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. 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