Partial Averaging in Classical S-Matrix Theory Vibrational Excitation of H2 by He BY W. H. MILLER AND A. W. RACZKOWSKI Inorganic Materials Research Division, Lawrence Berkeley Laboratory and Depart- ment of Chemistry, University of California, Berkeley, California 94720 Received 28th December, 1972 Within the framework of a general semiclassical theory that combines exact classical dynamics and quantum superposition it is shown how a certain averaging procedure allows one to treat some degrees of freedom in a strictly classical sense while others are quantized semiclassically. This enormously simplifies the application of the theory to three-dimensional collision systems and also leads to an interesting formal structure in the theory: the quantum-like degrees of freedom are quantized semiclassically via use of double-ended boundary conditions, while the unquantized classical-like degrees of freedom enter only through a phase space average over their initial coordin- ates and momenta.Preliminary results for vibrational excitation of H2 by He are presented and compared with available quantum mechanical calculations. 1. INTRODUCTION The last several years have seen an increasing use of classical trajectory calcula- tions in describing inelastic and reactive molecular collisions.l# The advantage of classical approaches is that the equations of motion can always be solved (at least numerically) without the necessity of introducing any dynamical approximations, whereas this is generally not the case for a quantum description. The shortcoming of a classical theory is, of course, that real molecules obey quantum rather than classical mechanics.The object of our research in recent years has been to show how exact classical dynamics (i.e., numerically computed trajectories) can be used as input to a general semiclassical theory.3* The principal physical idea is a natural extension of the Ford and Wheeler treatment of potential scattering, namely that one uses a quantum mechanical formulation of the scattering problem (so as to incorporate quantum superposition of probability amplitudes), but evaluates the dynamical parameters of the theory within the classical limit. For a collision between species which possess internal degrees of freedom, the result of this semiclassical theory is a prescription for how one uses classical mechanics to construct the classical-limit of S-matrix elements, the “classical S-matrix ”, which are simply the probability amplitudes for transitions between specific quantum states of the collision partners.In a number of examples it has been seen that this combination of “ classical dynamics plus quantum super- position ” accurately describes the quantum effects in molecular collisions. One of the most practically important aspects of classical S-matrix theory is the ability to analytically continue classical mechanics in such a way as to describe cZnssicaZZy forbidden processes, i.e., those which do not take place via ordinary classical dynamics ; Section 2 discusses such processes in detail. This paper describes a feature of the theory that is particularly useful when some of the internal degrees of freedom are very classical-like, but others are highly quantized.This is common in an atom-diatom collision, for example, where there is typically a large number of 4546 CLASSICAL S-MATRIX THEORY rotational states that are strongly coupled and may thus be treated by strictly classical methods, but only a few vibrational states are involved so that this degree of freedom is highly quantum-like. The " partial averaging " approach allows one to use a strictly classical Monte Carlo treatment of the classical-like degrees of freedom while the quantum-like degrees of freedom are quantized semiclassically-all without introducing any dynamical approximations into the theory. Preliminary results for vibrational excitation of H2 by He are presented in Section 3.2. SUMMARY OF THE THEORY A. CLASSICALLY ALLOWED TRANSITIONS All formulae in this paper will be written explicitly for nonreactive collisions of atom A and diatomic molecule BC. The cross section for collisional excitation of BC from initial vibration-rotation state (nl, j , ) to final state (n2, j2), summed and averaged over the m-components of the rotational states of BC, is given by where Sn2j2z2,nljlll(J, E) is the S-matrix element for the A+BC collision system ; El is the initial collision energy, kf = 2pE1/A2, E is the total energy, J is the total angular momentum, and 2 is the orbital angular momentum for relative motion of A and BC. The classical limit of a particular S-matrix element is given by where 4(n2j2Z2, nljlZl) is the classical action along the classical trajectory that is determined by the indicated double-ended boundary conditions (units are used such that PZ = 1); specifically, where (R,P) are the translational coordinates and momenta for radial motion of A and BC, and (qn, n) (qj,j), (qz, I ) are the action-angle variables for these degrees of freedom ; the initial and final values, (nljlZ,) and (n2 j2Z2), are required semiclassically to be integers.The pre-exponential factor in eqn (2) is the Jacobian relating the final values (n2j2Z2) to the conjugate initial values (qnl, qjl, qJ which lead to these specific final values ; i.e., with (nljlZl) fixed, one varies (qnlqjlqll) to cause (n2jJ2) to take on their desired final integer values.Typically, however, there is more than one classical trajectory that satisfies these double-ended boundary conditions; eqn (2) is then a sum of similar terms, one for each such trajectory. In forming the square modulus of the S-matrix element as it appears in eqn (1) interference terms thus result. In the co-linear A+BC collision these interference effects are quite prominent (and are accurately described by classical S-matrix theory), but it has been noted that the sums that occur in eqn (1) diminish their effect for a three dimensional A + BC system ; i.e., the interference terms are quenched. If the interference terms are neglected, then it is easy to see that eqn (1) and (2) giveW . H . MILLER A N D A . W . RACZKOWSKI 47 where it is assumed that enough integer values of J, ZI, and 1, contribute to justify replacing the sums over them by integrals.If many integer values of n, and j , are accessible from the initial state nljl, then it is also convenient to average eqn (4) over a quantum number increment about n2 and j , : where the limits of the j , and n, integral are the integers plus and minus 3. In eqn (9, one can now change integration variables from (n2j212) to (qnlq,lqzl) ; this elimin- ates the Jacobian from the integrand, giving J' d(qn1/2n) J' d(q j1/2n> J' d(q,1/2n)(l) (2.6) where the limits of the integral over qnl,qjl, and qzl are values such that the final values of n and j are in the increment (n, +$, n2 -3) and ( j , +*, j , -+), respectively. The simplest way to evaluate eqn (2.6) is to sweep qnl, qjl, and qzl through their complete domains (0,2n), putting the outcome of each trajectory into a quantum number " box " labeled by the closest integer value of the final values of n and j ; this is essentially what is done in a standard Monte-Carlo calculation for this type of quantity.If interference eflects are neglected, therefore, classical S-matrix theory for classically allowed processes reduces to standard Monte-Carlo methods. As an aside regarding the above strictly classical expression, it is actually more consistent if one also averages the cross section over a quantum number width of the initial quantum numbers n, and j , ; this gives the form of a complete phase space average over initial conditions ; the limits of the nl andj, integrals are the initial integer values plus and minus -5.This latter expres- sion, which treats initial and final states on an equal footing, satisfies microscopic reversi bility . B . CLASSICALLY FORBIDDEN TRANSITIONS In some cases there may be no classical trajectories (at the given energy) that con- nect the specific initial and final states (nljl) and (n&) ; the transition is then said to be classicdy forbidden which in practice simply means that the process is " weak ", i.e., has a small transition probability. Vibrationally inelastic transitions in low energy collisions of light diatomics with atoms are usually such processes. Other important examples of classically forbidden processes are tunnelling in reactive systems that have activation barriers and electronic transitions between different adiabatic electronic states .Although there are no ordinary classical trajectories that contribute to these48 CLASS~CAL S-MATRIX THEORY processes, it is in general possible to analytically continue classical mechanics and h d complex-vdued trajectories that do so. This can actually be accomplished by inte- grating the equations of motion with complex initial conditions and with a complex time variable. Along such complex-valued trajectories the action integral 4 is complex, so that the S-matrix element in eqa (2.2) has an exponential damping factor, exp( - Im4) ; classically forbidden processes are thus a generalization of the concept of tunneling in one-dimensional systems. Just as for classically allowed processes, there may be several different classical trajectories (complex-valued oms) that contribute to the specific S-matrix element in eqn (2.2) ; because of the sums that appear in eqn (2.l), however, it is still reasonable in most cases to disregard interference between these different trajectories. Further- more, even though the nl j , -tn& transition is classically forbidden for n, # n,, it will typically be true that many different j2 values have comparable probability. In such cases it is thus possible to average over a quantum number width ofj, as in Section 2A (replacing sums by integrals) but not far n2. Changing from integration over final values to integration over initial values as in Section 2A, thus gives where PnZcnl is essentially a one dimensional-like vibrational transition probability that depends parametrically on the initial conditions of the other degrees of freedom and is calculated by holding the initial conditions (Zljl, qllqjl) constant while qnl is varied to make n, equal to the desired integer value.The important practical advantage in this " partial averaging " scheme is that one must deal with double-ended boundary conditions (through a root search procedure) only for the vibrational degree of freedom, the one that is being quantized semiclassically, with the orbital and rotational degrees of freedom entering only through a phase space average over their initial conditions. The four dimensional integral in eqn (2.7) can now be evaluated by Monte Carlo methods. Furthermore, one can obtain all partial cross sections-i.e., the distri- bution in final rotational quantum number j2 and/or the distribution in scattering angle (the differential cross section)-in the usual Monte Carlo fashion by assigning the numerical value of the integrand in eqn (2.7) to the appropriate " box " labeled by j2 and scattering angle. In summary, the only approximations involved in eqn (2.7) and (2.8) beyond classical S-matrix theory itself are (i) neglect of interference between different traject- ories that lead to the same final values of Z2 and j,, and (ii) repIacement of sums over integer values of Z2 and j , by integrals. As has bem noted, the interference terms would essentially average to zero even if they were included, and one only needs a few integer values of l2 and j , to justify replacing the sums by integrals.The import- ant practical advantage of this partial averaging approach is that double-ended boundary conditions (and the related root search) are required only for the quantized degrees of freedom (i.e., vibration), while the other (unquantized) degrees of freedom enter only through a phase space average over their initial conditions.W. H , MILLER AND A . W. RACZKOWSKI 49 3. RESULTS FOR He+H2 COLLISIONS Calculations based on eqn (2.7) and (2.8) are being carried out for the He+H, collision system and its isotopic variants. The interaction potential is that of Gordon and Secrest,6 and the H2 potential is the accurate fit of Waech and Bernstein to the Kolos-Wolniewicz potential, Fig. 1 shows our preliminary results for the total 0- 1 vibrationally inelastic cross section as a function of the initial collision energy ; i.e., the quantity shown is 0 1 +oo(E,) = C 0 1 j z + O d E l ) * (3.1) i 2 Although these results are not final, they should be the correct semiclassical values to within at least a factor of 2.At 5 eV collision energy the O-, 1 transition is still classically forbidden ; i.e., with El = 5 eV, n, = j , = 0, there are no values of J, ZI, qzl, qjl for which a real-valued classical trajectory leads to n2 = 1. El lev FIG. 1.-The cross section for excitation of Hz (by collision with He) from its ground state (nl = jl = 0) to its first excited vibrational state (n2 = l), summed over all final rotational states as a function of initial collision energy ; the quantity shown is defined by eqn (3.1) and (2.1).The arrow indicates the energetic threshold for this transition. FIG. a t - \ 1 $ 4 1 0 1 \ \ -I - 0 2 4 G B j 2 2.-The distribution in final rotational states j 2 that accompany the O - t l vibrational in He + H2 collisions, u1 j z + O O , as defined by eqn (2.1). The initial collision energy is excitation 3 eV.50 CLASSICAL S-MATRIX THEORY Fremerey and Toennies have recently carried out coupled channel (i.e., quantum mechanical) calculations for this system and find a value of - 1.0 x A2 for the total 0+1 cross section at Eo = 1.09 eV. Within the uncertainty in our preliminary semiclassical results, therefore, there is excellent agreement with this quantum mechanical value. Fig. 2 shows the quantity ~ l j z + O O as a function ofj, at El = 3 eV collision energy ; i.e., this is the distribution in final rotational state that accompanies vibrational excitation from the ground state.Although j , = 0 is the single most probable final rotational state, there is a significant amount of rotational excitation which accom- panies the O+ 1 vibrational excitation. Further calculations for this system are in progress and more details of the calcula- tional procedure will be presented in a later report. This work has been supported by the National Science Foundation and the U.S. Atomic Energy Commission. Appreciation is also expressed to Dr. C. Sloan for helpful discussions. D. L. Bunker, Meth. Comp. Phys., 1971, 10,287. J. C. Polanyi, Acc. Chem. Res., 1972, 5, 161. (a) W. H. Miller, J. Chem. Phys., 1970, 53, 1949 ; (b) 1970, 53, 3578 ; (c) Chem. Phys. Letters, 1970,7,431; (d) J . Chem. Phys., 1971,54,5386 ; (e) C. C. Rankin and W. H. Miller, J. Chem. Phys., 1971,55,3150 ; (f) W. H. Miller and T. F. George, 1972,56,5668 ; (9) T. F. George and W. H. Miller, J. Chem. Phys., 1972,56,5722 ; 1972,57,2458 ; (h) J. D. Doll and W. H. Miller, J. Chem. Phys., 1972,57,5019. See also the related sequence of papers by Marcus et a!., (a) R. A. Marcus, Chem. Phys. Letters, 1970,7,525 ; (b) R. A. Marcus, J. Chem. Phys., 1971,54,3965 ; (c) J. N. L. Connor and R. A. Marcus, J. Chem. Phys., 1971,55, 5636 ; (d) W. H. Wong and R. A. Marcus, J. Chem. Phys., 1971,55,5636 ; (e) R. A. Marcus, J. Chem. Phys., 1972,56,311; (f) R. A. Marcus, J. Chem. Phys., 1972,56,3548 ; (9) J. Stine and R. A. Marcus, Chem. Phys. Letters, 1972,15, 536 ; (h) R. A. Marcus, J. Chem. Phys., 1972,57,4903. K. W. Ford and J. A. Wheeler, Ann. Phys. (N.Y.), 1959,7,259,287. M. D. Gordon and D. Secrest, J . Chem. Phys., 1970,52, 120. H. Fremerey and J. P. Toennies, private communication. ’ T. G. Waech and R. B. Bernstein, J. Chem. Phys., 1967,46,4905.