Faraday Discuss. Chem. Soc., 1982, 73, 257-273 Atom-Molecule Interactions from Muhiproperty Analysis An Integrated Study of the Dynamics for Oxygen-Rare-gas Systems BY FRANCO BATTAGLIA AND FRANCU A. GIANTURCO Gruppo di Chimica Teorica, Nuovo Edificio Chirnico, Universitii di Roma, 00100 Rorna, Italy AND PIERGIORGIO CASAVECCHIA, FERNANDO PIRANI AND FRANCO VECCHIOCATTIVI Dipartimento di Chimica, Universitg di Perugia, 06100 Perugia, Italy Receked 30th Nouember, 198 1 The interaction potential between the ground electronic state of the oxygen molecule and the helium atom is obtained through a numerical fit of data from collision experiments and is given via an effective, spherically symmetric analytic form. The latter expression, together with similar ones obtained previously for the 0,-Ar and 02-02 systems, is then used within a quantum-mechanical model to treat vibrationally inelastic processes and to generate various partial integral cross-sections at several energies.A comparison with experimental relaxation data is then carried out for the three cases examined; the results show generally good agreement for the He-02 and Ar-02 systems, suggest a greater influence of rotational inelasticity for the Or-02 case and allow us to assess the importance of Van der Waals complex formation on the relaxation mechanism. I . INTRODUCTION Collisional effects have constantly plagued the experimental studies of intramole- cular dynamics in “ isolated ” molecules. It is important to understand in these processes the interplay between intramolecular dynamics and intermolecular perturb- ations induced by collisions.In this context one should attempt to extract micro- scopic information regarding the selective effects of collisions, ultimately exploring monoenergetic collisions which affect intramolecular dynamics for either large molecules or weakly bound Van der Waals complexes. Significant progress in this direction has recently been accomplished, for example by exploiting some features of supersonic beam expansions which make it possible to study energy-resolved collisional effects in the proximity of the nozzle sources.1*2 Another useful classification of the basic molecular processes separates those processes occurring on a single electronic potential energy surface (PES) from those which involve at least two distinct electronic configurations.In the former case one focuses attention on intrastate dynamics, which deal with some forms of rotational and/or vibrational predissociation as well as the intramolecular vibrational energy re- distribution that takes place between bound vibrational levels of a large molecule or of a weakly bound adduct, as in the examples discussed below. The basic molecular processes provide a general framework for the elucidation of the nature of a variety of interesting chemical phenomena. It is well known that complex chemical events in large molecuIes can be described in terms of several258 02-R A R E-G A S COL L l SI ON A L 1 NT ER A CT I ON S parallel and/or sequential basic molecular relaxation processes. The effects of medium perturbations, for instance, on intermolecular and intramolecular dynamics are of importance in understanding the role of collision-induced processes in excited- state reactivity in the condensed phase.Moreover, collisional processes can be explored on the microscopic level by considering the role of Van der Waals bonding on vibrational prediss~ciation,~ which results in vibrational relaxation for one or for all of the weakly bound partner^.^ The induction of collision-controlled electronic relaxation can also be investigated by a similar approach. In particular, of con- siderable current interest are the intrastate predissociation processes that involve rotational and vibrational de-excitation of weakly bound complexes. Here the presence of an attractive well plays a key role, as shown by the simplest example of an atom and a diatomic molecule at an energy not sufficient to excite the vibration of the molecule.As the partners approach each other the attraction increases the available kinetic energy, with the result that the diatomic molecule can be excited while the atom is now bound by the attractive force. ’ The resulting species is only quasibound since it can predissociate by transferring the “ local ” vibrational excitation of the diatomic molecule back to the translation, thus heavily affecting V-T processes.6 This mechanism is quite general, since even in the absence of chemical attraction the long-range Van der Waals forces support bound states, and hence give rise to such “ local ” quasibound states as those described.In a predissociation via V-T or R-T processes, the energy required to promote the translation to a dissociative state is provided by the de-excitation of either the vibration or the rotation.’ Another possibility is provided by the inverse rotational predissociation (IRP), a process involving collisions between constituents with given angular momentum hJ and specific energy ca. En, which give rise to the emission of an infrared photon to a lower bound state of the complex. It can be envisioned as the decay from a trans- lational continuum via an intermediate state to a radiative continuum, of consider- able importance in a~trophysics.~ For IRP involving atom-diatom collisions, with the diatom being in a vibrationally excited state, the decay channel can involve non- radiative vibrational predissociation.An example is provided by the recent sug- gestion that the low-temperature vibrational relaxation of 1, by He occurs via an orbiting resonance of the complex HeI,(B 311, v) decaying into the dissociation channel He + I,@ 311, v - 1). In the present work we investigate all the above possibilities for collisional pro- cesses involving O2 molecules and rare gases (He, Ar) wherein the full interaction is obtained via a quantitative analysis of several experimental findings; moreover, the structure of the corresponding Van der Waals complexes is examined in order to provide a basic understanding of the low-temperature behaviour of the vibrational relaxation in the gaseous mixture. The integrated use of experiments and theoretical models will thus give us, in the examples discussed below, a novel method for the interpretation and prediction of the collisional behaviour of such Van der Waals complexes.Since some of the systems have already been discussed in the available literat~re,~.~ in section 2 only new results regarding the He-0, system are presented in detail and in section 3 the theoretical model that allows us to obtain vibrationally inelastic cross- sections under various collisional conditions is discussed. In section 4 we offer a comparison between the specific case examined here and also draw conclusions as to the present model’s capabilities in treating vibrational relaxation of Van der Waals complexes.F . BATTAGLIA et al. 259 2. POTENTIALS FROM MULTIPROPERTY ANALYSIS The potential functions used here for the Ar-02, 0,-O2 and He-0, systems are obtained by a combined analysis of different properties: for the Ar-0, and O,-O, systems the integral scattering cross-sections, the second virial coefficients and some spectroscopic features were analysed; for the He-0, system, the integral and total differential cross-sections together with the second virial coefficients were analysed.Note that all the experimental data are sensitive mainly to the spherical average of the potential: therefore they are able to provide the first term of the usual Legendre expansion of the PES for an atom-diatom system: co W , r , y ) = c VI(R,r)Pi.(cosy) (1) 1 = 0 where y is the orientation angle, that is the angle between the diatom axis, Y, and the position vector of the atom with respect to the centre-of-mass of the molecule, R.For the analysis of these three systems a multiparameter potential model, with enough flexibility to reproduce efficiently the complete set of analysed data, was chosen. The Exponential-Spline-Morse-Spline-Van der Waals (ESMSV) model was used: Vo(R) = &f(x); x = R/R, (2) f(x) = A exp[--(x - l)] x< x1 x2 < x < x3 x3 < x < x4 x4 < x = exp[al + (x - xl)(a2 + (x - x2)[a3 + (x - xl)a41>l x1 < < x2 = exp[-2/3(x - l)] - 2 exp[-p(x - l)] = b1 - (X - X3)(b2 + (X - X4)[b3 4- (X - X3)b4]} (3) -cbx-6 - c8x-8 - clOx'o - c6 = C6/&R:; c8 = Cs/&RA; Cl0 = ClO/&RI,O. While details of the analysis for the Ar-0, and 0,-0, systems have been reported and discussed The best-fit potential parameters for the spherical component of the interaction in the three systems are listed in table 1.Relative integral cross-sections for He-0, collisions were measured by Butz et a1.l' in the range of collision velocities between 0.2 and 2.6 km s-'. They are reported in fig. 1 as a function of LAB velocity. These results show a glory maximum located at ca. 0.6 km s-', and they provide experimental evidence for the existence of at least one bound state in the He-0, Van der Waals molecule, as will be discussed below. Luz- zatti ut al." also performed measurements of the absolute value of the integral cross- sections in the 0.8-2.3 km s-' velocity range. These cross-sections are also shown in fig. 1 as a function of the centre-of-mass (c.m.) velocity, and are the results of two independent experiments: one for the scattering of a helium beam by an oxygen target and the other for the scattering of an oxygen beam by a helium target.The very good agreement between these data and those of the previous investigation'O (see fig. 1) has allowed us to place the integral cross-section for He-0, over a wide range of collision velocities on an absolute scale. Recently Keil et a1.12 measured the total differential cross-section for the same system at a relative collision velocity of 0.56 km s-l and managed to resolve some of the diffraction oscillations. It is important to remember that all these scattering data are mainly sensitive to the low- energy repulsive branch of the potential curve, with the exception of the integral cross-section in the glory region which is sensitive to some characteristics of the potential surface near the equilibrium distance.13 The second virial coefficient for the He-0, gaseous mixture, BHe-02(T), is available the analysis for the He-0, system is described here.260 O2-RA R E - G A S CO L L I SI ONA L INTERACT I ON S TABLE 1 .-SPHERICALLY AVERAGED, EFFECTIVE POTENTIAL PARAMETERS (IN ATOMIC UNITS) FOR THE SYSTEMS EXAMINED.FOR THE MEANING OF THE PARAMETERS, SEE TEXT AND EQN (2) AND (3). parameter 0 2 - 0 2 Ar-02 He-02 4.19( - 4) 7.45 6.30 0.8490 12.53 7.81( +1) 1.56(+3) 4.06( $4) 0.63 0.75 1.05 1.48 Spline coefficients: a1 4.4724 a2 - 15.4739 a3 -24.5323 a4 94.9043 bl - 0.9270 b3 - 1.4387 64 - 3.8364 b2 1.8660 4.23(-4) 7.03 6.45 0.6246 13.69 7.50( + 1) 1.61(+3) 4.48( $4) 0.72 0.86 1.10 1.50 3.3626 -23.0071 - 66.5509 -418.3314 -0.7741 1.5351 5.2294 - 4.205 1 9.26( - 5) 6.50 6.00 1.3208 12.46 1.10( + 1) 3.55( +3) 1.73( +2) 0.61 0.85 1.10 1.45 5.1388 - 20.901 0 - 35.1579 - 152.2673 -0.7964 1.6878 4.1422 - 3.6674 only in the 90-80 K temperature range.14v15 However, it is possible to obtain BHe-02 ( T ) over a more extended range of temperature by using the following rela- tionship : B H e - 0 2 ( T ) = 3 [ B H e - H e ( T ) + B 0 2 - 0 2 ( T ) 1 - E H e - O z ( T ) and assuming for the values measured by Brewer and Vaughn l6 for He-Ar gaseous mixtures.The coefficients for pure He and 0, are taken from Levelt- Sengers et al.” This approximation, which also provides good agreement with the values that were directly measured for the He-0, ~ y s t e m , ~ ~ ” ~ is based on the fact that 0, and Ar show very similar gaseous properties, e.g.the second virial coefficients for Ar and 0, are practically equal over a wide temperature range.” It therefore becomes possible to have the required second virial coefficients available up to a temperature of 500 K. All these data were simultaneously analysed in order to obtain the potential para- meters for the spherical component of the interaction reported in table 1. Since all these data are minimally sensitive to the short- and long-range interaction, the A , a , C,, C, and C,, were kept at fixed values during the best-fit procedure. For the repulsive walls the values assumed were obtained following Smith’s work and taking the 02-02 short-range repulsion as proposed by Cubley and Mason.19 The c6 parameter was chosen at the value calculated by Starkshall and Gordon,” while c, and Clo were obtained assuming for the c,/c6 and c,oC6/c,’ ratios the values of 4.43 A2 and I .3, respectively, which are the theoretical values for the He-Ar system.The differential [I(O)sinO] and integral [ Q(u)] cross-sections were calculated within the JWKB approximation. For the fitting to the Butz et al. and Keil et al. data, theF . BATTAGLIA et al. 26 1 500 LOO 300 5 ; 200 8 0 .- Y I m c c! on + ._ 100 dplL- 1 2 3 collision velocity/km s-l FIG. 1 .---Integral cross-sections as function of collision velocity for the He-02 system. The circles of curve (a) report the Gottingen data lo as a function of LAB velocity and on a relative scale, fitted by the present potential (continuous line).The data of curve (b) refer to absolute measurements as a function of velocity and are also fitted via the present potential. centre-of-mass cross-sections were convoluted to the laboratory system taking into account the experimental conditions reported in the relevant papers. The best-fit Q(u) are reported as a solid line in fig. 1. In fig. 2, Z(0)sinO in the c.m. system cal- culated with the present potential (solid line) is compared with that calculated with the best-fit spherical MSV potential by Keil et al.', (dashed line). The slight dis- crepancy as to the maximum of the oscillations is actually an improvement of the fit quality. The B(T) coefficients were calculated for a spherical potential following Hirschfelder et al.,, with two quantum corrections. Finally, in fig.3 the derived spherical potentials for He-0,, Ar-0, and 0,-0, are plotted together around their well regions. The long-range tails for Ar-0, and 0,-0, are very similar, while in the short-range region the repulsions lead to a smaller R, for Ar-0,. The well depths for Ar-02 and 01-02 are almost the same. As expected, the He-0, system shows weaker interaction. The energy levels of the bound states for these systems were calculated using the WKB approximation. The Ar-0, system shows 7 vibrational levels with the rota- tional quantum number K = 0, while 0,-0, shows 6 vibrational levels for K = 0. The total number of the rotational levels is limited because of rotational predissoci- ation: 23 in the Ar-0, system, the maximum value of K allowed for the ground vibra- tional level, u = 0, is 44, which decreases when going to higher vibrational levels.One eventually gets a maximum K value of 9 for u = 6. Among all these roto- vibrational states, 41 levels are above the dissociation limit. For the 0,-02 system, if the electronic, nuclear and spin symmetries are neglected, the situation is very similar to that of Ar-0,, as is expected from the similarity of the potentials, while for262 O,--RARE-CAS COLLISIONAL INTERACTIONS I I 1 I .. .. the He-U, system only one vibrational state is possible. The existence of this state has been shown experimentally by the detection of a glory maximum in the velocity dependence of the integral cross-section measured by Butz et The maximum value of K allowed for He-0, is 6 and only K = 5 and K = 6 are above the dissocia- tion limit.While for the He-02 and 0,-02 systems all the available experimental data sample only the spherical component of the interaction, for Ar-0, there exist some scattering 24 and spectroscopic 25 measurements which are also rather sensitive to the anisotropic I_,-..L-I 3.5 10 4 5 RIA FIG. 3.-Effective potentials for all the systems studied here: Ar-02 (-1, He-02 (- - -), 02-02 (- * - -). Only the radial region arouna the Van der Waals wells is shown.F . BATTAGLIA id. 263 component of the interaction. This allowed us to obtain an approximate V2(R) component of the potential energy surface, as discussed el~ewhere.~’~ For the He-0, system, Keil et d .1 2 found a small discrepancy in the amplitude of the dif- fraction oscillations in the differential cross-section, between the experimental data and one of their best-fit curves. They attribute this discrepancy to the influence of the potential anisotropy. In our opinion this discrepancy is only an artefact of the potential model they assumed, since it is seen to have been reduced by our present potential. A better chance of obtaining an accurate angular dependence of the interaction in the He-02 system can be found by using the recent experimental re- sults of Faubel et aLZ6 on the rotationally inelastic cross-sections at thermal energy. A further study, which fits these highly accurate partial cross-sections and generates a fully anisotropic PES which includes the presentIy discussed effective spherical potential for He-02., has recently been completed 27 and confirms the good accuracy attained by the present fit of several sets of experimental data.3 . VTBKATIONAL INELASTlCITY AND COLLISIONAL RELAXATION Once such knowledge of the interaction potential for the systems under study here is obtained, it becomes interesting to assess their reliability by theoretical and com- putational means. In the case under discussion, one possibility is offered by the experimental data that are available on the vibrational relaxation, over extended ranges of temperature, of 0, moIecules 28 and of their mixtures with rare gasesat9 We therefore analyse the use of the above effective potential forms within a quantum- mechanical treatment of the collision problem, as well as the succeeding calculations of their corresponding relaxation times.Energy transfer processes for the case of a structureless particle colliding with a target with internal structure may readily be classified. This leads to translational- internal energy transfer (T-I), where I stands for rotational (R), vibration (V) and electronic (E) motions, or for a combination of these. Insofar as one restricts the study to thermal (E,,,, 5 0.1 eV) and hyperthermal (E,,,, 5 10 eV) collision energies, the Born-Oppenheimer separation of electronic and nuclear motions is an acceptable initial approximation. Adiabatic interaction potentials can therefore be used to provide the required dynamical coupling between the states which are employed to expand the total wavefunction : where, in an uncoupled representation, the xI.are diatomic vibrational functions, the mj are rotational functions, and the uJ are unknown translational functions describing the relative motion of the partners. If the index x ~ { j , u, J ) , then any initial condition xin is defined by a set of the above quantum numbers, and the relevant potential function causes transitions between target internal states and consequent energy transfer within the usual conservation energy requirements for the isolated system. I n the case of Van der Waals molecules, the uJ also describe locdly a set of bound states and pseudobound states of the weakly bound complex; hence J contains the required quantum numbers for their classification.Unfortunately, it is still not possible to carry out a rigorous calculation within the above quantum-mechanical scheme for those excitation processes which involve rotational and vibrationaI states of the target. The reason for this is that the com- puting time increases as ca. N6, N being the number of rotational states included in eqii (4) where all the energetically accessible states should be added. Even if we264 0,-RARE-GAS COLLISIONAL INTERACTIONS neglect the vibrational states and include only the rotationalIy open channels in a coupled representation, in order to go from low collision energies (E,,,, x 1.0 eV) to hyperthermal values (Ecoll w 2-3 eV) one must increase the computing time by a factor > 102.30 Moreover, to know the correct dependence of the computed PES on the internal coordinate Y as a function of the collision coordinate R, is by no means a simple problem even in the adiabatic approximation, and therefore only for a very few systems has such a systematic study been carried out in detaiL3’ Computing through the known PES the relevant matrix elements for vibrational coupling adds, in any event, more equations to the purely rotational problem and therefore makes matters even worse as far as computing time goes.Therefore, for all the above reasons, one must still rely on simpler theoretical models fur the prediction of vibrational excitation cross-sections, and extract those modeIs from the large body of experimental information already available from beam e~perirnents.~~ The simplest and one of the most successful models is based on a surprisingly obvious prediction for the quantum-mechanical transition probabilities of a forced harmonic o ~ c i l l a t o r , ~ ~ * ~ ~ where one starts by averaging the potential over all angles in order to simplify the scattering problem: where x = Y - reqJ and Y,, is the equilibrium distance of the oscillator.One could then numerically calculate the classical trajectory for elastic scattering (i.e. neglecting energy losses during encounters) using V(R,x = 0), hence yielding R(t). Substituting R ( t ) into eqn ( 5 ) generates Veff(t,x), which can then be expanded in a Taylor series: By neglecting all higher-order terms in the above series and including only the first- order terms, the dynamicaI equation of motion reduces to : ,iiX + kx = F,(t) (7) where One would have then obtained a prescription for the coupling between relative motion and molecular oscilIations.As pointed out by Carruthers and N i e t ~ , ~ ~ the most probable energy transferred to the oscillator also coincides with the total energy being transferred, which is given by : where Po + is the excitation probability from state lo} to state Ik} and Aw is the oscil- lator frequency. Thus a close correspondence exists between classical and quantum- mechanical results provided that second-order terms can be neglected, as appears to be the case in many situation^.^^ In a time-independent quantum-mechanical picture the driving force plays its role through the usual coupling matrix elements between target states: V”,*Oo = (4 I/,,f(RJ)I~? (9)F .BATTAGLIA et al. 265 where the integration is over the internal coordinate, x. Since one is now dealing with an averaged potential, the rotational coupling is disregarded for the time being and the expansion (4) contains only vibrational functions to represent target states. The total angular momentum is now a good quantum number for the whole system, hence the unknown radial coefficients can be written as: %, Z(RPl(R4 which gives rise to the familiar set of coupled radial equations: Each coefficient is obviously subject to the usual boundary conditions that ensure flux conservation and provide the correct free-wave behaviour beyond a radial value R = Rfin where the potential Veff has essentially vanished. In the present case the latter matching was performed beyond R z 15.0 a.u.and the relevant numerical procedure employed was one which has been used by us p r e v i o u ~ l y . ~ ~ * ~ ~ It is clear from the above model that the dynamical influence on vibrational energy redistribution during collisions is controlled by the size and radial dependence of the matrix elements on the r.h.s. of eqn (10). In order to extract these from the effective potential forms generated in section 2, let us rewrite the truncated series of eqn (6) as follows: Veff(R,O)ex~ [aeff~l where the averaging of eqn ( 5 ) has been performed without following it with a classical choice for R(t), i.e. the linearized forced oscillator is here represented via an exponential coupling term in the full effective potential.The corresponding stretching parameter, cceff(R), is the logarithmic derivative taken around req; now the relevant matrix ele- ments of eqn (9), once the target functions are chosen to be either harmonic oscillators or Morse oscillators, are rapidly evaluated over many eigen~tates.~~ Previous comparisons with other atom-diatom systems 40 indeed showed that the present model employed for each II component in eqn (1) constitutes a good approxi- mation to the true form of that part of the PES which presides over the coupling between vibrational states, at least for transitions involving the lower-lying eigenstates. Moreover, the construction of an effective potential for oxygen molecules interacting with rare gases seems to indicate that a major role is played by the spherical part of the PES in low-energy collision^,^^^^^^ hence suggesting that the possible knowledge of aeff(R) could already provide useful indications on the vibrational low-energy behaviour for Ar and 0, as projectiles, together with the generally “ softer ” nature of the He-0, interaction (dashed line).Because of the smaller size of the latter rare gas, one also sees that the oxygen target, when interacting with it, behaves as a smaller- sized target. Each S-matrix element obtained by propagating the radial coefficients u,,,(R) of eqn (10) up to the outer matching can now generate the corresponding inelastic opacities at each collision energy examined : of -, ,,(kZ) = ( ~ ~ ~ i / k t ) ( 2 1 + l ) ~ ~ u , u t - sf -+ ut(kZ>12 k t = 2PU(Evo - E, + kZ0/W = 2P(Ec0,1 - 0 (12) (1 3) where266 0 2 - ~ A R E - G A s c o L L I s I o N A L INTER A c T I o N s The initial state of the target has been assigned here the arbitrary subscript v,.The unitarity of the full S-matrix was checked via the usual detailed balance accuracy, while numerical convergence was tested with the v'-summation on the r.h.s. of eqn (10) by including all open channels available up to a maximum of eight eigenfunctions. The corresponding 0 can be considered accurate within 1-2% of their value. Finally, each partial integral cross-section was obtained in the usual way by summing over contributing partial waves: In order to compare computed quantities with some experimental measurements, one needs now to perform the corresponding kinetic analysis.30 One of the standard approaches focuses on a two-state model, whereby the rate constants pertaining to direct and inverse processes between the initial and final vibrational levels can be related to each other when dynamical equilibrium is reached.For a gas in a trans- lational equilibrium, the rate constants are then obtained as the Laplace transforms of a function proportional to the relevant inelastic cross-sections : 37,38 where F(p) = EQ i +f(E). At a pressure of 1 atm, the vibrational relaxation time z is thus given, in units of s, by the following simple expression: p~ == ( y i 4 f + y J - - + i ) - l (16) where Ji} and If) are the two vibrational levels considered. In the classical study by Landau and Teller,41 a simple expression relates the logarithm of p z to the temperature T of the gas undergoing sound dispersion experiments. A standard LT plot would then give linear behaviour for p z as a func- tion of T-113.Any observed departure from such a model in the real-life collisional transfer should then show up as a non-linear dependence in specific ranges of tem- perature. Moreover, the specific features of the PES employed, like its dependence on A, on r and on their relative orientation 7, will also affect results in the linear region by yielding relaxation of these particular systems. It is also well known that the repulsive region of the potential surface is the one mostly responsible for the largest values of the required matrix elements of eqn (9), at least as far as neutral species arc.concerned, where only weak dispersion forces act in the long-range regions. Vibrational energy transfers require substantial distorsion of the molecular charge distribution by the incoming atom, and therefore inelastic events are most likely to happen within the region of charge overlap between bound electrons. An approximate stretching parameter could therefore be obtained by considering theshape of the PES at the classical turning point, Rcla, for the collision energy under study. If one now defines the logarithmic derivative of the effective potential plus the centrifugal barrier that pertains to the chosen impact parameter at a specific RC1,, one can write: where the Veff is now V,,, + Z(1 4- 1)/2yR2 for each partial wave appearing in eqn (10).This new approximate coupling parameter obviously depends on the collisionF. BATTAGLIA et al. 267 1 .o 0.8 2 0.6 . e s W 0 . 4 0.2 0 I' I I \ ;i \ He-02 \ - \ \ \ L . J '. 3.0 3.5 ._ '. 1 - . - I RIA r- - -1, 0 2 - 0 2 (- ' - *). FIG. 4,-Repulsive walls for the effective potentials discussed in this work : Ar-0, (-), He-0, energy as well as on the shape of the PES within the repdsive region. A possible refinement could be introduced by modifying Ecoll according to the value of the well depth, ahhough this modification is Iikely to have little effect for Van der Waals complexes with rather shallow wells and for which the repulsive wall often appears in a logarithmic scale as linearIy dependent on R. The existing differences in the shape of the repulsive regions for the systems studied here are evident from fig. 4, where a linear scaIe clearly shows the similar computed relaxation times which are too fast or too slow with respect to the values found through measurements, or which exhibit very different slopes when compared with experimental data.In conclusion, both the detailed mechanism by which energy is transferred during collisions and the specific features of the PES at various geometries will greatly affect the magnitude and slope of computed LT plots, which in turn will provide an additional test for the reliability of the full potential surface considered. 4. RESULTS AND DISCUSSION A comparison between some of the computed excitation cross-sections is shown The energy E in fig.5 for oxygen molecules interacting with He and Ar projectiles.268 02-R A R E - G A S C 0 L L I S 1 0 N A L I N T ERA CTION S is here defined as E = Ec0,, - E t h , where E t h is the threshold energy value for the final level; in the case of If) = 11) it corresponds to 0.1923 eV. As seen in fig. 5, helium is a more effective projectile in transferring energy to the vibrational modes of the target, in spite of the fact that its coupling between molecular levels induced by the potential surface is rather small, in fact smaller than in the case 1 . c n c, ._ g 10-2 '2 0 w W W Q 7 lo"* 6 10-6 I I 0 2 I I 4 6 EIeV FIG. 5.-Partial integral cross-sections for vibrational excitation of O2 molecules by He and Ar projectiles. The energy E corresponds to the energy available above the relevant thresholds.Ar-02 (- - -), He-02 (-). of Ar projectiles. Some selected values for the stretching parameter at various collision energies and for different partial waves are shown in table 2 for the two rare gases and indicate the direct influence, within the present model, of the shape of the potential on vibrational coupling. Table 3 also reports full matrix elements, at a given value of R, between various levels and for the three different collision projectiles. In spite of the smaller range of its potential and the weaker vibrational coupling, one sees from the tables and figures that the He projectile manages to produce larger cross-sections, mainly because of its lighter mass and therefore shorter collision times that enforce more effectively the sudden condition on the relevant c ~ o s s - s ~ c ~ ~ ~ ~ s .~ Fig. 6 presents similar calculations for the 02-0, system and compares them with the results for argon projectiles. In spite of their similar values of D, and of matrix elements, as seen in tables 2 and 3, the above systems show marked differences in their cross-section behaviour at intermediate energies (ca. 1.0 < E/eV < 4.0). The 0, projectile appears to be less efficient in causing vibrational excitations of the 0, target, especially within the ranges of energies where the higher repulsive part of the potential is sampled.F . BATTAGLIA el a/. 269 TABLE 2.--STRETCHING PARAMETERS, D1(Ecoll), AS A FUNCTION OF PARTIAL WAVE AND COL- LISION ENERGY, FUK He AND Ar PROJECTILES ON 0 2 TARGETS (IN ATOMIC UNlTS) ~ ~ ~~ ~~ E 0.5 eV 2.0 eV 3.0 eV 1 Ar We Ar He Ar He 0 I0 20 40 60 80 120 130 260 0.952 0,951 0,947 0.930 0.876 0.805 _- -- - 0.954 0.940 0.880 0.690 0.479 0,240 - - 0.968 0.967 0.966 0.959 0.941 0.91 7 0.849 0.708 - 0.957 0.952 0.93 I 0.853 0.737 0.598 0.343 0.173 I 0.970 0.969 0.968 0.963 0.950 0.932 0.882 0.776 0.595 0.958 0.953 0.93 8 0.880 0.79 I 0.680 0.445 0.235 0.146 '"t 1 I I I I I 0 2 I, 6 EleV FIG.6.-Same as in fig. 5 for 0, and Ar as projectiles.270 Oz-RA RE- GAS COL LIS IONA L INTER A CTTONS T K 8000 I03 : - - 1-0 2-1 E c 2 10’ r - 6 - . 3 - 4 I -L-L 0.05 0.10 0 15 TJIIC3 FIG. 7.-LT plots of computed and measured relaxation times for Ar-Q and (- - -) Ar-02 (computed), (-) 0 2 - 0 2 (computed), (- 0 --) Ar--02 (measured), (measured ).O,-O, mixtures: (- --) 02-02 Since numerical accuracy was achieved within similar levels in both cases, a pos- sible explanation could be provided by the small differences in the repulsive walls of their respective PES; at higher collision energies the repulsive potential used here for the OZ--O, interaction becomes “ softer ” than in the Ar-0, case. The corresponding differences in stretching parameters cause the matrix elements to become smaller in the case of the former than in the latter, hence the slower increase of O,-O, inelastic cross-sections when the collision energy increases. The marked variations seen in fig. 6 can then be attributed to their changes in the form of the effective potential and to their strong influence in the present model, where no other physical effects are introduced in the more complex molecule-molecule relaxation process.The corresponding relaxation times are compared with experiments 28*29.43 in fig. 7 , for both Ar and O2 projectiles, over the temperature range 1000-8000 K. One TABLE 3.-cOUPLYNG MATRIX ELEMENTS, vU,”t(R) (IN ATOMIC UNITS), FOR THE THREE “ STRUCTURELESS ” PROJECTILES STUDIED IN THE COLLISlONAL EXClTATlON OF 0 1 TARGETS vu,LA R ) R He Ar 0 1 v 0 , m 4.7 0.043 64 0.061 62 0.061 58 v o , ,(R) 4.7 0.136 1(-2) 0.920 S(-3) 0.922 6(-3) vo ,d R) 4.7 0.104 1 ( - - 3 ) 0.60021-4) 0.600 8(-4) v1,m 4.7 0.062 62 0.092 48 0.092 40 V2,d R ) 4.7 0.077 82 0.1 I5 22 0.115 21F. BATTAGLIA et al. 27 1 sees that the computed p z values for Ar-0, are very close to measured values, both in magnitude and slope, and follow the simple linear behaviour of the LT plot.The 02-02 results, on the other hand, produce relaxation times that are slower than ex- pected and therefore indicate that a simple spherical interaction is not sufficient to describe vibrational relaxation in this system. In fact, the faster relaxation of pure 0, molecules, as opposed to their behaviour in the presence of argon, indicates the greater availability in the first case of rotational energy exchanges that speed-up the slower vibrational decay which would occur uia only V-T transfer processes.44 Moreover, the highly anisotropic interaction of O2 molecules forming weakly bound adducts is disregarded in our model, which thus underestimates relaxation times even more as the temperature decreases.The possibility of adduct formation at low T is, however, taken into account in our model for those cases where the effective potential used is a more realistic description of the whole interaction. This is shown in fig. 8 and 9, where the computed results for He-0, and Ar-0, are compared with the available measurements both in the high-T region and at low temperatures. Fig. 8 reports relaxation times for He and Ar colliding with oxygen and shows very good agreement between computed values and experimental findings. This is lo3 102 E c, v) 3 10' Q 100 TIK 8000 1000 300 I I I / ' I ,/P T+/K+ FIG. 8.-Same as in fig. 7 for the Ar-02 and He-02 systems. true for the He-0, mixture down to ca. 500 K, where the presence of pseudobound states (resonances due to potential shape) increases the interaction and favours V-T transfer as the temperature decreases.Experiments seem to show departure from linear behaviour and our calculations confirm this effect. Such an effect is shown even more clearly by the results plotted in fig. 9, where the low-Tdata for Ar-0, are well reproduced by our calculation. On the other hand, the272 02-R A R E- GAS C 0 L L I S I 0 N A L INTER ACT I 0 N S T+/K+ FIG. 9.-Low-temperature behaviour of computed relaxation times for 02-02 and Ar-0, compared with available measurements: (-) O,-O, (computed), I- - -) Ar-02 (computed), (- -) OrOz (measured), (- 0 -1 Ar-O.! (measured). 0,--0, data remain linear in the range observed and are only qualitatively given by our computed p z for (1 +O> relaxation.Here again, the spherical effective potential is missing the orientational effects between approaching O2 molecules and therefore yields computed cross-sections that remain too small down to very low values of T. In conclusion, we feel that the effective potentials obtained by fitting a large body of data regarding simple systems could be used for computing vibrational inelasticity during collisions and can provide interesting information on the mechanism which presides over these processes. 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