Semi-Classical Theory for Non-separable Systems : Construction of “Good” Action-Angle Variables for Reaction Rate Constants* BY WILLIAM H. MILLER-^ Department of Chemistry, and Materials and Molecular Research Division of the Lawrence Berkeley Laboratory, University of California, Berkeley, Ca 94720 and University Chemical Laboratory, Lensfield Road, Cambridge, England Received 3rd May, 1976 A semiclassical expression for bimolecular rate constants for reactions which have a single activa- tion barrier is obtained in terms of the “ good ” action variables of the (classical) Hamiltonian that are associated with the saddle point region of the potential energy surface. The formulae apply to non-separable, as well as separable saddle points. 1. INTRODUCTION A semiclassical expression for reaction rate constants has recently been derived which is of interest because it has the form of a generalization of transition state theory which correctly takes into account non-separability of the Hamiltonian about the saddle point of the potential energy surface.lg2 Application3 of this theory to the collinear H + H2 reaction gives reasonably good agreement with (numerically) exact results of quantum scattering calculations4 over a wide range of energies near and below the effective threshold for reaction, much better agreement than the usual (i.e., separable) quantum mechanical version of transition state theory.This is of some practical importance, since for chemical reactions with significant activation energy it is the threshold region that primarily determines the thermal rate constant.This semiclassical reaction rate theory 193 involves certain periodic classical trajec- tories of the molecular system and is very similar in its mathematical structure to the periodic orbit theory of semi-classical eigenvalues derived by Gutmiller (see also the important modification by Miller). As has been discussed,6 however, the quantum condition produced by periodic orbit theory, although qualitatively correct, contains dynamical approximations in addition to the semiclassical approximation itself, and semiclassical quantization of non-separable systems is in general more accurately achieved by the Hamilton-Jacobi approach of the old quantum theory.’ In this approach it is necessary to construct the total non-separable Hamiltonian as a function of the complete set of “ good ” action variables (or adiabatic invariants) of the system, and recently a great deal of progress has been made in finding ways to do this non-perturbatively (Le., numerically).The preceding paper in this volume, * Supported in part by the U.S. Energy Research and Development Administration, and by the t Camille and Henry Dreyfus Teacher-Scholar ; J. S. Guggenheim Memorial Fellow on sabbatical National Science Foundation under grant GP-41509X. leave from the Department of Chemistry, University of California, Berkeley, California 97420.WILLIAM H. MILLER 41 for example, describes an application of the Hamilton-Jacobi approach to semi- classical eigenvalues and also gives more references to this topic.Just as the periodic orbit theory for eigenvalues involves implicit dynamical approximations, so does the periodic orbit of the semiclassical reaction rate theory. To overcome these dynamical limitations of periodic orbit theory one would like to construct a Hamilton-Jacobi theory for reaction rate constants analogous to that for the eigenvalue problem. Such is the object of this paper. To motivate the form of the general result section 2 first discusses the separable case. Section 3 then shows that the expression suggested there is also the correct semiclassical result for the general, nsn-separable case. The relation between this new semiclassical rate theory and the earlier one1*3 based on periodic orbits is dis- cussed in section 4. 2. SEPARABLE CASE For notational simplicity, formulae below are all written without taking explicit account of total angular momentum conservation; the modifications necessary to account for this are minor and have been given bef0re.l~~ The rate constant for a bimolecular reaction, such as A + BC --+ AB + C, can be written in the k(T) = (2nh QJ-l dE e-P"N(E); (1) Irn where Qr is the partition function (per unit volume) for reactants, is related to tem- perature in the usual way, and E is the total energy. The exact expression for N(E) is 1,3 N E ) = 2 ISnt,n(E>12, (2) n'.n where the reactive S-matrix elements must in general be determined by carrying out a quantum mechanical scattering calculation.n( nl,n2, . . ., nF-1) and n' are the quantum numbers for the F - 1 internal degrees of freedom of reactants and products, respectively, and F is the total number of degrees of freedom of the complete system in its overall centre of mass.If motion along a reaction coordinate (the F'th degree of freedom) is separable from the P - 1 internal degrees of freedom, then the S-matrix is diagonal, ISn*,n(E>l2 = Jn,,nP(E - &If), (3) where P(EJ is the one-dimensional tunnelling probability for a translational energy E, along the reaction coordinate; En is the energy of internal state n. Since the semiclassical approximation for P(EJ is P(EJ = [l + e2Q(Et)]-1, (4) where 8(Et) is the usual barrier penetration integral, eqn (2) becomes where N(E) = 2 [l + e2e(E-E,) I- ? n=O 2 ° C 2 ... 2 . n=o n,=O n,=O np-,=O42 SEMI-CLASSICAL THEORY FOR NON-SEPARABLE SYSTEMS The action variable for the F'th degree of freedom, n F , is defined in the usual one- dimensional fashion (setting R = I), = iO(E,), (7) so that the expression for N(E) finally takes the form N(E) = 1 11 + e2n Im n~(E,n)]-l, n=O To see the meaning of eqn (8) in a simple example, consider a particle first in a two-dimensional harmonic potential well.The total (classical) Hamiltonian is given in terms of the two action variables (Le., semi-classical quantum numbers) in this case by E(n,,n2) = hw(n1 + 4) + hu2@2 + $1. (9) Making the replacement m2 + ico2 corresponds to inverting the harmonic potential of this degree of freedom, so that the two dimensional well becomes a saddle point region, a harmonic potential well in one direction and a parabolic barrier in the other.The hamiltonian in eqn (9) thus becomes E(n,,n2) = kw,(n + 3) + ifio,(n, + $1, (10) and one notes the important point that there are '' good " action variables associated with a saddle point region of a potential surface just as there are those associated with a minimum. Rather than requiring both action variables to be integers, as is done for the semiclassical eigenvalue problem, one solves the following equation, E = E(n,,n2), (1 1) n2(E,nl) = -3 + iCfim1(nl + - El/(fiwJ, (12) to express n2 as a function of E and n,; for eqn (10) this is and eqn (8) in this case gives J For this harmonic-parabolic saddle point eqn (13) is actually the exact quantum mechanical result, the " bonus " one often obtains with harmonic potential functions in semiclassical theory; i.e., if eqn (13) is substituted into eqn (l), the result is where Q * is the partition function of the " activated complex " and I' is the tunnelling correction : Q* = +hm1/3/sinh (ihml/3) I? = +hco2p/sin ( ~ h u 2 p ) .(1 5 4 (1 5b) The next section shows that eqn (8), derived here for the separable case, is also the correct semiclassical expression for N(E) for the general, non-separable case. In the non-separable situation, however, the total (classical) Hamiltonian is of course not a separable function of the P " good " action variables as in eqn (lo), and theWILLTAM H. MILLER 43 partition functioii for the activated complex and the tunnelling correction thus do not enter as separate factors as they do in eqn (14), and to construct the function E(rz,, n?, .. ., nF) is tlic primary task of the theory. It can be done, for example, using the procedure devised by Chapman, Garrett, and Miller9 for solving the Hamil- ton-Jacobi equation in action-angle variables. (Interestingly, it does not appear that the approach of Eastes and Marcuslo would be possible because the classical trajectories about a saddle point are not quasi-periodic.) One action variable, nF say, is identified as the one associated with the " reactive direction ", and one then solves the equation E = E(n,,n2, . . . , nF> (16) to obtain nF(E,nl,n2, . . . , n,-,). Eqn (8) then gives N(E) and eqn (1) k(T). 3. NON-SEPARABLE CASE Another formally exact expression for the rate constant, completely equivalent to eqn (1) and (2), is', k(T) = Qr-' tr[e-D"s(f)fP], (17) wherefis an operator that defines a surface which divides reactants and products, f is the Heisenberg time derivative off, and 9 is a projection operator that projects onto all states which have evolved in the infinite past from reactants.ll It is easy to see that eqn (17) can be written as eqn (1) with N(E) given by N(E) = 2nh tr[G(E - H>S(f)f?Y], (19) an expression formally equivalent to eqn (2) but more useful for present purposes.The plan is to evaluate eqn (19) semiclassically. For notational simplicity consider the case of two degrees of freedom, i.e., a collinear A + BC + AB +- C reaction. If n, and n2 are the two " good " action variables, then the quantum mechanical states In,n,) are eigenstates of the total hamiltonian, Hln,n,) = E(n1,nJ l W 2 ) .(20) Evaluating the trace in eqn (19) in this representation gives where J , q, and q2 are the angle variables conjugate to n1 and n2, and E(n1,n2) is the total (classical) hamiltonian, in general non-separable, expressed in terms of the " good " action variables. [Superposition of amplitudes from the different 271 intervals of q1 leads to n, being integral, and this is why the domain of q1 is limited to (--n,n) rather44 SEMI-CLASSICAL THEORY FOR NON-SEPARABLE SYSTEMS than (- 00, wavefunctions are l2 Since (nl,n2) and (ql,q2) are conjugate variables, the semiclassical (23) tynln1(q1,q2) = d(27c1-2 e i(nlql + n2q2), so that eqn (21) becomes where the fact has been used that nl is real (an integer) and that n2 may not be, and wherefhas been taken to involve only the (n2,q2) degree of freedom.(The final result cannot depend on the specific choice for$)'' The first factor in eqn (24) is clearly unity, and the second factor is essentially a one-dimensional flux for the (n2,q,) degree of freedom that depends parametrically on nl. The flux through a one- dimensional barrier has the standard form of the frequency v of the (n2,q2) motion, multiplied by the tunnelling, or transmission probability P, p = (1 + e2n Im (1 + CZrr Irn 2)-1 . - - e-2n Im n2 Eqn (24) thus becomes Before proceeding to the final result, it is interesting to note the simple illuminating interpretation eqn (27) has if one expands the tunnelling probability in a geometric series : where Eqn (27) then reads (284 (28@ (29) P = e-20 - e-4Q + e-60 .. . 0 = ~t Im n2. ( n l n 2 [ 6 ~ f P l n l n 2 ) = ve-,O - V ~ - ~ O + ve-@ . . ., and each term can be thought of as the flux contribution of one of the classical trajectories that comes from reactants and crosses the dividing surface f = 0. The trajectories (classical trajectories in complex time 12) are pictured in fig. 1 ; the reactive ones contribute positive flux and the non-reactive ones negative flux. The Appendix shows how eqn (29) results directly from eqn (24) if one makes a particular choice for f. The final expression is now obtained by substituting eqn (27) into (21), where n2(E,n,) is determined by E = E(n1,n2). This is eqn (8) for the case F = 2.WILLIAM H. MILLER 45 FIG. 1.- Sketch of the potential energy along a reaction coordinate through a saddle point region and the different classical trajectories [in complex time-see ref.(12)] that come from reactants and pass through the saddle point region at least once. P is the probability associated with the trajectory in each case. 4. RELATION TO PERIODIC ORBIT THEORY It is easy to see how the periodic orbit expression for the rate constant is recovered if one uses eqn (8) with the periodic orbit quantum condition6 for determining E(nl,n2, . . . , nF). Periodic orbit theory determines E(nl,n2, . . . , nF) implicitly by the equation6 where @(E) is the action integral for one pass around the periodic orbit and (coi(E)) are the F - 1 stability frequencies. For the case of a saddle point 0 is pure imagin- ary 9 @(E)/ti 2ie(E); (33) so that eqn (32) gives F- 1 27C(nF + *) = i[2e(E) - 28'(E) 2 hui(E)(ni +)I, i= 1 or F- 1 which, with eqn (8), is the exprsssion obtained' via periodic orbit theory for N(E). 5.CONCLUDING REMARKS This paper has shown how a semiclassical approximation for the rate constant for a bimolecular reaction with a single activation barrier can be constructed in terms of the complete set of " good " action variables associated with the saddle point of the potential energy surface. This new expression for the rate constant bears the same relation to the earlier semiclassical rate expression in terms of periodic orbits that the old quantum theory for semiclassical eigenvalues bears to the periodic orbit quantization rule. Since the expressions based on the " good " action variables46 SEMI-CLASSICAL THEORY FOR NON-SEPARABLE SYSTEMS of Hamilton-Jacobi theory should in general be more accurate than their periodic orbit approximations, this new expression for the rate constant is expected to be more accurate than the one based on periodic orbit theory. APPENDIX If in eqn (24) f is chosen as f= - sin q2, then the effect of 9 is simply to require q, > 0. The operator f is 1 8 . where (n& =- i aq2' This then gives (setting h= 1) which is eqn (27). W. H. Miller, J. Chem Phys., 1975, 62, 1899. Mechanical Transition State Theory, 1976. S . Chapman, B. C. Garrett and W. H. Miller, J. Chem. Phys., 1975, 63, 2710. D. G. Truhlar and A. Kuppermann, J. Chem. Phys., 1972, 35, 2232. M, C. Gutzwiller, J. Math. Phys., 1971, 12, 343. W. H. Miller, J. Chem. Phys., 1975, 63, 996. N. C. Handy, S. M. Colwell, and W. H. Miller, Faraday Disc. Chem. SOC., 1976, 62, pre- S. Chapman, B. C. Garrett, and W. H. Miller, J. Chem. Phys., 1976, 64, 502. 'See also, W. H. Miller, Accounts. Chenz. Res., Importance of Non-Separability in Quantum ' M. Born, The Mechanics of the Atom (Ungar, New York, 1960). ceding paper. l o W. Eastes and R. A. Marcus, J. Chem. Phys., 1974, 61,4301. l1 W. H. Miller, J. Chem Phys., 1974, 61, 1823. l2 W. H. Miller, Adv. Chem. Phys., 1974, 25, 69; 1975, 30, 77.