GENERAL DISCUSSION Dr. G. D. Barg, Dr. H. Fremerey and Prof. J. P. Toennies (Gottingen) said: To examine the validity of classical mechanics in calculating differential inelastic cross sections we have compared the results of classical trajectory and close coupling calculations of rotational excitation of He-H, ( j = 0) at E,, = 1.09 eV. The po- tential hypersurface used included the repulsive potential model of Krauss and Mies l with the Gordon and Secrest parameters,2 to which was added the long range aniso- tropic potential of Victor and Dalgarn~.~ A Morse vibrotor potential was used to describe the H2 molecule. 2000 trajectories were used in the Monte Carlo calculation and a n = 0, j = 0, 2,4 and n = 1, j = 0, 2,4 basis set (18 channels) was used in the close coupling calculation.In the case of the j = 0-2 differential cross section, the close coupling calculation shows a gradual drop off from a maximum of da/dw E 6 x lo-' t f 2 sr-' at 8,, = 0" to 8 x t f 2 sr-l at 180". The classical cross section is zero for 8,, < 30" and then rises up sharply to the quantum mechanical value. Agreement is good at larger angles, except at 180" where the classical cross section drops off somewhat. For the j = 0+4 cross section a similar behaviour is observed. The classical cross section vanishes below 90" and good agreement is achieved at larger angles. This comparison suggests that whereas the classical torque in the weak potential well ( E - 1 meV) is apparently too small to produce a transition, the wave mechanical uncertainty makes such a transition possible.This picture is consistent with the case of Li+-H2 (E-0.25 eV) where the agreement between quantum and classical mech- anics is good down to small angles (8,, - 5"). Dr. G. G.Balint-Kurti (University ofBristoZ) and Dr. B.R. Johnson (Ohio State Univer- sity) said : We would like to comment on a method of extending the range of validity of the distorted wave Born (DWB) approximation beyond that mentioned by Gordon. The conservation of particle flux in nonrelativistic quantum mechanics requires that the S matrix be unitary. In most perturbation approximations such as the Born and DWB approximations this property of unitarity is destroyed. A unitary matrix S may be written in the form S = (1) where A is a Hermitian matrix. The Born and DWB approximations both expand S directly in a perturbation series, using the strength of the potential as a perturbation parameter.If, instead, however, the matrix A is expanded in a perturbation series, then the resulting S matrix will always be unitary, so long as the approximate A matrix is Hermitian.'-' It can further be shown, by comparing the Born series for the S matrix with the expansion of elA, that evaluating A to first order only in the poten- tial is equivalent to including a part of all orders in the Born series.8 Thus the M. Krauss and F. H. Mies, J. Chem. Phys., 1965,42,2703. M. D. Gordon and D. Secrest, J. Chem. Phys., 1970,52, 120. G. A. Victor and A. Dalgarno, J. Chem. Phys., 1970,53, 1316. Comment in this Discussion by G. M. Kendall and J. P. Toennies.R. D. Levine and G. G. Baht-Kurti, Chem. Phys. Letters, 1970, 6, 101. G. G. Balint-Kurti and R. D. Levine, Chem. fhys. Letters, 1970, 7 , 107. ' R. D. Levine, MoZ. Phys., 1971,22,497. R. D. Levine, J. Phys. By 1969, 2, 839, also G. G. Balint-Kurti, to be published. 5960 GENERAL DISCUSSION exponential Born and exponetftial distorted waue approximations may be regarded as partial resummations of the Born and distorted wave Born series respectively. As an example let us consider the case of atom-rigid rotor collision. The potential is taken to be of the form where R is the distance from the atom to the centre of mass of the diatomic and 0 is the angle between the axis of the diatomic and the line joining the atom to the centre of mass of the diatomic.Following Arthurs and Dalgarno the total wavefunction is expanded in terms of angular functions which are eigenfunctions of the total angular momentum. The problem is reduced in this manner to a set of coupled, second order differential equations in R. In the exponential distorted wave approximation these equations are uncoupled in some manner, the uncoupled equations are solved for the distorted waves and these are then used to evaluate the perturbation expressions for the A matrix elements. In the present case we use the distortion decoupling procedure to define the distortion potential (i.e., the distorting potential is taken to be the diagon- al part of V ) . The S matrix can then be written in the form : roo The functions qEl(RP) are the total angular momentum eigenfunctions defined in ref. (9, and J, j and I are the quantum numbers corresponding to the total, the rotational and the orbital angular momentum respectively.The distorted wave &(R) is the solution of the differential equation : which is regular at the origin and becomes a sine wave at large distances &(R) - sin (kjR-ln/2+Sjl) R+CO ( 5 ) The approximation outlined in eqn (3)-(5) is the exponential distorted wave distortion (EDWD) approximation which has previously been examined. It has also been shown that an identical perturbation expression arises if the A matrix is expanded in a double perturbation series and evaluated to lowest order in both the A. M. Arthurs and A. Dalgarno, Proc. Roy. SOC. A, 1960,256,540. R. D. Levine, B. R. Johnson, J. T.Muckerman and R. B. Bernstein, J. Chern. Phys., 1965,49,56. R. D. Levine and G. G. Balint-Kurti, Chern. Phys. Letters. 1970, 6, 101. G. G. Balint-Kurti unpublished, work ; see also R. D. Levine and B. R. Johnson, Chem. Phys. Letters, 1970, 7, 404.GENERAL DISCUSSION 61 potential and in I?. lt is therefore, completely in keeping with the EDWD approxi- mation to use a semi-classical JWKB approximation to evaluate the distorted waves x$(R). This additional approximation reduces the task of evaluating the functions x ~ ~ ( R ) to that of performing a quadrature, and greatly simplifies the computation of the A matrix elements. To evaluate the JWKB wavefunction we use in essence the prescription given by Bernstein : x;~(R) = 0 k j "0.629 271 +a458 7452 = [ g ] - 0.104 878z3 - 0.038 229z4] = Fc-* sin (kjReff+n/4) where 3 z = e) kj(R- Ro), and Ro is the classical turning point of the motion. k-; = The phase shift is given aj, = in the semi-classical approximation - l < z < l < -1 (6) z > l dR'.by 1) dR'. (7) The use of the JWKB approximation to evaluate the distorted waves (eqn (6)) together with eqn (3) to calculate the S matrix is called the exponential semi-classical distorted wave distortion (ESCDWD) approximation. The quadratures were per- formed by a simple trapezoid rule integration. All of the quadratures, including the evaluation of the A matrix elements (eqn 3) were performed simultaneously. The potential energy parameters chosen were suitable for describing an Ar + TlF collision. As the potential, however, contains no odd Legendre polynomials it is inore appropriate to a homonuclear diatomic.One of the distinguishing character- istics of the model potential is the large short range anisotropy (al2 = 0.6). It is on account of this large anisotropy that the exponential Born approximation gives poor results for this case. Fig. 1 compares the exact and various approximate opacities for t h e j = O+j' = 2 first order allowed transition, at an energy of E/E = 1.7, as a function of the orbital angular momentum. The calculations include 9 coupled channels and the exact calculations were performed using the amplitude density m e t h ~ d . ~ As may be seen from fig. 1 , the distorted wave results (SCDWD) start to deviate appreciably from the exact ones for I values below 54.The exponentiated (ESCDWD) and exact results are in much better agreement over the whole range of 2 examined. The present model R. B. Bernstein, A h . Cliern. Pliys., 1966, 10, 75. * The potential energy parameters used were : B = 2p&R;/fi2 = 1200, &h/& = 0.01 where Eth is the lowest threshold energy ( j = 0+2), a12 = 0.6 and a6 = 0.2. G. G. Balint-Kurti and R. D. Levine, Chem. Phys. Letters, 1970, 7, 107 D. Secrest and B. R. Johnson, J. Chem. Phys., 1966,45,4556 : B. R. Johnson and D. Secrest, J. Cliem. Phys., 1968, 48, 4682.62 GENERAL DISCUSSION I I I 1 30 40 5 0 6 0 70 I FIG. 1.-Exact (dots) and approximate opacities for j = 0+2 transition calculated using 9 coupled channels. I FIG. 2.-Exact (dots) and approximate opacities for j = 0->4 transition calculated using 9 coupled channels.GENERAL DISCUSSION 63 potential, with its large anisotropy parameters, forms a specially severe test for any perturbation approximation.It is seen, however, that even at low I values the ESCDWD approximation gives reasonable values for the opacity. The EBDWD approximation fails completely due to the omission of distortion effects in this method. 5 0- 40- s J. 2 30- !% + - 20- 4 4 e-4 10- 6 0 . b I I:2 1:4 I :6 l:8 2.0 El€ FIG. 3.-Variation of the P44(2t0) opacity with energy. Fig. 2 shows the opacities for the j = 0-jf = 4 first order forbidd n transition, and fig. 3 shows the effect of varying the energy on the j = 0-j’ = 2 opacity. In both these cases the ESCDWD method yields reasonable results. In general it seems true to say that by the use of the exponential method the range of validity of a perturbation approximation, such as the Born or distorted waue Born, may be significantly extended with relatively little extra computational effort.We thank Prof. R. D. Levine for suggesting much of this work. Dr. M. D. Pattengill and Prof. J. C. Polanyi (University of Toronto) (contributed): In his introductory remarks, Marcus referred to a number of simple models for chemical reaction-the impulsive model, the DIPR model, and so on. There exists a category of model (a sub-category in Marcus’s scheme) which can be loosely designated as “ retreat-coordinate models ”, since these models lay their major stress on the forces that operate as the products separate.2 Models of this type tend to be based on the concept of a forced oscillator.The new chemical bond is regarded as, in some sense, existing at the outset. The repulsion between the products forces some vibration into G. G. Baht-Kurti and R. D. Levine, Chem. Phys. Letters, 1970, 7, 107. J. C. Polanyi and J. L. Schreiber, in Physical Chemistry-An Adcanced Treatise, Vol. VT, Kinetics of Gas Reactions, eds. H. Eyring, W. Jost and D. Henderson (Academic Press, New York, 1973) Chap. 9.64 GENERAL DISC U SSION this new bond ; the balance of the reaction energy goes into product translation and rotation. A feature of the chemical process that has been omitted from the simple forced oscillator models is the fact that the new bond is in the process of being formed while the product repulsion is being released.This is, in fact, a distinctive feature of chemical reaction, as compared with energy-transfer between an atom and a stable molecule. We have been developing a classical model of the retreat-coordinate variety which explicitly includes the effect of ‘‘ tightening ” in the new bond, AB, concurrently with the release of repulsion along the coordinate of separation, BC. The model (termed FOTO, for Forced Oscillation of a Tightening Oscillator) in its present form assumes that reaction proceeds collinearly through a selected intermediate configuration A--B.C. The dynamics of energy-release are followed in the retreat from this configuration. The extended A--B bond is treated as an harmonic oscillator of diminishing equilibrium separation and increasing force constant, being “ forced ” by a B-C repulsion of constant magnitude and finite duration.As the repulsion is released, A- -B evolves from a fractional bond to a normal one. Several assumptions,2 along with the use of empirical relationships (Pauling’s bond-order relationships, and Badger’s rule), suffice to parametrise the initial A- -B.C configuration and the required time dependences in terms of a single input parameter, namely the initial fractional bond-order of A- -B. The model has been applied to ten reactions for which some experimental data and trajectory results exist. The correspondence is encouraging. The model is sufficiently complete to embody recognisable analogues of “ attractive ”, “ mixed ” and “ repulsive ” energy relea~e.~.Prof. R. A. Marcus (University of Illinois) said: The agreement of the results of Balint-Kurti and Johnson with the exact values is most encouraging. I believe that for their system, the reduced moment of inertia I/pa2(in the notation of ref. (5)) is still in the region where the adiabatic distortion of the rotor is not too serious, judging from the results in ref. (5). It might be interesting to compare ESCDWD with exact results for a system with a substantially smaller I/pa2 (e.g., HX + M or H2 + M), where the “ static ” classical approximation seemed to show some signs of breaking down,5 although the latter results were quite incomplete and should be extended. Dr. R. G. Gilbert (University of Sydney) and Prof. T. F. George (University of Rochester) said : We have investigated the exponential Born distorted wave approxi- mation (EBDW) for collinear H+H2(u = O)+H2(u = O)+H at energies where only the ground vibrational state of H, is open? The S-matrix in the distorted wave approximation (DW) is given as S = I -iB, where Bll = BZ2 are the DW elements of B for elastic scattering and BI2 = B,, are for reaction.The EBDW form of S is given as S = exp(i6,) exp( - i6’) exp(i6,), where 8‘ is the off-diagonal part of B and A simple quantum model which stresses the strengthening of the new bond is that of G. L. Hofacker and R. D. Levhe, Chem. Phys. Letters, 1971, 9, G17. cf. F. E. Heidrich, K. R. Wilson and D. Rapp, J. Chem. Phys., 1971,54, 3885. J. C. Polanyi and J. L. Schreiber, in Physicuf Chemistry-An Advanced Treatise, Vol.VI, Kinetics of Gas Reactions, eds. H. Eyring, W. Jost and D. Henderson (Adademic Press, New York, 1973) Chap. 9. For a full account see M. D. Pattengill and J. C. Polanyi, Chem. Phys., 1974, in press. A. 0. Cohen and R. A. Marcus, J. Chem. Phys., 1970,52,3140. R. G. Gilbert and T. F. George, Chem. Phys. Letters, 1973.GENERAL DISCUSSION 65 2&, with elements 2a1 = 2d2 = 26, is the diagonal part of 6.l. Expanding the matrix exponentials in terms of the projection operators of the matrices, we find the elements of S to be Sll = = cos(Bzl) exp(2i8) and S12 = SZl = -i(sin(Bz1) exp(2i6)). The EBDW probability of reaction is then lS2,12 = sin2(Bzl). We obtain distortion surfaces for reactants and products by fitting the lowest adiabatic Porter- Karplus surface to a Morse curve at the local minimum (at fixed separation of H from the centre of mass of H& with the constraint that this curve support at least " 0.4 0.5 0.6 energy (eV) FIG.1 .-The EBDW probability (0) for collinear H+ H2(u = O)+H2(u = 0)+ H as a function of total energy and the probabilities from the coupled-channel calculations of Diestler (D) and Wu and Levine (WL). A are the results of the Condon approximation applied to the DW S-matrix. one vibrational state. Wave functions in B2 are evaluated in the perturbed-station- ary-state approximation. The results (0) are shown in fig. 1 in comparison with the coupled-channel calculations of Diestler (D) on the Porter-Karplus surface. (A are the results for Szl = -iBzl (DW) with the Condon approximation applied to Bzl : B21 is assumed to be proportional to the nuclear overlap integral between reactant and product wave functions.) The EBDW S is unitary, whereas the DW S yields reaction probabilities as high as four.6 However, the EBDW probability drops off too soon near a total energy of 0.6 eV, which is most likely due to the first vibrational state becoming locally accessible.and Wu and Levine (WL) Dr. S. Bosanac (Bristol University) said: Without loss of generality, we can take the potential of a three-particle system to be of the form G. G. Balint-Kurti and R. D. Levine, Chem. Phys. Letters, 1970, 7, 107. R. D. Levine, Mul. Phys., 1971, 22,497. R. N. Porter and M. Karplus, J. Chem. Phys., 1964,40, 1105. D. J. Diestler, J. Chem. Phys., 1971, 54, 4547. S.-F.Wu and R. D. Levine, Mol. Phys., 1971,22,881. R. B. Walker and R. E. Wyatt, Chem. Phys. Letters, 1972 16, 52. 55-c66 GENERAL DISCUSSION For multichannel scattering, (C collides with (AB)) this form reduces to the we11 known (2) where Y, R, and 8 are the standard coordinates. We now discuss multichannel scattering. The standard way of solving the Schrodinger equation with the potential (2) is to reduce the six dimensional differ- ential equation to only one dimension by expanding the wave function in the complete set of solutions for Vl(r). The members of the complete set we designate by $"(r) where n stands for quantum numbers defining the solutions. After neglecting the contributions from the continuum states, we get a set of M coupled differential equations in one variable only, the coupling being the matrix elements of V4(r, R, cos 8) between the eigenfunctions &(r).One way of solving the set of the equations is to uncouple them and solve M separate two-body problems. The solutions we designate by $A@) and $i(R), where again n refers to the quantum numbers and the indices 1 and 2 designate the solutions with different asymptotic behaviour for large R. A solution of the multichannel equations is now given in the form of an integral equation vf = vl(r)+ v4(r, R, cos e) $(R) = $'(R)+ dR'lc(R, R')$(R') (3) 0 where and K(R, R') is the kernel of the integral equation, given by is a diagonal matrix of the boundary conditions imposed on $(I?), and $2(R)$1(R')V;(R'); R 2 R' $'(R)$2(R')V;(R'); R < R'. K(R, R') = The prime on V4(R') denotes the potential matrix with no diagonal elements and C is a constant diagonal matrix.Iterating the integral equation once, we get a distorted wave Born solution to the wave equation. However, the S-matrix from this solution is not unitary, and we make it such by exponentiating the Born approximation Let us discuss the validity of such a method. We do this by developing the exponential function in the power series S - 1 +LJ * dRf$'(R')V:(R')$'(R') + c o (6) + J* dR'$ '(R')V:( R')$ dR"$' (R") V i ( R")$' ( R") c.2 0 and compare the result with the iterated solution of the integral eqn (3) S , - l+f dR'$'(R')V&(R')$'(R')+ c o dR"$l( R") V;( R")$'( R") + dR"$2( R")Vi(R")IC/ (R"). (7)GENERAL DISCUSSlON 67 The third term in the series for S could be treated as an estimate of the exact The estimate is good, provided value given by the third and fourth terms in (7).Yk(R’) is small. In other words, the quantity $ = jJic(R, R’) ic(R‘, R) dR’ dR (8) is a measure of validity for the exponential distorted wave approximation. However, in the case of reactive scattering, Yk(R’) is not in general small. If we neglect the term V3(rac) in (l), (in the case of reactive scattering A+ (BC)+C + (AB)) the potential assumes the form v’(r, R , cos 0) = Vl(r)+ V4(r, R , cos O)+V2(JR2+a2r2-2arK cos 0) (9) - Vi(r)+ V:(i., R)+ V,”(JR2 +a2r2)+P,(cos O)(V:(r, R)+ V$(r, R)). The term Vi(r, X) which enters in (9) and contributes to $ (8) cannot be taken as small because the expansion in P,(cos 0) of a two-body potential is poorly convergent.The difficulty could be resolved by solving the complete three-body problem for the potential and then using the exponential distorted wave approximation. This, however, would involve solving the Fadeev equations. v’’ = vl (rAB) + V 2 ( r B C ) + v3(rAC> (10) Dr. J. P. Simons (University of Birmingham) said : The successful use of a Franck- Condon modulation method for treating the multi-curve crossing problem discussed by Child has been demonstrated in a semi-classical “ impulsive half-collision ” calculation for energy transfer from Hg(63Po) into vibrational levels in CO or NO. V 0 2 4 6 B 1 0 42 14 16 18 0-3 0 0.12 I I I I I I I 1 I 1 i I 1 0 1 2 3 4 5 6 7 8 9 V FIG. 1.-Vibrational distributions in the quenching of Hg(63Po) by CO and NO.Experimental data : filled circles, CO, open circles, NO, (ref. (5)). Calculated distributions : broken lines, (assum- ing an exponential repulsive potential curve and a decrease in bond order of 0.6 in C-0 and N-0, ref. (2)). Intersystem crossing from a linear triplet Hg-CO or Hg-NO collision complex transfers the system onto a steeply repulsive part of the singlet potential surface and the subsequent recoil forces the molecular species into oscillation. If the equilibrium M. Child, this Discussion. J . P. Simons and P. W. Taker, Mol. Phys., 1973.68 GENERAL DISCUSSION C-0 or N-0 bond lengths remained constant throughout the entire process the recoil would force the molecules from an initial vibrational state i = 0 into a range of final statesf, the relative populations of which, would follow a Poisson distribution.'.If their bond lengths were extended at the instant of intersystem crossing a range of initial states i> 0 would be populated ; it was assumed (cf. ref. (3)) that their relative populations were governed by the appropriate Franck-Condon factors for the radia- tionless transition. The distribution over final states was obtained by calculating the set of probabilities Pi,, weighting each one by the corresponding Franck-Condon factor S& and then summing over the range of initial states i. Very good agreement with the experimental data was obtained assuming a decrease in bond orders of 0.6 in the triplet complexes of both CO and (see fig. 1). Prof. K. F. Freed (University of Chicago) said: I would first like to comment on Marcus' discussion concerning the contribution of the quasi-periodic classical orbits to the semi-classical bound states.Below is a non-rigourous proof which illustrates the the nature of the quasi-periodic classical orbits which can contribute to the quantum mechanical bound states and which demonstrates the existence of these types of orbits at all the classically allowed energies. (Of course, the quantization conditions are not necessarily satisfied at all these energies.) Following Gutzwiller and others,6* let $i(q) and Ei be the eigenfunctions and eigenvalues of a multi-dimensional Hamiltonian, and consider the sum over states where E is a complex parameter. As is well known, the poles of G in the E-plane are the eigenvalues and the residues at these poles provide the eigenfunctions.If we are just interested in the eigenvalues, then it is possible to focus attention on the spectral function 5 * G(E) dqG(q, q; E ) f C ( E - E i ) - ' , s 1 which has poles at the eigenvalues corresponding to the bound states. Gutzwiller and others 6* investigated the semi classical description of the bound states of multidimensional nonseparable systems by introducing the semi-classical approximations G,,(q, q; E) into (2). Upon evaluating Jdq in (2) by stationary phase, Gutzwiller '9 was lead to the conclusion that only the periodic classical orbits provide contributions to the semi-classical approximation to (2), and hence to the quantum bound states. (see below) on the full G,,(q, q' ; E ) showed that the periodic or multiply periodic orbits could contribute to the semi- classical quantization of bound states.Since the multiply periodic orbits can have incommensurate frequencies, this class includes the quasi-periodic orbits which Marcus emphasizes in his paper as yielding the semi-classical bound states. Our previous work K. E. Holdy, L. C. Klotz and K. R. Wilson, J. Chem. Phys., 1970,52,4588. R. D. Levine and R. B. Bernstein, Chem. Phys. Letters, 1972, 15, 1. M. Child, this Discussion. G. Karl, P. Kruus and J. C. Polanyi, J. Chem. Phys., 1967, 46,224 ; G. Karl, P. Kruus, J. C. Polanyi and 1. W. M. Smith, J. Chem. Phys., 1967,46,244. M. C. Gutzwiller, J. Math. Phys., 1967,8, 1979; 1969, 10, 1004; 1970,11,1971 ; 1971, 12,343. W. H. Miller, J. Chem. Phys., 1972, 56, 38.' K. F. Freed, J . Chem. Phys., 1972, 56, 692.GENERAL DISCUSSION 69 Let 6 g be a small volume in configuration space which is centred at point q. Consider the coarse grained average of G over the region 5q, Provided 5q is small and q is not near a mode of some $i(q), the contribution from the ith state in (1) to (3) will approximately be $i(q)$T(q)(E-Ei)-l. So, for a given q, we might lose a few states from the coarse graining in (3), but by varying q all should be found as poles in Ga,(q; E). At this juncture, we introduce the well-known semi- classical approximation 9 G,,(q', q"; E ) = (ih)- ID(q', q"; E)J' exp [iA(q', q"; E)/h - ivn/2], (4) classical paths where the sum in (4) is over all the classical trajectories leading from q' to q" with energy E , A is the classical action 4 q ' , qtt; E ) = l: p(zi ; ElD (5) ph; E ) is the classical momentum for the system at q with energy E, lZ2A(q', q"; E ) a2A(q', q"; E)l - _ _ _ ~ ~ ~ and v is the number of times the classical path encounters a caustic.into (3) and express the ldq' and jdq" as the limit of a summation to yield [G,,(q; E)ISc = lim Aq-+u { ( i h ) - ' ( d q ~ - ~ C C ID($, q"; E)I* x Substitute (4) ni3 x classical Aq',Aq'' paths exp [iA(qf, q"; E)A/-ivn/2]Aq'Aq".). (7) Eqn (7) has contributions from all the trajectories of energy E which start in the region 6q about q and end in that same region. By Poincark's theorem, this class of traject- ories is infinitely dense, since all trajectories starting in the region of phase space 6q and any allowed momentum must return to this region infinitely often in the course of time.As noted previously, this must lead to destructive interference unless the infinite number of different paths in the sum have actions which bear some relationship to each other, i.e., the paths must be parts of periodic or multiply periodic orbits. For 6q small and for trajectories which have the same number of traversals n through the region 6q, D,(q', q" ; E ) is slowly varying provided q is not near a caustic as we now assume. Thus, the only differences between such contributions can be phase differ- ences which arise from the action differences Consider now the difference in the contributions of various terms to (7). = s ( q f , qrt; E ) . 6, +;--"(qt, i3A q"; E ) . 62 3- .. . w aq" l M. C. Gutzwiller, J. Math. Phys., 1967,8, 1979; 1969,10, 1004; 1970, 11, 1971 ; 1971,12, 343. P. Pechukas, Phys. Rev., 1969, 181, 166, 171.70 GENERAL DISCUSSION As 6q is small, terms in (6)2 can be ignored in (8), and and 6, can be replaced by dq, a typical radius vector of 6q, in order to obtain an order of magnitude estimate. Thus, we have where are the final and initial momenta, respectively, for trajectories of energy E going from q‘ to q”. Thus, the phase difference between ‘‘ neighbouring ” classical trajectories in (7) is approximately 6p*6q/h. So, for cases in which the paths starting in and returning to 6p6g will all contribute to (7) and not destruct- ively interfere, i.e., all trajectories in the “ quantum box ” Gp*Sq<A contribute. The classical trajectories satisfying (1 1) are generally distinct from those which contribute to the normalization (6) for a periodic trajectory that traverses 6q, as periodic trajectories with energy E need not pass through a 6q (for some q) that the quasi-periodic trajectories do.In the limit 6q-+O, the operator (1/6q)J dq’ becomes Jdq’d(q-q’), where 6 is the Dirac delta function, and (12) Hence, all the quasi-periodic trajectories passing through q and satisfying (1 1) contri- bute to the semi-classical bound states so long as they satisfy the auxiliary quantization conditions. Another way to obtain the result (11) is to note that in the stationary phase evaluation of the integral in the semi-classical approximation to (2), the phase need only be stationary to order ti as this is “ effectively ” zero in the phase classical limit.I would also like to mention some work that John Laing and I have been doing to investigate the general semi-classical limit of multichannel scattering. We are interested in cases in which certain degrees of freedom must be treated quantum mechanically, while others may be treated semi-classically as in WKB approximations. Such cases arise, for example, in photochemical dynamics of polyatomic molecules where the molecule can rattle around on more than one nested potential energy surface before dissociation. A special case of this problem is to investigate the purely classical limit, if any exists, of motion on several potential surfaces when transitions may occur over very wide regions and not just along the intersections of the surfaces.2 I shall not go into the details, but instead just note that the time-dependent Schrodinger equation for multichannel scattering can be represented in a Feynman path integral from involving only the particles’ masses, the potential surfaces and the matrix elements connecting these surfaces.This removes a deficiency of Pechukas’ approach which required transition-matrices for arbitrary motions of the non- P. Pechukas, Phys. Rev., 1969, 181, 166, 174. J. C. Tully and R. K. Preston, J. Chem. Phys., 1971,55,462 ; W. H. Miller and T. F. George, J. Chem. Phys., 1972, 56, 5637. R. P. Feyntnan and A. R. Hibbs, Quantum Mechanics and Path Integrals (McGraw-Hill, New York, 1965). 6p. sqpi < O(1) (1 1 ) 6q GSqh ; E)+G(q, q ; E).GENERAL DISCUSSION 71 quanta1 degrees of freedom.In our case, what was an action integral in the single channel case now becomes a matrix, corresponding to a classical mechanics with internal degrees of freedom. However, this action matrix is nonlocal in time and can only be represented as a Magnus type expansion.' Consequently, the classical equations of motion have forces which depend on the whole trajectory of the particle. We are currently investigating the applicability of this formalism in the natural approximation that the Magnus expansion is truncated in lowest orders. Prof. R. A. Marcus (University of Illinois) said: I should like to report on some recent results on the semi-classical calculation of eigenvalues and resonances by Dr. Walter Eastes of this laboratory, using a method outlined in my paper at this Discussion.Eastes considered the collinear problem of a harmonic oscillator interacting with an atom via a Morse potential and calculated the bound states and the resonances quantum mechanically (numerically). The semiclassical results obtained thus far are encouraging. For example, when the quantum mechanical ground state (0,O) energy was - 1.806, and a value of zero was assigned to one quantum number, the following values were found for the other semiclassical quantum number n for various energies : n E n E - 0.064 - 1.83 + 0.01 6 - 1.80 - 0.01 1 - 1.81 + 0.043 - 1.79 - 0.0004 - 1.806 One sees that the trajectory giving the correct n (namely zero) serves to locate the eigenvalue of - 1.806.(At each energy several trajectories were used to find the ones giving the constant values of n listed in the above table.) Extension to larger inter- actions is being made and the results will be submitted for publication elsewhere. In response to a question posed by Freed regarding Gutzwiller's belief that periodic trajectories are needed to calculate eigenvalues, the arguments in my paper (and really those of Keller given earlier) show that such trajectories are not needed. None of the trajectories used to construct the above table were periodic. Prof. W, H. Miller (University of California) said: Regarding the point made in the last paragraph of Section A of Miller and Raczkowski's paper, S. M. Hornstein and I have carried out such a calculation for the collinear H + H2 reaction on the Porter- Karplus potential surface; the interest was to see if this symmetrical averaging procedure could describe the energy region near the classical threshold for reaction more accurately than the " quasi-classical ", or unsymmetrically averaged reaction probability.For the collinear H+H, system let x(nl, q1 ; E ) be the function such that x(n1, 41 ; E ) = 1 if the trajectory with the indicated initial conditions [(n,, ql) for the action-angle variables of the vibrational degree offreedom, and R1 = large, PI = - {2p[E-&(n,)])* for the translational degree of freedom] is reactive, and zero otherwise. It is easy to show, then, that the usual Monte Carlo, or " quasi-classical '' reaction probability is given by Punoym(E) = ( 2 W 1; dq,x(O, 41; El, (1) P.Pecliukas and J . C . Light, J . Cbem. Phys., 1966, 44, 3897.72 GENERAL DISCUSSION whereas the symmetrically averaged quantity is (For energies E in the threshold region all final quantum numbers correspond to the n2 = 0 " box ".) Thus the unsymmetrical procedure quantizes the initial vibrational state but not the final one, whereas the symmetrical procedure quantizes neither ; the classical S-matrix approach quantizes both initial and final vibrational states (via the boundary conditions on the trajectories). Fig. 1 shows the results : the solid line is the classical S-matrix result,' which is essentially the same as the exact quantum mechanical result, the dotted line is the unsymmetrical reaction probability [eqn (l)], and the dashed line the symmetrically averaged reaction probability [eqn (2)].(The abscissa is the nominal initial transla- tional energy, Eo = E - e(O).) The symmetrically averaged reaction probability, which would be preferred on any a priori theoretical grounds, is actually seen to be poorer than the unsymmetrical averaging procedure; it is difficult to give any solid theoretical reason why this should be so, and it may in fact be fortuitous. 0.4 - h Y .C( - 2 0.3- 0.12 0.14 0.16 0.18 0.20 0.22 0.24 collision energy/eV FIG. 1. The moral of this example presumably is that if there are so few quantum states involved that it matters how trajectories are assigned to " boxes ", then the only reliable way of using trajectories is via the semiclassical theory, which quantizes the appropriate quantum numbers initially and finally.Mr. J. L. Schreiber (Uniuersity of Toronto) (communicated) : Miller and Raczkow- ski have observed that cross sections calculated by the currently popular classical trajectory method do not satisfy a microscopic reversibility relation, due to the dis- symetric way of handling initial and final conditions. They have proposed a further averaging over initial internal state indices (n) as a " fix " for this situation ; however, the resulting cross sections still do not satisfy a microscopic reversibility relation. In the spirit of the quasi-quantum boundary conditions applied to the trajectories, it seems desirable for the resulting cross sections to satisfy a quantum-like algebraic microscopic reversibility relation.The well-known microscopic reversi bility relation between forward and reverse T. F. George and W. H. Miller, J . Cliem. Phys., 1972,57, 2458.GENERAL DISCUSSION 73 cross sections (averaged over degenerate levels of the initial state, and summed or integrated over the degenerate levels of the final state) 6, may be written where p, p' are the initial and final momenta, and gn is the degeneracy (or density of degenerate states) associated with the internal state index n (which may be a collection of indices, in which case integration over n is a multiple integration over all indices). For a given transition ni+n; the quantitiesp andp' are related by conservation of total energy. Cross sections calculated by the conventional classical trajectory method involves grouping the (continuous) n' values in a certain range n; - 3 to n;+ +, and associating these with the quantum transition into the state n;: Typically, pr and nf are fixed, and the cross section 0, for the ni+n; transition is the integral of all n' values in this range.Since the total energy is constant there is an implicit integration over p' as well n;+3 a, c,(pini -+ p;n;) = 1 dn' 5 dp' 6(E(p', n')-E(p>, n;))xo(pini -+ p'n') (2) where E(p, n) is the total energy of the system with the indicated internal state and momentum. It is clear that the forward and reverse nc's do not satisfy a simple (algebraic) microscopic relation of the form (1). n f -t 0 In order to obtain the simple relation, we define a new quantity, 0, 1 n i + f co Cc(pini -+ p;n;) = dn gn dp p2 x S(E(P, n>-E(pi, ni>)ac(pn + pin>)* PI gni J n i - * J* f, = E,gni s"'" n i - $ This average over the classical manifold of states associated with the quantum state p i n , is at a fixed total energy.Doing the integral overp and denoting ET = p:/2p, the (fixed) nominal translational energy, we get dn gn(ET+E(ni)-E(n)} x cc(ETn -+ Ein') (3) where ET is the quantity in curly brackets in the integrand, the translational energy required by conservation of total energy, and E(n) is the internal energy of the state n. This form for the classical cross section is easily applied, and results in a simple relation of the form (1) between forward and reverse cross sections so defined. a,(pini -+ p' n' Prof. R. D. Levine (Jerusalem) (communicated): The conclusions of this paper support earlier expectations by Miller In particular, one can show that for a certain class of properties, namely those which have a classical analogue, the loss of interference upon averaging is expected on a rigorous basis and is not an approximation.and by myself.3* Prof. J. N. Murrell (Sussex University) and Dr. S . Bosanac (Bristol University) said : We have been applying the classical S-matrix theory of Miller and Marcus and The classical cross sections u are continuous functions of their arguments, and as such satisfy this relation for a differential element of phase space, but not generally for an entire " correspondence principle " interval. * W. H. Miller, J. Chern. Phys., 1971, 54, 5386. R. D. Levine, Abstracts of Papers, VII ICPEAC (North-Holland, Amsterdam, 1971) p.912. R. D. Levine, J. Chern. Phys., 1972, 56, 1633.74 GENERAL DISCUSSION co-workers to the atom-diatom vibrational relaxation over a strongly attractive potential surface. The potential used as a model for the co-linear triatomic system A-B-A was based upon the spectroscopic properties of the ground state of C 0 2 and consisted of a CO Morse potential and a repulsive 0-0 Hulthkn potential. This combination produced a potential with an activation barrier to reaction of 2 eV and a well depth of 5.4 eV relative to CO +O. From our provisional results we wish to make some comments about the difficulties encountered in this type of calculation. 4 3 (c.2 B 0 0 I 2 3 4 5 6 FIG. 1. 4i Fig. 1 shows the results of trajectories obtained at an initial total energy of 1.3 eV greater than the activation barrier.Full lines represent reactive collisions and dotted lines non reactive. As was shown by Rankin and Miller for the H+C12 linear collision [a potential with no stable triatomic] the trajectories may be divided into " direct " and " complex ". In the figure we have only shown those that are direct, and in the regions of qi [the initial phase of the diatomic] not covered by the direct trajectories one obtains an apparently random set of final vibrational quantum numbers, nf, both reactive and non reactive. It is the problem of the separation of the direct and complex regions that we wish to comment upon. Given complete accuracy in the integration procedure all trajectories would be found to lie on continuous segments of nf[qi) curves.In practice it is possible to distinguish direct and complex regions according to the range of qi spanned by these segments. For example in the case ni = 0 there is one reactive segment that spreads from qi = 4.30 to qi = 2n + 1.16, and two shorter non reactive segments. A scan of Aq = 0.025 over the whole range, with an integration procedure that left the total W. H. Miller, J. Chem. Phys., 1971, 54, 5386.GENERAL DISCUSSION 75 energy accurate to better than 1 x eV, showed no other continuous regions with a width greater than 0.2, all other regions were therefore designated as complex. Because of the time reversibility of the classical trajectories, or because of the unitarity of the S-matrix, there must be connections between the continuous regions for different values of ni.For example, the region labelled A in n, = 0 shows trans- itions are possible over the specified scan of qi to nf = 0-4. It follows that there must be a nonreactive continuous region in each of ni = 1-4 in which the transition to nf = 0 is allowed. These are indicated by the letter A in each scan. We note however, again because the S-matrix must be unitary, that the slope [dnf/dqi] must be the same for n, = 0, n, = 4 as for ni = 4, nf = 0. Given that the continuous curves are approximately parabolic, it follows that the parabola must become steeper as ni increases. In other words one cannot make an absolute judgement on whether a region of trajectories is direct or complex based on the range of Aq over which a continuous curve is found.The region A in the scan ni = 4 must be designated direct even though the curve is continuous only from 6.02 to 6.10. The figure gives a labelling to each of the short continuous regions which shows their correspondence for different ni. If one takes region D for example this appears in ni = 2 but as it does not allow for nf = 1 or 0 there is no such region in ni = 1 or 0. In D the transition 2+3 occurs at four values of q1 so that there will be two curves D1 and D, in ni = 3 at which the transitions 3-2 are allowed. However, the transition 2+4 in D only occurs at two values of qi so that although there are two curves D, and DZ, in only one of these [Dl] is the transition 4-2 allowed. Prof. R. A. Marcus (University of IZZinois) said: The paper of Connor on the evaluation of multi-dimensional semi-classical integrals is very interesting.In most instances the mapping of the phase of the integrand onto quadratic or cubic functions is quite adequate for obtaining reasonably accurate results. However, we have noticed a number of instances where the phase along one coordinate is so slowly- varying that a cubic or quadratic mapping is inadequate, even in one dimension. TRANSITION PROBABILITY exact Airy Bessel uniform parameters (a, p, E ) transition quantum uniform (Stine) 0.114, &, 3.8 1-1 1 .00 4.7 1 .00 0.300, 3, 3 1-1 0.98 1.46 0.97 0.300, 3, 6 1-1 0.22 0.22 0.23 0.300, 3, 10 0-1 0.22 0.21 0.21 0.300, 3, 4 0-1 0.11 0.11 0.10 0.1 14, 3, 3 1-0 7.1 x 10-4 7 . 6 ~ 10-4 6 .8 ~ Recently, therefore, J. Stine in our laboratory has developed a new uniform approximation, involving a mapping of the phase onto a sinusoidal function and an appropriate mapping of the pre-exponential fact0r.l He has obtained encouraging results, a few of which are listed in the following table. The a, p and E refer to the parameters in the Secrest-Johnson problem of a collinear collision of a diatomic molecule with an atom (exponential repulsive interaction potential). Complex- valued trajectories While Bessel functions have been used before in semiclassical perturbtion calculations, they J. Stine and R. A. Marcus, J. Chem. Phys., 1973, 59. J. Stine and R. A. Marcus, Chem. Phys. Letters, 1972, 15, 536 ; W. H. Miller and T. F. George, J. Chem. Phys., 1972, 56, 5668, 5772.were used for the last two rows in the table.76 GENERAL DISCUSSION apparently haven’t been used to construct uniform approximations. Dr. Kreek is now applying the method to the two-dimensional integral arising in collisional rotational-translational energy transfer problems, and we hope to have results shortly. Dr. J. N. L. Connor (University of Manchester) said: I have an addition to make to my paper.l The integral for which a uniform asymptotic approximation is derived is a special case of the n-dimensional integral The position of the saddle points (classical trajectories) depends on the set of para- meters a. No concrete example of such an integral in semiclassical collision theory was given. However, an example can be found in the paper by Marcus,2 specifically eqn (2.7) where A = F,(wR, nE)-Fi(wR, mE)+3(ln+Zm+ 1 ) ~ .The correspondence between the two integrals is as follows. For a fixed energy E, the set of parameters a is equal to m = (rn,}, the set of final quantum numbers for the collision. When the energy E is allowed to vary as well, there is then an additional parameter on which the position of the saddle points (classical trajectories) depends. Mr. Y-W. Lin, Prof. T. F. George and Prof. I(. Morokuma (University of Rochester) said : The analytic continuation of classical mechanics for the description of classically forbidden processes, as discussed in the paper by Miller and Raczkowski, involves the analytic continuation of potential energy surfaces for complex values of nuclear coordinates. Such analytic continuation is necessary for a semiclassical description of electronic transitions in molecular collisions between adiabatic surfaces of the same symmetry, if the description is carried out strictly in the adiabatic representation.We have applied the semi-classical theory of Miller and G e ~ r g e , ~ which has been developed in the adiabatic representation, to the collinear rearrangement H+ + D2 + HDf + D, as proceeding through a transition between the two lowest singlet, adiabatic surfaces of HDZ. A trajectory started asymptotically on surface 1 with vibrational quantum number nl as as action variable for D2 propagates on the same surface in real time steps until AE, the absolute energy difference between the two surfaces, reaches a local minimum. Allowing time steps to become complex, we continue the trajectory to the nearest (complex) point of intersection, where the trajectory switches continuously to surface 2.Hence we assume the transition to be localized at the point of intersection. Our method can be viewed as an extension of the method of Stueckelberg for a single nuclear degree of freedom to several nuclear degrees of freedom. Unlike his method, however, ours must deal with many transitions due to the many local minima in AE. Fig. 1 shows the complex r-motion of a trajectory from nl = 0 on surface 1 to n2w0 on surface 2, which has propagated on surface 1 into the product valley, where r = tr+iri is the complex internuclear distance of HDf (strictly speaking, t is the internuclear distance of HD when the system is on surface 1, J.N. L. Connor, this Discussion. R. A. Marcus, this Discussion. W. H. Miller and T. F. George, J. Chem. Phys., 1972,56, 5637. E. C. G. Stueckelberg, Helv. Phys. Acta, 1932, 5, 369.GENERAL DISCUSSION 77 and of HD+ when the system is on surface 2). The four branch points, corresponding to the zeroes of AE(r, R), and their extended branch cuts are drawn for R = (6.9+ 0.03i) Bohr, where R is the complex distance from the D in HDf to the atom D. The r-motion in fig. 1 is in the vicinity of R = (6.9 +O.O3i) Bohr. AE goes through a minimum when r, goes through a turning point. Near this minimum the trajectory 0.3. 0.2. 0.1 - 2 5 0.0. i" -0.1 . -0.2. -0.3. R = (6.9+0.03i) Bohr I / - 3 t 3.0 I I- FIG. 1.-The complex r-motion of HD+, D for real time steps in the vicinity of R = (6.9+0.03i)BohrY where Y = r,+ ki is the complex internuclear distance of HDf and R is the complex distance from the D in HD+ to the atom D.The four branch points and their extended branch cuts for the function AE(r, R) are drawn for R = (6.9+0.3i)Bohr. can proceed in complex time steps to either of the two nearest branch points to switch continuously to surface 2, and it need only cross the branch cut without actually passing through the point itself. This is illustrated by the thin line crossing the cut and continuing on surface 2 (dotted line). Hence there are two different branching procedures for continuing on surface 2 from this local minimum. Likewise there are two branching procedures for remaining on surface 1 : crossing no branch cuts or crossing both branch cuts.In this manner we see that many trajectories with different branching procedures can end in n2 = 0 on surface 2. If A ( j ) is the complex action along such a trajectory j , the classical limit of the S-matrix element for the transition from n, = 0 on surface 1 to n2 = 0 on surface 2 is given by a sum over j of terms each proportional to exp(iA/(j)/A). Preliminary calculations show the probability of this transition to be of the order of The analytically continued surfaces used in this calculation are obtained through the insertion of the appropriate complex values of r and R into the elements of the electronic Hamiltonian matrix, where the elements are derived as functions of real r and R for HD; in the diatomics-in-molecules method used by Tully and Preston.' This method of analytic continuation is subject to error, especially when the selected real points of Y and R used in the drivation of these function elements are not close to J.C. Tully and R. K. Preston, J . Chern. Phys., 1971, 55, 562. for an initial relative translational energy of 3 eV.78 GENERAL DISCUSSION each other and when the imaginary parts of Y and R are large. The most powerful means of analytic continuation involves the complete diagonalization of the electronic Hamiltonian, He,, for complex values of r and R , and we are currently carrying this out for the HD; system. As a test case we have diagonalized He, for the one-electron system HeH++, focusing on the potential energy curves, E36 and E4,, corresponding to the 30 and 40 molecular ion states. Due to the Wigner noncrossing rule,' AE(R) = E,,(R) -E,,(R) can never be zero if R is complex. The state wave functions which diagonalize He, are expressed as linear combinations of atomic orbitals on both the helium and hydrogen nuclei (LCAO MO), where the atomic orbitals are Gaussian orbitals with four s type on hydrogen and eight s type and four p type on helium. + 4 + 5 + 4 + 3 + 2 - 2 - 3 - 4 -5 - 6 0 1 2 3 4 5 6 AEr/10-3 Hartree FIG. 2.-The complex energy difference AE(R) = E4,(R)-E3,(R) = 4Er+iAEi for HeH++ as a function of complex internuclear distance R = R,+iRj. The thick solid line is for real R, i.e., Ri = 0. Since our LCAO MO calculations are in excellent agreement with the exact results of Bates and Carson for real R, we do not include d type orbitals. The electronic Hamiltonim matrix is complex and non-Hermitian, so that the resulting eigenvectors, i.e., states of HeH++, are members of a biorthogonal basis. AE(R) has a minimum at R = 3.6440 Bohr (for real R) of 0.002 94 Hartree, and AE(R) = 0 (i.e., JAEJ < for R = (3.643 408 8k0.037 475 326i) Bohr. For complex R we can write AE = J . von Neumann and E. Wigner,. Z . Phys., 1929, 30,467. S. F. Boys, Proc. Roy. SOC. A , 1950,200,542. D. R. Bates and T. R. Carson, Proc. Roy. SOC. A, 1956,234,207.GENERAL DISCUSSION 79 AE,+iAEi and R = R,+iRi, and fig. 2 shows AE for lines of constant R, and of constant Ri. Since AE is an analytic function of R except at the branch point where AE = 0, the mapping is conformal so that each line of constant R, crosses each line of constant Ri perpendicularly. Since the electronic Hamiltonian and the atomic orbitals are real functions of R, AE is a real function of R, so that AE(R*) = {AE(R)) * and we need show only lines of positive Ri. The lines of constant Ri are not sym- metric about the line AE, = 0, which is anticipated since AE for real R is not sym- metric about the line corresponding to the real part of R for which AE = 0. Our results are quite encouraging, and the task of finding surfaces for HD; for complex values of nuclear coordinates does not seem very difficult.