84 86.0 3.60 64.39 3.77 42.0 1.67 3.9 84.0 3.66 63.63 3.83 a Estimated with the Ho Hamiltonian. u = 1).The theoretical result, 5.9 ps, falls within the error of the experimental data, 8 & 3 VS.'~ Hence, our ground-state interaction potential provides a good reference point for estimations of the band origins of the B tX transition spectrum of the complex. Excited State: QCT Results Let us first consider results from the QCT approach which we thought to be a useful guideline for the preliminary PES characterization. Table 2 compares computed predissociation widths with the low-resolution data from ref. 10 and 12. Obviously, the 3D classical decay rates for PES I strongly underestimate the experimental ones.Calculated vibrational product state distributions appear to be colder and narrower than those measured. For example, the branching ratios for Au = u'-u = -1, -2, -3 and -4 channels at u = 27 are estimated as 1 :0.32 :0.07 :0.007 in contrast to 0.7 : 1 :0.6 :0.3 deduced from dispersed fluorescence studies.I2 In agreement with experimental findings,12*13 the QCT method indicates the insignificant role of the rotational energy transfer in VP dynamics (energy transferred into the product's rotation amounts to less than 10% of the total available energy). To eliminate the shortcomings of the above results and to gain more insight into the applicability of the classical approach we perform QCT calculations for several trial inter- action potentials of the form given in eqn.(XII) with par_am- eters D, a and 9varying within 40-45 cm-', 1.65-1.80 A-' and 3.83-3.85 A ranges, respectively, which seems most prob- able in terms of the Hamiltonian function Ho. Among these potentials we choose one providing the closest agreement Table 2 Predissociative widths of the Nee ..Br,(B) complex (cm-I): QCT calculations and experimental values ~~~ ~ quasiclassical trajectories' experimental 0 PES I PES I1 valuesb 25 0.290 f0.001 - 1.09 f0.2 26 0.344 f0.002 - 2.38 f0.4 27 0.421 f0.002 0.65 & 0.02 3.12 f0.4 28 0.310 f0.001 0.647 k0.001 2.50 f0.4 29 0.537 f0.001 0.926 f0.002 2.88 f0.4 30 0.665 f 0.003 1.100 f0.003 1.88 f0.4 'Indicated errors are the standard deviations of the exponential fits and do not account for statistical uncertainty.Experimental values are taken from ref. 10; revised error bars are taken from ref. 12. J. CHEM. SOC. FARADAY TRANS., 1994, VOL. 90 with the measured decay width at u = 27 and repeat trajec- tory calculations on the Ne. .Br2(B) predissociation dynamics for u = 27-30. This 'improved' potential is exactly PES I1 as described above. Although the decay rates for PES I1 are twice as large as those for PES I, they still fall far below the experimental data, see Table 2. Markedly better agreement is, however, attained for the vibrational branching ratios which are 1 :0.98 :0.20 :0.04 for u = 27. Rotational product state distributions become also somewhat hotter than those for PES I.Excited State: Quantum Results Energies and Structural Parameters The positions of the lowest Nee .-Br,(B, u) resonances -Dt) computed by the VDD/RIOSA) method with all three PESs (Reported in Table 3) may be compared with the experimen- tal estimate Do= 61.2 & 2.0 cm-' (established from the vibrational energy distributions,I2 this value should be referred to u = 27, 28). PES I underestimates the dissociation energy, whereas PESs I1 and I11 give values within the experi- mental error limits, but closer to the upper bound. Note that Do values derived from the oversimplified Hamiltonian Ho (Table 1) reproduce the results of the more rigorous calcu- lations (Table 3) with a maximum deviation of <2 cm-'.The analysis of the B tX transition frequency shifts in the complex relative to those in the free halogen molecule is more informative, since it is the only directly measurable character- istic of the resonance positions. With the dissociation energy of the ground-state complex D,(X) = -71.25 cm-' (see above), we may easily estimate the frequency shift as (XVII) Calculated and measured lo*'' frequency shifts are presented in Table 3 and Fig. 2. The results for PES I1 are in excellent agreement with the band origins of low-resolution excitation spectrum over the whole studied range of u, whereas PES I11 reproduces them within a 1 cm-error. In contrast, the shifts obtained with PES I are nearly twice as large as the mea- sured ones. This discrepancy is much larger than the uncer- tainty of the experimental estimations for the dissociation energies and, therefore, cannot be attributed to the inaccurate determination of ground-state PES.The calculations with PESs I, I1 and I11 yield for the vibra- tionally averaged intermolecular separation, R, (at u = 10) values of 3.43, 3.70 and 3.77 A, respectively, whereas the Table 3 Energies of the lowest metastable levels, -Dt)/cm-', of the Ne. .Br,(B) complex and shifts of B cX transition frequencies, Am/cm-':quantum calculations and experimental values 10 -57.49 13.76 -65.04 6.21 -64.23 7.02 6.045' 14 -57.23 14.02 -64.77 6.48 -63.99 7.26 6.59 16 -57.06 14.19 -64.60 6.65 -63.84 7.41 6.66 17 -56.98 14.27 -64.51 6.74 -63.76 7.49 6.82 20 -56.66 14.59 -64.19 7.06 -63.48 7.77 - 22 -56.41 14.84 -63.93 7.32 -63.25 7.99 7.50 25 -55.94 15.31 -63.45 7.80 -62.83 8.41 7.38 26 --55.76 15.49 -63.26 7.99 -62.66 8.58 8.01 27 -55.55 15.70 -63.05 8.20 -62.48 8.77 8.80 -28 -55.32 15.93 -62.81 8.44 -62.28 8.97 9.10 29 35 --55.07 -54.80 16.10 16.45 -- -- -- - 9.59 9.62 J.CHEM. SOC. FARADAY TRANS., 1994, VOL. 90 0. c 9.0-I 3.0U 4 1TE 71 I TJ:< 8.0i 1'Ilv) * r r >. s 7.0: c 2.0 .-a Gi!: .i 1 x6.0i 5.05 1 8 12 16 20 24 28 32 V :I -\, I,,Fig. 2 Frequency shift of the B tX transition in the Ne. * .Br, complex relative to free Br, molecules. (a * -) (-) 8 10 12 14 16 18 20 22 24 26 28 30 32 calculated with Nee ..Br,(B) PES 11; (---) calculated with PES 111. V Fig. 3 Predissociative widths of the Nee .Br,(B) complex. (0)and ( x ) Data of high-I2 and low"-resolution experiments, respectively; analysis of the excitation spectrum lineshapes estimates Ro as calculated with PES I(----); I1 (---); I11 (-)3.65 f0.01 A." The PES I result is again the worst. Predissociat ion Widths mates the decay rates by two. None of the PESs, however, are Table 4 and Fig. 3 summarize the VDD/RIOSA results for able to reproduce correctly the critical excitation at which the predissociation widths of the Ne. * SBr complex for three predissociation width falls off: PES I shifts it to u= 30, excited-state PESs. They are compared with two sets of whereas for PES I1 the Au = -1 channel appears to be com-experimental data available from high-resolution excitation pletely closed at u= 27, but the two lowest rotational states, spectroscopy for u= 10-20,12 and from low-resolution j' = 0, 2, belonging to this channel are energetically accessible measurements" (with improved error bars presented in ref.at u= 26. PES I11 again provides the most acceptable12) for u= 10-30. Only the former provides a good reference picture, placing the complete locking of the one-quantum point for elucidation of the PES quality because the reliablity decay channel at the correct value of u= 28. However, the of the latter seems questionable. Indeed, at low excitation maximum of the decay rate is shifted to u= 26 owing to its energies (u< 20) low-resolution data not only quantitatively partial locking associated with thej' > 10 asymptotic states.deviate from high-resolution linewidth estimations, but also The decrease in the fragmentation rate above the critical exhibit qualitatively incorrect behaviour. Consequently, low- excitation energy is too sharp for all potentials. This reflects resolution results should be adopted with caution at high uas the inaccuracy of the computational scheme, described above, well. However, the predissociation rate fall-off at the critical which completely decouples scattering channels, disabling excitation u= 28 indeed occurs, being caused by the locking population transfer among them.Being unreliable for the two of the dominant decay channel Au = -1, as is confirmed by quantum decay processes above, this approximation should measurements on the vibrational product state distribu-also disturb the energy dependence of the predissociation tions.12 The nature of the second observed rate maximum at width just below it. u= 29 seems unclear although the analogous effect arising from resonant interactions among open and closed asymp- Product State Distributions totic channels has been theoretically predicted by Roncero et Suppressing the asymptotic state interaction strongly affects al. for the Ne- * -12(B) vdW complex.28 the vibrational energy distributions of the product molecule. The agreement with high-resolution data is best for PES As a consequence, they appear to be very cold for all PESs, 111.PES I yields slightly worse results and PES I1 overesti-accumulating 97-100% of the population in the dominant Table 4 Predissociative widths of the Nee * .Br,(B) complex (cm-I); quantum calculations and experimental values theory experiment 0 PES I PES I1 PES I11 low resolution" high resolutionb 10 0.019 0.028 0.017 -0.015 f0.005 14 0.058 0.080 0.052 0.16 0.051 f0.005 16 0.101 0.138 0.089 0.09 0.057 & 0.006 17 0.130 0.178 0.1 16 0.06 0.081 f0.008 20 0.285 0.368 0.253 -0.151 & 0.015 -22 0.453 0.570 0.403 0.75 f0.2 -25 0.948 1.073 0.755 1.09 f0.2 -26 1.180 1.079 0.994 2.38 k0.4 -27 1.397 0.073 0.780 3.12 f0.4 28 1.491 0.057 0.035 2.50 +_ 0.4 --29 1.408 --2.88 f 0.4 30 0.105 -1.88 f 0.4 From ref.10. From ref. 12. 3234 Table 5 Rotational temperatures of the Br,(B, u -1) predissocia-tion products of the Ne-.-Br,(B, u) complex, K: quantum calcu- lations and experimental values theory U PES I PES I1 PES I11 experiment' 10 4.1 4.4 4.4 6b 14 3.9 4.2 4.7 -17 3.6 3.9 4.0 -20 3.4 3.5 3.5 -22 3.1 3.2 3.3 5 25 2.9 2.2 1.9 -26 2.5 1.1 1.9 -27 2.0 0.2 0.5 <1 C C C28 1.1 -C C C-29 0.5 --From ref. 12. The value of 6.9 K is given in ref. 11. The u' = u -1 channel is closed. exit channel. Only the qualitative trend in the product vibra- tional temperature T,,(PES I) < T,,(PES 11)x T,,(PES 111) is therefore of value. In contrast to the vibrational decoupling, the rotational infinite order sudden approximation does not influence rota- tional product state distributions so dramatically, although it somewhat overcools them.,' All rotational distributions peak at j' = 2 and cannot be fitted by a Boltzmann distribution.To estimate the Br,(B, u') rotational temperature in the domi- nant vibrational decay channel v', the simple relation (Erot) x ikTotmay be used, where (Erot) denotes the first moment of rotational distribution with the Br,(B) rotational constant, B,, and k is the Boltzmann constant. Computed and meas~red'~.'~ T,,values are presented in Table 5. One can again see the superiority of PES 111. Discussion Excited-state Interaction Potential The possibility of deriving a definite conclusion on the reli- ability of the empirical potential surface depends on the quality of the experimental data and on the accuracy of their theoretical evaluation.Uncertainties in both quantities are pertinent to the present study of the Nee -.Br, complex so we find it instructive to give here a brief account of the main issues of comparative analysis of the B-state Ne-Br, inter-action potential. In principle, the most stringent test for the attractive well region of the PES is provided by energy-level calculations. However, for vibronic transitions it implies that one of the relevant PESs is known with definite accuracy. So far, our guess for the ground-state Ne- -Br, potential satisfactorily fits all the available spectroscopic information, it seems rea- sonable to estimate its uncertainty to be that of the experi- mental data used for parametrization.Taking the corre-sponding error as 2-3 cm-l, one can discard PES I but cannot distinguish PESs I1 and I11 on the basis of the line- shift analysis (Table 3). If the VDD/RIOSA approach is able to find resonance positions with an accuracy of at least several times smaller than the experimental uncertainty irrespective of the excita- tion energy, the situation with regard to the predissociation width is more delicate. With increasing u, the decay channel interference effect, which increases the contribution of the multiquantum channels, prevents our method from yielding a uniformly accurate energy dependence of the VP rate.At the same time, as discussed above, the measured lifetimes also have different error limits. Hopefully, our approach appears J. CHEM. SOC. FARADAY TRANS., 1994, VOL. 90 to perform sufficiently well just in the excitation energy domain, where the high-resolution experiments have been carried out (v = 10-20), since the Au = -1 decay channel correctly treated by the VDD-RIOSA method, strongly dominates therein. Analysis of the linewidths calculated for this range unambiguously shows that PES I11 is superior over the two others (see above). To extend the comparison for higher excitations it is instructive to use the well known energy gap law relating the predissociation width to the vibrational quantum number of the diatomic fragment (see, for example, ref.9): (XVIII) where EA, = Eu+Au -E, is the vibrational energy release of the diatom and A and B are adjustable parameters. Being derived by Beswick and J~rtner,~ within a 2D FGR model, this relationship also neglects final-state interactions. There- fore, the extrapolation of high-resolution data for T, (in which the dominant Av = -1 decay channel amounts to at least 90% of the population'2) to higher u allows one to pick up the single-channel contribution to the decay rate. Fig. 4 compares the energy gap law extrapolation with the partial widths for the Av = -1 channel. Evidently, PES I11 behaves better than the others at higher excitation energies too. This qualitative analysis provokes us to estimate the total 'high-resolution linewidths' for high energy taking the results of the energy gap law extrapolation as a Av = -1 partial width and using experimentally measured vibrational product-state distributions.12 The total decay widths for v = 22 and v = 26 so obtained are 0.4 and 2.0 cm-', respec-tively, values remarkably smaller than the low-resolution ones, 0.75 and 3.12 cm-'.lo Being rather approximate, these speculations nevertheless may be treated as pointing to an overestimation of actual Ne..Br,(B) fragmentation rates in low-resolution experiments. The same conclusion follows from common consideration of the deconvolution of a line profile, whose width is much smaller than the bandwidth of a pumping laser." As concerns the critical quantum number u* at which the Av = -1 channel closes, its value may be seriously affected by the inaccuracy of the computational scheme, which cannot predict the resonance positions exactly.Even very small errors in this quantity, which may be neglected in the lineshift calculations, may result in the open or closing of several rota- V Fig. 4 Predissociative widths of the Ne. -.Br,(B) complex. (0)Data of high-resolution experiments;I2 ( x ) their extrapolation by the energy gap law, eqn. (XVIII); calculated with PES I (-- - -) ;I1 (---); I11 (-). J. CHEM. SOC. FARADAY TRANS,, 1994, VOL. 90 tional exit channels,j’, owing to the small Br, rotational con- stant and, therefore, significantly disturb the behaviour of the predissociation rate near the critical energy.However, on the whole, PES I11 reproduces the critical behaviour better than the other potentials. To quantify these qualitative considerations, more rigorous theoretical approaches, which account for the close coupling between the vibrational decay channels, should be imple- mented (see, for example, ref. 5, 6 and 24). In particular, they should reliably treat the product state distributions and eluci- date the nature of the second predissociation rate maximum observed in low-resolution measurements. Classical and Quantum Dynamics Correspondence A somewhat unexpected conjecture of the present study is the failure of the quasiclassical trajectory method to reproduce the decay rates of Nee * .Br,(B) complexes, which contrasts with experience from classical dynamics calculations on the Ne.-C12(B),17 He. -.Br,(B),30 He. -.12(B)31932 and Nee -.12(B)33complexes. The most reasonable explanation for this is the inaccuracy of the selection procedure for the initial conditions which may overestimate the bending contribution to the initial vdW zero-point vibrational energy and, there- fore, lead to an underestimation of the fragmentation rate. At the same time, note that QCT calculations represent the vibrational product state distributions reasonably well, working much better than the decoupled FGR approx-imation. Besides, classical dynamics exhibit the correct sensi- tivity to the parameters of the PES: the differences in predissociation widths and product distributions computed classically for PESs I and I1 follow the same trends as those derived from quantum calculations.Another remarkable feature of the QCT data is the irregu- lar dependence of classical VP rate on the excitation energy. Interestingly, the fall-off in the classical rate occurs at the same energy as the actual retardation observed experimen- tally (Table 2). This coincidence, also found for I,(B) com-plexes with helium3’ and neon,33 seems surprising because the real irregularity originates from a purely quantum effect. Indeed, there are strong indications for its accidental nature, e.g. the maxima of QCT rates are the same for both PESs being sufficiently far from the critical excitations established for PESs I and I1 in quantum calculations. Actually, the clas- sical irregularity should be attributed to the onset of strong non-linear resonances, a reason entirely distinct from the exit-channel energy balance controlling the quantum critical behaviour.Comparison of Two-and Three-dimensional Dynamics The data obtained in the present 3D calculations for PES I may be compared with those derived within a restricted T-shaped model for the same potential and very similar theo- retical approaches.I6 Fig. 5 depicts the ratio of 2D and 3D predissociation widths as a function of excitation energy for QCT and FGR methods. These ratios are essentially constant over the whole range (except for the points u = 28, 29 near the critical excitations which are, of course, different in 2D and 3D cases).This finding is consistent with the observation of the negligible effect of rotational energy transfer in VP dynamics and implies that the variation in the predissocia- tion rates originates almost completely from the energy shift by the zero-point bending contribution. Indeed, appealing to the decay width-squared shift ~orrelation,’~ one can find that the average ratio of two- and three-dimensional widths for the quantum approach (1.19) is very close to the average squared ratio of the corresponding dissociation energies Do (1.23). 4.0I -0 3.0; 25o^ 2.0:z?. L I .-**---.-------Cb 0.0 8 12 16 20 24 28 32 V Fig. 5 Ratios of Ne. . .Br,(B) predissociative widths calculated for PES I within and three-dimensional models of the complex by (0)FGR and (0)QCT approaches Conclusions Implementing an approximate quantum-mechanical approach, we performed detailed tests on three model PESs describing vibrational predissociation of the Ne- .Br ,(B) vdW complexes.An empirical potential constructed within a 2D model (PES I) was discarded on the basis of B +-X tran-sition frequency shift calculations, whereas the potential func- tion originating from the adjustment of quasiclassical trajectory data (PES 11) yielded too high fragmentation rates. The most acceptable agreement for all available spectro- scopic and dynamical information for the Ne. * .Br,(B) complex was attained with the Walter-Stephenson PES I11 deduced from interactions in rare-gas dimers.A reasonable approximation for the ground-state Ne-Br,(X) PES was also suggested. 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