Franck-Condon Transitions in Multi-curve Crossing Processes BY M. S. CHILD Department of Theoretical Chemistry, 1, South Parks Road, Oxford Received 24th January, 1973 The high energy solution of the multi-curve crossing problem involving many vibronic levels is shown to reduce to that of a single curve crossing modulated by Franck-Condon factors. A quanti- tative validity criterion is derived. While the dynamical problem associated with a single crossing between potential curves is essentially solved,’-’ certain molecular processes may be described in a multi-curve-crossing form, and here the situation is less satisfactory. Important examples involve interaction between covalent, M +XY, and ionised, M+ +XY-, species, with each vibronic state being represented by a different curve, as shown below.\ FIG. 1 Theoretical difficulties then arise from overlap between adjacent crossing regions, because a given crossing has a dynamic width,4 given in the linear approxiniation Vi(r) = Vo-Fi(r- R), Ar,, = [hu/21Fi - F’1]f, where u is the velocity at the crossing point, R. This width may be seen either as the range within which the “ classical ” momentum uncertainty at distance Ar from the crossing point, cannot be accommodated within the uncertainty principle, or, from a mathematical viewpoint as the region outside which rapid oscillations in the equations of motion 30M. S. CHILD 31 damp out any contribution to the transition probability. The possibility of such overlap clearly restricts the use of formulae based on the accumulation of single crossing probabilties lo-ll to the low velocity region.At the same time, eqn (2) shows that while Ar increases with v it does so less rapidly than the velocity itself. Hence the interaction time, Ar/u decreases as v-* and must in the limit fall to a value small compared with the time period of the internal motion. Under these conditions we might expect to revert to a single electronic level crossing model modulated by Franck-Condon populations in the final vibrational states.’ The purposes of this note are first to justify this expectation and secondly to provide quantitative conditions on its validity. We assume for this purpose a model based on unperturbed internal states xln(p) and xzm(p) with the energies E l , and Ezn in channels 1 and 2 respectively, in which the electronic interaction term, V1 2(r), is independent of the internal coordinates p (this is an essential pre-condition of any Franck-Condon result). Furthermore, the translational velocity is assumed sufficiently high to allow the use of a time dependent classical trajectory formulation,’ the relation between time and distance being of the form r = ro+u(tl.(4) The resulting equations for the amplitude coefficients cln(t) therefore become where the elements define the (orthogonal) internal state overlap, or Franck-Condon amplitude, matrix. Clearly a first-order perturbation solution, valid for small VI2(r), subject to the bound- ary conditions C l n ’ ( - 00) = 6,”*, - a) = 0, (7) = T 2 1 S n n 9 (8) will always yield a solution of the desired Franck-Condon form where in this case with t , used to denote the time at the crossing point.channel 2 to obtain new coefficients The theory in the more general case relies on an orthogonal transformation in a 2 n ( t ) = C S m n ’ ~ 2 m ’ ( t ) , (10) nr32 MULTICURVE CROSSING PROCESSES in terms of which by the orthogonality of S c,m(t) = C Srn'adt). (11) n' Note the use of the label n to denote the channel one level to which the transformed channel two states are coupled. Substitution in (4) now yields m These equations are, apart from the terms in Xnn.9 appropriate to a single curve cros- sing between (14) (15) where W i is the energy at the crossing point, the final forms in (14) and (15) being justified by the linear approximations (1) and (4). Furthermore the terms in Xnnt in eqn (12) may be recognised as those responsible for spontaneous decay of the non-stationary states represented by (11).Hence we wish in physical terms to find conditions under which the time constant for this spontaneous decay is large compared with the as yet unspecified interaction time, to. The argument is facilitated by the substitutions W&) = E,,+V,(t) E! W;-F,o(t-',) W2n(t) = E,,+V,(t) 21 W",F&-t,) and the appropriately averaged curve and the introduction of a new dimensionless variable z = ( t - t,)/to. Eqn (12) then take the forms . dbln - 'm) exp [Iia(z)]bzn(z) h dz ' db2n dz I2 I - - I-=- '''' 2 ( t ) exp [ - ia(z)]bIn(z) + whereM . S . CHILD 33 The onset of rapid oscillations, la(z)l> n, leading to widespread destructive inter- ference on integrating (1 8) may therefore be set at the points z = * 1, by defining the interaction time as The final term in (18) therfore provides a contribution to b2,, of order twice the coefficient ( t o X n n r / A ) when (18) is integrated over the interaction zone, - 1 <z< 1.The condition for neglect of this term, for moderate values of [toV12(t)/?i] is therefore to = [27EA/Up71-F214. (20) Cases for which (toVI2/h) < 1 are seen to be covered by eqn (9). Hence, if the condition (21) is satisfied, the solution of (18) is equivalent to that for a single curve crossing between Wln(r) and the appropriate mean curve wzn(r) ; and the resulting transition amplitude, given in the Landau-Zener approximation by (22) is readily converted, byt the use of (1 1) and (16) into a set of Franck-Condon trans- ition amplitudes Finally, by virtue of the closure relation ITi"z'l = I~z,(.o)l = exp { - ~Cv,z(t,)12~~ul~1 - & I 1 9 I K m t = ICAW)I = ISmnTC;lI.(23) C Snmsmn = 1 7 m P , , =c ITnm12 = lT'1"1I2. the total probability of scattering into channel two is given m The validity of (23) and (25) does not however rest on the Landau-Zener approxi- mation. The overall conclusion is that a single curve crossing, Franck-Condon, model is justified at velocities sufficiently high that (21) is satisfied, provided that the asymptotic is calculated from the correct near vertical electron affinity The validity criterion (21) is seen to depend on the ratio of the interaction time, to, to the time period, (E2m-E2n)/lZ, of the internal motion, and also on the extent of overlap between the covalent and ionic vibrational states. E(X2) = E(X)+D(X,)-D(X,)+Eln--~n. (27) l L. D. Landau, Phys. Z . Sow. Union, 1932,2,46. C. Zener, Proc. Roy. SOC. A , 1932, 137,696. E. C. G. Stuckelberg, Helu. Phys. Acta, 1932, 5, 369. D. R. Bates, Proc. Roy. SOC. A, 1960, 257, 22. V. K. Bykhovskii, E. E. Nikitin and M. Ya. Ovchinnikova, J. Expt. Theor. Phys., 1965,20,500. G. V. Dubrovskii, J. Expt. Theor. Phys., 1964,19, 591. ' M. S . Child, Mol. Phys., 1971,20, 171. * E. E. Nikitin, Opt. Spektr, (trans.), 1961, 11, 246. H. Hartmann, Chemische Elementarprozesse (Springer, Berlin, 1968), pp. 43-77. V. I. Osherov, Zhur. Exp. Teor. Fiz., 1965,49, 1157. Yu. N. Demkov, Dokl. Akad. Nauk. S.S.S.R., 1966,166,1076. G. M. Kendall and R. Grice, Mol. Phys., 1972,24, 1373. l 3 J. B. Delos, W. R. Thorson and S. K. Knudson, Phys. Reu. A, 1971, 6, 709. 55-B