Semiclassical Theory for Collisions Involving Complexes (Compound State Resonances) and for Bound State Systems t BY R. A. MARCUS Department of Chemistry, University of Illinois, Urbana, Illinois 61 801 Received 23rd January, 1973 Semiclassical theory for bound states is discussed and a method is described for calculating the eigenvalues for systems not permitting separation of variables. Trajectory data are supplemented by interpolation to connect open ends of quasi-periodic trajectories. The method is also applied to quasi-bound states. Previously, semiclassical S-matrix theory has focused on " direct " reactions. Processes involv- ing complexes (compound state resonances) are treated in the present paper and an expression is derived for the S-matrix. Use is made of the above analysis of quasi-bound states and of trajectories connecting those states with open channels.The result deduced for the S-matrix has the expected factorization property, and expressions are given for computing the quantities involved. Some extensions and applications will be described in later papers. An implication for classical trajectory calculations of complexes is noted. 1. INTRODUCTION A semiclassical S-matrix theory for " direct " inelastic and reactive collisions has been developed in recent papers.l* (Ref. (3) contains related studies.) On the other hand, many collisions and other processes involve short- or long-lived vibra- tionally-excited intermediates 4* : unimolecular reactions, molecular beam reac- tions involving complexes,6 other bimolecular reactions (e.g., possible at a thre~hold),~ and intramolecular energy transfer in general.In the present paper we consider the dynamics of coupling between open channels and quasi-bound states (" compound state resonances "9), and formulate a semi- classical S-matrix theory for such collisions. The theory of direct collisions is first summarized in Section 2. A method utilizing classical mechanical trajectory data is proposed in Section 3 for calculation of eigenvalues of bound states. It is non- perturbative and is directed toward systems for which one cannot separate variables. It is adapted in Section 4 to quasi-bound states. The principal result for quasi-bound states is given by eqn (4.9), (4.12), (4.16) and (4.17). A method for calculating the quantities is described. One-dimensional (" shape ") resonances, such as those occurring in orbiting, have earlier been treated semiclassically.o-l 2. SEMICLASSICAL THEORY OF DIRECT COLLISIONS The semiclassical wave function $+(q, nE) is a function of the coordinates, denoted For direct inelastic (2.1) -f This research was supported by a grant from the National Science Foundation at the University collectively by q, the quantum numbers n, and the total energy E. collisions, this wave function at large separation distances R is $+(q, nE) = (qlnE+} = [A' exp{iF2'} + A exp{i(F2 -3n}] exp{iZnn/2}, of Illinois. 34R. A . MARCUS 35 where the term with superscript i describes the incoming particles, and the second term is for the outgoing ones.* The A’s and F’s are functions of q, n and E ; - n/2 is the usual phase loss for a reflection ; Znn/2 is present by convention, where In is the orbital quantum number contributing to n.Units of A = 1 are used. F2 in (2.1) is a classical mechanical generating function l3 and serves to make a canonical transformation of variables from q, p to momenta in nE and to conjugate coordinates). Illuminating discussions of the relation between classical canonical transformations (e.g., embodied in Fz(q, nE)) and quantum mechanical unitary transformations (e.g., embodied in (qlnE+) have been given. 14-1 8’; for a partial wave, when the q’s consist of a radial coordinate R and angle coordinates w, is given by (2.2). [The value when the q’s are conventional coordinates appears later in (4.5).] Fi(wR, nE) = -knR+2nnw, (2.2) The w’s are canonically conjugate to 2nn and are frequently employed for the internal motions.‘.2* 17-19 Their properties are very convenient and have been summarized in Part 1V.l nw is an abbreviation for I=njwj, the summation being over the various internal degrees of freedom (including orbital, if any). F2(wR, nE) is the phase integral calculated from the classical trajectory passing through the cited q [the point (w, R)], beginning at the given n and E and hence at some initial point (wo, R,) : R W RO wo F,(wR, nE) = -k,Ro+2nnwo+ 1 pRdR+2n1 i i d w , (2.3) where Z is the instantaneous n along the path. vation 1 s 20-22 or by normalization to a 6(n - n’)[2n6(E- E’)] it is the determinant t The amplitude A in semiclassical solutions can be evaluated either by flux conser- I5-l6 When normalized to A = ]PF2/dadql* (2-4) = on-+, A = v-+jawo/awl+, (2.5) where the a’s are E and (since h = 1) 2nn.One finds from (2.2)-(2.4) that where u, and li denote the magnitude of the initial and final radial velocity for the trajectory specified by the final point (wR) and the initial momenta present in nE. The semiclassical S-matrix Smn can be defined via the expression for t+h+ at large R : v;*(exp (2nim w})[8,, exp { -i(k,R --+Zmn)] - $+( wR, nE) = m S,, exp {i(kmR-3Zmn))]. (2.6) Comparison of (2.1) and (2.6) yields (Part 11, ref. (1)) .1 S,,, = 1 18 wo/a wl*(exp iA) dw (to be stationary phased), 0 using the fact that only the stationary phase points contribute (whence ZJ = Un,).’ given by A is (2.8) A = F2( wR, nE) - Fi(wR, mE) ++(1, + 1, + 1)~. * The wave function given by eqn (2.1)-(2.6) is for a volume element dq.If the volume element is gt dg, e.g., R2 dR dw,’ then (2.1) and (2.6) are multiplied by g-$, e.g., R-I. t This is equivalent in (2.5) to the normalization used in ref. (l), where each A was normalized to unit radial flux at a given R.36 RESONANCES A N D BOUND STATES A stationary phase evaluation of (2.7) yields evaluated at fi = m. The summation is over stationary phase points wt, i.e., over the one or several traject- ories leading from the initial momenta in nE to the final momenta in mE. (The Riemann sheet discussion in Section 3 and 4 provides an understanding of the fact that several trajectories lead from nE to mE, namely one per sheet.) F4 is a gener- atorI3 far a canonical transformation from nE to mE, and was denoted by At, apart from the z-terms, in ref.(1). F,(mE, nE) = - w d i = - J q dp. (2.10) A uniform approximation of (2.7) can be made. It usually involves Airy functions and has wider validity than the stationary phase value, (2.9). 9 The amplitude A in (2.5) becomes infinite on certain surfaces ((‘caustics ”). This infinity in A is detected by the intersection there of neighbouring trajectories. (E.g., see later, “ surfaces ’’ AB, BC, CD and DA in fig. 1) : the resulting vanishing of the “cross-sectional area” between the trajectories causes the amplitude A to become infinite to “ conserve probability flux.” In such cases one can still usually use (2.7) to obtain a stationary phase or uniform approximation for Sm,, since the stationary phase points are themselves usually nonsingular.To have a useful integral expression for S,, one can change the coordinate repre- sentation to one which is sometimes singularity-free, by a canonical transformation (Part III of ref. (1)) from wR to Wz, where the i7 are constant and z is a time variable. This representation leads to the same result as (2.7), but with dw amd dw replaced by a@ and dK, and with A having an added term 2n(ii-m)@. The uniform and station- ary phase values of the new expression agree exactly with those of (2.7)-(2.9), but the new integral is now sometimes also of particular use when asymptotic evaluation methods become poor. The form of this integral expression had been predicted by intuitive arguments.3. SEMICLASSICAL TREATMENT OF EIGENVALUE PROBLEMS To obtain the phase F2(q, n) of the wave function for a bound state at a point q, one may integrate along a trajectory, as in (2.3). Except in the case of degeneracy, accidental or intrinsic, this trajectory does not close on itself, i.e., is not periodic. Moser 23 and Arnold 23 have proved an important theorem for celestial mechanics and thereby for the present nonlinear mechanics. Under certain conditions for systems not permitting (or permitting) separation of variables, the motion is quasi- periodic (multiply-periodic) rather than ergodic. That is, the p’s amd q’s can be re- presented as functions of time by Fourier series, e.g., Z aml,. .mN exp(i Z 2nm,vit), where the m’s denote the integers from -co to co and where the coefficients a decrease exponentially with (Irn,l+ .. . +lmN[).23 The contrast between ergodic and quasi- periodic is seen in fig. 1 : the former would occupy the whole space within the line of constant energy, while the other would be more confined spatially. In a nondegenerate system, N is the number of degrees of freedom. The vi are the frequencies of the true angle variables wf for this problem, i.e., those canonically conjugate to the actions, N m l . . .mN i = 1R . A . MARCUS 37 2nnI, the classical counterparts of the quantum numbers. Hamilton’s equations of motion yield (3.1) d(21tnl)/dt = -aH(n)/awi = 0, dw,/dt = ~ H ( ~ I ) / ~ ( ~ z T Z ~ ) = v,(n). Quasi-periodicity implies that there is a canonical transformation from (q, p) to (w, 2nn) and hence a generating function F,(q, n) * for this transformation. In turn, F,(q, n) defines a congruence of trajectories, directed along VF,.Later, F2 will be the phase in a semiclassical wave function, each “ surface ” of constant phase serving as a wave front and the trajectories serving as rays along the normal to the front. FIG. 1 .-“ Box-like ” orbit formed by a simple trajectory in a bound or quasi-bound state. (The actual figure was made for two uncoupled oscillators, using conventional coordinates.) Caustics are AB, BC, CD, DA. The elliptical curve is a constant energy curve, with energy equal to the total energy. In the nondegenerate situation depicted in fig. 1 a single trajectory, for the case of two degrees of freedom, generates four congruences of rays, corresponding to the four possible algebraic signs of the two components of the momentum p, as seen in fig.1 and as emphasized in fig. 2 and (in Appendix) fig. 4. The corresponding VF, (= p) FIG. 2.-Congruences of rays present in fig. 1, each corresponding to a branch (Riemann sheet) of the function p(q). then has four branches (Riemann sheets).20 [When there are N coordinates instead of 2 there are 2N branches.] The rays in fig. 1 do not cross when they are on the same sheet, except at the boundaries, i.e., at the caustics AB, BC, CD, DA. * The symbol F2(q, n) is shorthand notation for F2(q, 2 4 . An analogous remark applies to F2 in Section 2.38 RESONANCES AND BOUND STATES The semiclassical wave function is a linear combination of the solutions, A(q) exp(iF2(q, n)], one per sheet ; F2 is Jp dq.In the several-dimensional wave diffrac- tion literature, the formula for connecting such asymptotic expansion terms is found by assuming a local separation of variables near the caustic and solving the local problem exactly.24 The local solution usually involves the Airy function and its derivative. When this procedure is applied to the present problem in the vicinity of caustics AB and AD, near point A, one obtains (3.2) for the case of two vibrations. (It is easily generalized to N-coordinates, and then has 2N terms.) Each sheet of F2 is described by a Roman numeral : (3.3) (3.4) a denotes 2nn1 and 2nn2 ; Cis a normalization constant. The pre-exponential factors, which are absolute values of the determinants, are equal in the region near A.One sees from (3.4) that congruence II(1V) is related to I(II1) by time-reversal. If one similarly obtains a local solution in the vicinity of D, and uses it to obtain the connection formula.relating the ( A exp iF2)'s, the resulting t,b is similar to (3.2), but with A replaced by D and with a different arrangement of the n/2's, namely 0, 0, -7r/2, n/2 for branches I to IV. Since @ is single-valued these two solutions can differ at most by a multiplicative constant. A further analysis (Appendix) then establishes (3.5). A similar comparison of (3.2) with the t,b resulting from a local solution near B yields (3.6) : 2n(n,+3) =fc, Pdq = /;Pldq+ lP P' Plrrdq (3.6) where 0, 0', P, P' are arbitrary points on their respective caustics, as in fig. 2.It may be emphasized for this derivation that the local separation of variables near a caustic is used only to obtain the connection formulae of the ( A exp iFJs. The global nature of the ( A exp iF2)'s as solutions, for points not near caustics, is the principal tenet (asymptotic expansion) of semiclassical theory. Eqn (3.5) and (3.6) have been obtained earlier by a different argument 2o : each ( A exp iF2) term was assumed to be single-valued and A was allowed to vary on passage through a caustic. (This appears to be a type of phase integral argument.) The present discussion and that in Section 4 avoids a phase integral approximation, though the latter frequently suffices. F2 satisfies the Hamilton-Jacobi partial differential equation and the method of characteristics (classical trajectories) is a convenient method for solving it.However, that method is not the only one. Thus, it should be emphasized that the Jpdq's need not be along dynamical trajectories, that is, along a ray congruence. In fact, in the case of a nondegenerate system the motion is not periodic and so one cannot compute the C1 and C2 path integrals merely by integration along classical trajectories.R. A . MARCUS 39 Instead, one can use an " interpolation '' of the exact trajectory data to close the ends of an open ended path. For example, the jpdq data obtained from the single trajectory can be represented by a Fourier series. One may then use that series to join two ends of a trajectory and so compute $p dq along the two independent paths.Fourier series, suitably chosen to avoid the " small divisor '' problem when necessary, have been extensively used in the astronomical l i t e r a t ~ r e . ~ ~ Fig. 1 describes an orbit which has been termed " box-like " in the computer-simulated stellar dynamics literature.26 Other classes of orbits exist (" shell-like," " tube-like "),26 readily understood physically, and can be analogously treated. The need for supplementing trajectory data by interpolation for calculation of $p dq has been missed, incidentally, by a number of who accordingly but incorrectly insisted on periodic dynamical trajectories. (The classical Feynman propagator, with its usual dynamical associations, was used as a starting point.) Two other approaches should also be recalled : perturbation theory 28 and a proposed mapping of the nonseparable problem onto a separable one.2 Quasi-periodicity implies incidentally, that there exists a canonical transformation for a mapping of the nonseparable one onto a separable one, since the problem in (3.1) has become separable.- ' =+ & 4. SEMICLASSICAL THEORY OF COMPOUND-STATE RESONANCES The description of the quasi-bound (q-b) state, like the bound state, involves caustics. The two states differ, in that the q-b state is connected via (real or complex) trajectories to other states, e.g., to open collision channels. We consider, by way of example, the situation depicted in fig. 3, where AB, BC, DC and DA are caustics bounding the q-b state. The semiclassical wave function for collisions involving compound-state resonances is constructed below so as to satisfy the boundary conditions at R = 03, and in the vicinity of the various caustics.I@ -~ IB,40 RESONANCES A N D BOUND STATES As in Section 3 a solution based on local separation of variables near A, to find the connection formula, yields (3.2). Comparison with a solution determined near D yields (3.5). A local separation of variables is made in a segment near caustic EB, followed by use of the one dimensional solution for the coordinate normal to EB and the two turning-point one dimensional solution l 1 for the curvilinear coordinate parallel to EB. (The two turning points occur on EF and on BC.) When the latter is fitted to the solution (3.2) one obtains the desired connection formula, whence a standing wave solution (q(rt2E) for the q-b state InzE) is found to be <qIn2E) = 231d2Fi/dqda()(exp {iG') + exp (iG"))+ where 21j82F',"/aqdalt(exp (iG"') + exp (iG")) (4.1) GI = F\ +a,, - &bS - &T, G" = 3'; - 6, + &5s + $71 (4.2) GIv = Fiv - a,., + $+s GI11 = FIII 2 + 6,s - 3 4 s , F: = J' p,dq (y = I , .. ., IV). E (4.3) The pr's are related as in (3.4), and so F" = -Pi and Piv= -F!. Thus, (4.1) is a standing wave solution. a,, and c $ ~ are independent of n and m and are discussed later. s denotes n,E. Eqn (4.1) contains two radially ingoing terms (branches I1 and IV) whose sum is denoted by (qln2Bi), and two radially outgoing terms (branches I and 111), whose sum is denoted by (qlnzE9, so that (qln2E> is the sum of these.Each sum is separately normalized to 6(nz-n2)[2n6(E-E)] (and hence to unit radial flux). . We turn now to the $+(q,nE) satisfying the appropriate boundary conditions for a collision. The incident term in (2.1), denoted by (qlnE'), is given by (4.4) using conventional coordinates, (qlnE') = 2-*[la2Pk1'/i3qi3a,J3 exp (iFk"++in) + where 1d2~$1v/aqda,l* exp (iF$IV - tin}] exp (i~"n/2}, (4.4) (4.5) FLY = s' py dq-k,R ( y = 11, IV). BPI B,, is the vibrational turning point in state nE at the given R (fig. 3). In branch I1 the vibrational momentum points toward and in branch IV away from B,,. Similarly, the wavefunction (qlnEf) for an outgoing wave is identical with (4.4)-(4.5) but with I1 and IV replaced by 111 and I, respectively. The unperturbed wave function is the sum (qlnE') +(qlnEf).To construct the desired wave function $+(q, nE) analogous to (2.1) with (4.4) as the incident term, it is necessary to follow the two congruences of rays in (4.4) during the collision until they have become outgoing rays at large R. To do this we match an ingoing ray(s) of (q1.E') with one(s) of (qln,Ei) by finding the stationary phase vaIue of J<nzEilq> dq'(q1nE'). (The q' indicates integration at a fixed large R.) The stationary phasing serves to match a p(q) in InE'} with one in InsEi) and so provide a smooth trajectory to the q-b state. The outgoing ray(s) emerge as <qlnzE'>. Thus, the rays lead from the initial state nE to a congruence in the q-b state. Then, from another congruence in the q-b state they go out to some q at large R.Upon summing the contributions from all q-b states and including the contribution (if any) from anyR. A. MARCUS 41 direct collision trajectories not involved in the caustics of fig. 3 ((qlnE+),) one obtains $+(q, nE) (qlnE+) = (qlnE'> + (qlnE+)d+ (qln2Ef><n2EilnEi>, (4.6) n2 where the first term is the incident term, as in (2.1), and (n,E1lnEi) denotes the stationary phase value of l(n,Eilq) dq' (qlnEi). Comparison of eqn (2.6) and (4.6) shows that -Smn = (mE'lnE+)d + (mEfln2Ef)(n2EilnEi),I (4.7) n2 where (rnEfln2Ef) is obtained by stationary phasing J(rnEfJq> dq'(qln2E?. The latter also serves to match an outgoing ray(s) in <qlnzEf) with one(s) in (qlml?). Eqn (4.1)-(4.5) yield (after some manipulation related to that in eqn (7) of ref. (1 5)) (n2E'lnE') = P n s exp W r s - - + + s - + 4 } 9 (4.8) Pns = 3 c Iiaw;la4+ exp (i[~1;+3(ln+3)n]>, (4.9) where y = 1I.W and F i = P,,dq-k,R = - q dp, ( y = 11, IV), (4.10) where wY, is canonically conjugate to 2nn2, being equal to aFqYp(2~n~), and the inte- gration over py in (4.10) is over the path from B,, to E.s 1- s (mE'InE') = where P i s = and Fj; = The a,, in eqn (4.8) and p,, dq- kmR = - q dp, (7 = I, 111). (4.13) Jt J (4.1 1) is given by and, when EF and BC are real caustics, (4.14) (4.15) C3 is a contour encircling BC and EF. & a function of OS given in ref. (1 l), is close to zero unless the system is near the top of the barrier between BC and EF. 0, is given by a related equation l 1 when BC and EF are complex caustics. The exponential, exp{ 1, in (4.8) can be written as (exp( } - 1)+ 1 and, it can be shown, that stationary phase value of X&pns has the same form as (2.11).It constitutes contribution from the direct collision trajectory involving reflection from caustic EF. If the sum of this contribution and of S:,, is written as Szn, eqn (4.7), (4.8) and (4.1 1) yield (4.16) nz Snl" = - C' flAsPnsKexP W r s - i4.D - 11,42 RESONANCES AND BOUND STATES where the sum over s denotes the sum over n2 (at the given E). If the exponent is expanded about E-E, one obtains l1 S m n = G'm - C P~sPnsrsI(E-E,+ir,/2), (4.17) where the E, are the E's for which nl is an integer (when +s w 0) and where r, is given by eqn (24) of ref. (1 1). When exp( - OJ 4 1, we have (4.18) where vl, is the frequency i3E/i3(271n1) for the mode corresponding to n1 in the q-b state s.iOs can be calculated as the difference of the $p dq overa path (from any nE to any mE) which encircles caustic EF and one over a path which encircles both EF and BC. (The ends of the paths at large R are joined analytically, and care is taken to compare paths on the same ingoing branch y.) To apply eqn (4.16) or (4.17) in a form utilizing exact trajectories it is necessary to obtain the F2 for the q-b state, as deduced from Section 3. From caustic BC one then calculates trajectories leading to the open channels. The usual analytic argu- ments at large R then yield the integrals in /Ins and #&.* S L P w v,sexp{ - OS), 5. DISCUSSION Eqn (4.16) has the appropriate factorization pr~perty,~ one which reflects the " loss of memory " of" indirect " collisions occurring in the q-b state.It also: has the appropriate time-reversal symmetry of S,,, since Bt is a time-reversed j?. When state In2E) couples strongly with some state, the #I for that state will be close to and can be replaced by unity or evaluated by an integral expression (suitably expressed in terms of W's) used to obtain (4.8). The small P's can still be evaluated by the semiclassical expression (4.8) and (4.11). When the collision system is purely elastic, the sum over s reduces to a single term pf& (= l), and the formulation reduces to that l1 for elastic collisions, as it should. One implication of (4.17) for purely classical calculations involving complexes may be noted. When the relevant I'Gs are large, one may anticipate that classical tra- jectories will reasonably well reproduce the quantum dynamical behaviour.When, however, a relevant rs is small one should, at least, exclude from such calculations certain trajectories-those for which an individual vibration associated with the r, has, for any significant time, an energy substantially less than its zero-point energy. Otherwise the calculations could not approximate in this respect the corresponding quantum mechanical behaviour. Some extensions and applications, computational and perturbative, will be des- cribed elsewhere.§ As in the case of direct collision trajectories some partial averag- ing 30 should serve to isolate aspects of the collision which are predominantly quantum mechanical.* E.g., to calculate Pns the analytic expression for py(q) in state nE at large R is used to calculate - jq dp, from Bn to a point on the trajectory leading to BC. The pr(q) data for that trajectory are then employed, and the q-b py(q) data are used to obtain the contribution to reach B. Subtraction of i0J2 then yields the value to reach E. J Eqn (4.3)-(4.13) refer, for simplicity, to a vibration-translation problem, but are readily extended to include rotational and orbital problems by including appropriate generating functions, functions which disappear in the second halves of (4.10) and (4.13). Certain +7r terms, arising from passage through vibrational turning-points, are also omitted for brevity.R. A . MARCUS 43 APPENDIX DERIVATION OF EQN (3.5) A N D (3.6) The path from A to D on any branch y can be deformed to lie along the caustic AD.The component of p normal to AD is zero, and so (fig. 2) pr = prv = --pn = --pru on AD. The integral lpydq from A to D for y = I or IV thus equals that from D to A for y = I1 or 111. This fact is used in the comparison of the solution (3.2) and the corresponding one originating from D to show that the magnitude of the above integral equals n(n2+3), whence f p dq equals 27t(n2+$) if the cyclic path C2 lies along AD. To prove (3.5) one may suitably deform the above path, and to do this it is necessary to remove the multivaluedness of p. While p (and hence VF2) is a function of q, one may introduce a Riemann surface on which it is single-valued.20 As usual the pair of sheets which have the same p at a branch cut (caustic) are joined.When one does this at all caustics, one obtains for the Riemann surface a torus,’O as in fig. 4, where the torus is an open “cylinder”, whose rims are the “ellipse” passing through C and D and the one passing through A and B. A I FIG. 4.-Riemann surface for the system in fig. 1 and 2. The two lines joining D to C coincide spatially, as do the two joining A and B. The congruences in fig. 1 and 2 are indicated, as are the cyclic paths C2 and C1 in eqn (3.5) and (3.6). The momentum p(q) is now a (single-valued) analytic function of q on this surface, and so one may deform the contour C2 for f p dq, which originally went from A to D and back to A along AD, to be any other equivalent cyclic Cz path as indicated in fig.4. Mathematically, this path is equivalent to the second half of (3.5). Eqn (3.6) is obtained similarly. In three dimensions, points A and D lie at the corner of a cube, instead of a square, and AD lies along an edge. Once again all pr’)~ are equal in magnitude on AD, since the com- ponents of p normal to the edge vanish. Thus, the preceding argument can be generalized. For an N-dimensional system, it leads to p dq equal to 2n(ni+3), for the N topologically independent paths Cj. Fig. 2 is, essentially, a deformed square. An equilateral triangle has also been treated, by a different method.20 $ R. A. Marcus, Chem. Phys. 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