Semiclassical Effects in Heavy-Particle Collisions

ADVANCES IN ATOMIC AND MOLECULAR PHYSICS,

1

VOL. 14

SEMICLASSICAL EFFECTS IN HEAVY-PARTICLE COLLISIONS M . S . CHILD Department of Theoretical Chemistry University of Oxford Oxford, England

I. Introduction.. . . . . . .... ............... A. Experimental Bac ........................................ B. Theoretical Developments ........................................ C. Scattering in the Semiclassical Limit . . . . 11. Elastic Atom-Atom Scat A. Scattering Amplitude and Differential Cross Section. . . . . . . . . . . . . . . . . . B. Total Cross Section. . C. Semiclassical Inversio 111. Inelastic and Reactive Sc A. Integral Representations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B. Stationary Phase and Uniform Approximations ..................... ........... C. Classically Forbidden Events . . . . . . . . D. Numeridal Applications and Conclusio IV. Nonadiabatic Transitions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A. One-Dimensional Two-State Model. ......................... ...................... B. Inelastic Atom-Atom Scattering .......... C. Surface-Hopping Proc V. Summary . . . . . . . . . . . . . .......... ....... ............................. Refersnces

225 226 221 234

247 252 251 262 263 268 271 274 215

I. Introduction The past 15 years have seen major developments in the study of atomic and molecular scattering processes both from the experimental and theoretical points of view. The recent book by Levine and Bernstein (1974) offers a readable introduction. Among the most interesting of these has been the growing recognition of the semiclassical nature of the processes involved. This was first made apparent by Ford and Wheeler (1959a,b) in the case of elastic scattering, but its full significance has only recently been demonstrated by the work initiated by Pechukas (1969a,b), Miller (1970a,b), and Marcus 225 Copyright @ 1978 by Academic Press, Inc. All rights of reproduction in any form reserved. ISBN 0-12-003814-5

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M . S. Child

(1971).Subsequent developments have led to a coherent conceptual structure, the main elements of which appear sufficiently well established to justify the present review. There are, however, certain computational problems limiting general application of the theory, and it is also recognized that experimental conditions may lead to the averaging out of semiclassical effects in complex reactions. It may therefore be valuable first to give brief reviews of recent experimental developments, and of other important lines of theoretical research before turning to the main subject under review. A. EXPERIMENTAL BACKGROUND

One important class of experiments involves the scattering of molecular beams (Ross, 1966; Schlier, 1970; Fluendy and Lawley, 1973). Early limitations imposed by difficulties in detector design have been overcome by the development of high-intensity beam sources coupled with mass and spectroscopic analysis of the scattered particles. Atom-atom scattering crosssections derived in this way are illustrated in Figs. 5, 7, and 8, and analyzed in the text. Similar structure may also be evident in nonreactive atommolecule collisions, particularly if the reactant molecules are oriented by electric fields (Reuss, 1976).The chemically reactive scattering cross sections can seldom be resolved in such detail, but the measurement of product distributions as functions of velocity and scattering angle, from velocityselected reactants such as that illustrated in Fig. 1 is now possible for a large number of systems (Grice, 1976). This will be increasingly supplemented in the future by laser fluorescence analysis of the reaction products to provide information on the final internal energy distribution (Cruse el al., 1973; Pruett and Zare, 1976). Similar information on the product internal energy distribution may also be obtained by the infrared chemiluminescence techniques pioneered and largely developed by Polanyi for the study of very low-pressure gas reactions (see Polanyi and Schreiber, 1973, for a recent review). Figure 2 shows the type of information currently available by this technique (Ding et al., 1973). This shows product intensity contours as a function of final vibrational and rotational energy for two different reactant vibrational states (v = 0, 1). A number of systems studied in this way are now in use as commercial chemical lasers. The study of chemical laser intensities from one line to another also provides information on the relative populations of the product internal states (Berry, 1973). Another important application of laser technology has been to the study of energy transfer processes, particularly those involving the transfer of vibrational energy. Here one follows the quenching of either laser-induced fluorescenceor stimulated Raman scattering as a function of pressure (Moore,

SEMICLASSICAL HEAVY -PARTICLE COLLISIONS

90'

I

227

HCI from

0'

180 9 0'

FIG. 1. Contour maps of angle-velocity flux distributions in c.m. co-ordinates for the reaction product HX in the H + X, reactions. Direction of the incident hydrogen atom is designated 0 . [Taken from Herschbach (1973) with permission.]

1973; Bailey and Cruickshank, 1974; Lukasik and Ducuing, 1974). It is normally necessary to assume a thermal velocity distribution in the gas, but the accessible temperature range is considerably increased over that obtainable by shock tube and other more traditional techniques (Burnett and North, 1969). Studies of rotational relaxation leading to information on intermolecular anisotropies by spectral line broadening, molecular beam, and other methods (see Neilson and Gordon, 1973; Fitz and Marcus, 1973, 1975), have recently been supplemented by direct spectroscopic analysis of weakly bound Van der Waals complexes formed in the gas phase at artificially low translational temperatures (Klemperer, 1977). B. THEORETICAL DEVELOPMENTS The molecular scale of these events raises two types of problems for the theory. The first arises from the need to include quantum mechanical

IhI

’-4(k -0.097)

T;,-

300 K

FIG.2. Product internal state distributions for the H + C1, (ti = 0) and H + CI, ( L ‘ 2 I ) reactions. Contours give the measured rate constants as functions of the product rotational R’ and vibrational V ‘ energies. Note the bimodal character for u > 1. [Adapted from Ding rt al. (1973) with permission.]

SEMICLASSICAL HEAVY-PARTICLE COLLISIONS

229

effects in situations where the number of significant channels is large. For example, even the elastic scattering of two atoms may involve 100-loo0 significant partial waves, but the magnitude of the total cross section is determined by the uncertainty principle, and various readily observed interference effects, containing valuable information, can only be described quantum mechanically. The problem of the number of coupled channels becomes overwhelming for molecular collisions involving all but the lightest atoms. This is balanced to some extent by the averaging out of interference effects in the differential cross section, but interference might well remain significant in determining the vibrational distributions of chemical reaction products. The bimodal structure of Fig. 2b could be a case in point. There are also processes such as quantum-mechanical tunneling at the chemical reaction theshold and the transfer of vibrational and translational energy (Shin, 1976) under thermal conditions that are dynamically forbidden in classical theory and hence can only be accounted for by quantum mechanics. The second difficulty arises from the strength of chemical interactions compared with the relevant energy separations, and the resultant strong coupling between many channels. The cases of vibrational to translational energy transfer cited above and certain processes involving electronic energy or charge transfer are almost unique in being amenable to perturbation theory. The unifying concept in all approaches to these difficulties is the potential energy surface or, more accurately, the electronic energy surface in nuclear coordinate space visualized as being obtained by solution of the electronic Schrodinger equation within the Born-Oppenheim fixed nucleus approximation, although some processes of chemical interest involve nonadiabatic transitions from one surface to another (see Section IV). The present state of the theory is such that the principal qualitative features of at least the lowest energy surface can be reliably determined both for reactive (Baht-Kurti, 1976; Kuntz, 1976)and nonreactive (Gordon and Kim, 1972,1974)processes, but quantitative reliability can be expected only for systems involving the lightest atoms. An additional complication in the dynamical theory is therefore the need to employ a flexible functional form for the surface, consistent with the known qualitative form, which can be adjusted in order to bring the dynamical results into agreement with experiment. For the reasons given above, this scheme is feasible at present only within the realm of classical mechanics. Such Monte Carlo classical trajectory calculations have been the most important single aid to interpretation of chemical reactions studied by the molecular beam and infrared chemiluminescence techniques (Bunker, 1970; Polanyi and Schreiber, 1973; Porter and Raff, 1976). The other main lines of theoretical development bear on the question as to whether such a purely classical treatment can be justified. At one extreme

230

M . S. Child

there have been a number of accurate numerical solutions of the exact close-coupled quantum-mechanical equations for a variety of realistic model systems. The numerical techniques are discussed by Lester (1976). These benchmark studies relate in order of complexity to elastic-scattering phase shifts (Bernstein, 1960), the collisional excitation of harmonic and Morse oscillators (Secrest and Johnson, 1966; Clark and Dickinson, 1973), the scattering of rigid rotors (Shafer and Gordon, 1973; Lester and Schaefer, 1973), and more recently a spate of calculations on reactive systems, which have been reviewed by Micha (1976a). The latter are complicated by the necessity for a coordinate transformation from the reactants to products frame during the calculation, which raises acute problems in any full threedimensional study. At the time of writing, the only three-dimensional reactive calculations including both rotational and vibrational open channels have been for the H + H, reaction (Elkowitz and Wyatt 1975a,b; Schatz and Kuppermann, 1976). The most serious general complication in these exact calculations is the strong coupling between angular momenta associated with the internal and relative motions. Attention has therefore been concentrated on the development of decoupling schemes to reduce the number of coupled channels without serious loss of accuracy (McGuire and Kouri, 1974; Pack, 1974; De Pristo and Alexander, 1975, 1976).The spirit of this approach is similar to that which inspired Hunds’ cases in diatomic spectroscopy (Herzberg, 1950).Another general trend has been to reduce the labor of the calculation by the use of exponential approximations to the S matrix (Pechukas and Light, 1966; Levine, 1971; Balint-Kurti and Levine, 1970).The effort is little more than that required for a distorted wave perturbation calculation (Child, 1974a),but the unitarity of the S matrix is preserved. The sudden approximation (Bernstein and Kramer, 1966) is the simplest member of this family. A third device, borrowed from nuclear physics, is to introduce an imaginary “optical” term into the potential to suppress the need for calculation of quantities irrelevant to the process under investigation (Micha, 1976b). Finally there are several methods included under the general heading “semiclassical” that seek to retain the computational simplicity of classical mechanics without losing any essential quantum-mechanical characteristics. Conceptually the most interesting of these, to which the major part of this review is devoted, is the semiclassical S matrix method, stemming from the Feynmann path integral approach to quantum mechanics (Feynmann and Hibbs, 1965), as developed in the present context by Ford and Wheeler (1959a,b), Pechukas (1969a,b), Miller (1970a,b), and Marcus (1971). This differs from classical mechanics only by inclusion of a phase determined by the classical action (Goldstein, 1950) for the trajectory in question, and the development of special techniques to handle the resulting interference pat-

SEMICLASSICAL HEAVY-PARTICLE COLLISIONS

23 1

tern, which depend on the topological structure of the caustics of the classical motion (Connor, 1976a; Berry, 1976). The reader is referred to reviews by Berry and Mount (1972), Miller (1974, 1976b), Connor (1976b), and Child (1976a), which complement the account given here. A much older “classical p a t h procedure whereby the relative motion is assumed to follow a known mean classical trajectory while the internal motion is treated by quantum mechanics has recently been examined by Bates and Crothers (1970)and Delos et a/.(1972).This has the computational advantage of separating the internal from the orbital angular momentum and of reducing the equations of motion to a time-dependent form, with only one first-order equation for each channel. Use of this method is, however, restricted to situations in which the changes in the translational energy and angular momentum are small compared with their absolute values. Intermediate between these two philosophies there is a method due to Percival and Richards (1970) derived from the correspondence principle. Here the quantum-mechanical matrix elements appearing in the classical path equations are replaced by Fourier transforms taken over a fixed mean classical orbit for each internal motion in question. Thus the dynamical motion is again purely classical, but two mean trajectories, one for the internal and one for the relative motion, appear in place of the exact trajectories of the classical S matrix. The justification for this procedure lies in the use of classical perturbation theory, which may be particularly applicable to rotational energy transfer. Details of the method have been reviewed by Clark et a/. (1977). This completes the present brief review of developments in the dynamical theory. One other important but quite different, type of analysis, discussed at length by Levine and Bernstein (1976),concerns the information content of any given calculation or experiment, and the relation between statistical and dynamical behavior. The argument is that on purely statistical grounds the outcome of any event may be predicted from knowledge of the distribution of accessible phase space in the products region. Deviations from this distribution may therefore be attributed to dynamical considerations. Experience shows that these deviations may frequently be characterized by a small number of so-called surprisal parameters, which constitute the dynamical information content of the process. Similar arguments have also been used to extend the results of collinear collision calculations to threedimensional space. Detailed coverage of these developments may be found in the books by Levine (1969),Nikitin (1970), Eyring et a/. (1974, 1975),Fluendy and Lawley (1973),Child (1974a), and Miller (1976~). There are also a number of valuable review volumes covering both experimental and theoretical developments edited by Ross (1966), Hartmann (1968), Schlier (1970), Takayanagi (1973),

232

M . S. Child

and Lawley (1976). A number of recent reviews of the molecular collision theory literature by Levine (1972), Secrest (1973), George and Ross (1973), and Connor (1973e),may also be cited. The aim of the present report is to follow developments in the semiclassical S matrix version of the theory in relation to experimental measurements and exact quantum-mechanical calculations, where these are available. This will serve to emphasize the close interplay between classical and quantummechanical behavior required in the analysis of modern experiments. The semiclassical S matrix theory itself currently has some computational disadvantages, but its underlying philosophy has been of overriding importance in clarifying the nature of molecular collision processes. C. SCATTERING IN THE SEMICLASSICAL LIMIT Modern applications of semiclassical methods to heavy-particle scattering date from the work of Ford and Wheeler (1959a,b). The theory has been reviewed in detail by Berry and Mount (1972) in the context of elastic scattering and more generally by Miller (1974,1976b)and Child (1976a).The achievement has been to obtain quantum-mechanically accurate transition probabilities and collision cross sections by integrating the classical equations of motion. An obvious, but not necessary, starting point is the Feynmann path integral formulation (Feynmann and Hibbs, 1965), according to which the scattering amplitude may be represented as an integral over all possible phase-weighted classical trajectories relevant to the experiment in question, with the phase expressed in terms of the classical action (Goldstein, 1950). Recent progress lies in methods for the evaluation of this integral. The stationary phase (or saddle point) approximation yields a sum over the particular trajectories leading from the desired initial to the desired final state of the system. This “primitive semiclassical’’ approximation is adequate to account for most simple interference effects, but problems arise at the caustics or thresholds of the classical motion due to coalescence of two or more trajectories. This leads to divergence of the primitive semiclassical approximation, but the topology of the caustics obtained may be used to suggest a suitable mapping for a uniform evaluation of the integral by the methods of Chester et al. (1957), Friedman (1959), and Ursell (1965, 1972). The theory is particularly well developed for caustics with the structure of one or other of Thom’s (1969)elementary catastrophes (Berry, 1976; Connor, 1976a), but other situations can also be accommodated (Berry, 1969; Stine and Marcus, 1972; Child and Hunt, 1977). The difference between these uniform results and the primitive semiclassical approximations lies in the use of special rather than trigonometric functions to handle the interference,

SEMICLASSICAL HEAVY-PARTICLE COLLISIONS

233

but in every case the relevant classical information is derived from the particular initial- to final-state “stationary phase trajectories” for the problem in hand. The significance of these special trajectories has been further underlined by detailed investigation of classically forbidden events characterized by stationary phase points that are complex. The discovery of corresponding complex trajectories, obtained by integrating Hamilton’s equations from complex starting conditions along a complex time path (Kotova et al., 1968; Miller and George, 1972a,b, Stine and Marcus, 1972), has extended the concept of quantum-mechanical tunneling from coordinate to quantum number space. This is particularly important for the theory of molecular energy transfer because single vibrational excitations from the thermally occupied levels are dynamically forbidden by classical theory for almost all molecules at normal temperatures, in the sense that the classical maximum energy transfer is less than a vibrational quantum. Developments in the semiclassical theory of electronic energy transfer (nonadiabatic transitions) has also led to a complex classical trajectory treatment of systems with several degrees of freedom for the heavy particle motion. The intention of the following sections is to illustrate the main features of the theory in comparison, where possible, with available experimental results. The fundamental concepts are outlined in Section I1 by application to the theory of purely elastic scattering. The reader is referred for more detailed coverage to important recent reviews by Pauly and Toennies (1965, 1968), Bernstein (1966), Bernstein and Muckermann (1967), Schlier (1969), Beck (1970), Toennies (1974a), Pauly (1974), and Buck (1976). Section I11 describes extensions of the theory to cover molecular energy transfer and chemical reactivity, initiated by Pechukas (1969a,b), Miller (1970a,b), and Marcus (1970,1971). Here there is less scope for comparison with experiment although the measurement of inelastic differential cross sections is now becoming possible for favorable systems (Toennies, 1974a). Finally, developments in the theory of nonadiabatic transitions are discussed in Section IV. Related experimental measurements have recently been reviewed by Kempter (1976) and Baede (1976). Nikitin (1968), Crothers (1971),Delos and Thorson (1972), and Child (1974a) review in detail the most important theoretical lines of development.

11. Elastic Atom-Atom

Scattering

The semiclassical theory of purely elastic scattering is very fully developed. The general techniques are illustrated below by application to the scattering amplitude and the differential cross section. Short accounts are also given

234

M . S. Child

of the theory of the total elastic cross section, and of semiclassical techniques for direct inversion of experimental data to recover the scattering potential.

A. SCATTERING AMPLITUDE AND DIFFERENTIAL CROSSSECTION The theory relies on reduction of the standard expression for the scattering amplitude

by the methods of Ford and Wheeler (1959a,b), as extended by Berry (1966, 1969). It is assumed, unless otherwise stated, that the energy lies above the limit for classical orbiting (Child, 1974a).The first step is to use the Poisson sum formula to replace the sum by a combination of integrals,

jM(e) =

(ikl-1

Jox

1[exp(2iq,-

-

11 exp(2i~271)~,_,~,(cos0)di,(3)

where I is related in the semiclassical limit to the classical impact parameter b by the identity I = 1 + 3 = kb (4) The detailed semiclassical analysis relies on introduction of the WKB phase shift, the accuracy of which is well attested (Bernstein, 1960), and the following asymptotic approximation for PA- 1/2(cosO), valid for 2 sin 6 >> 1 :

P A - 1,2(cos0)

- (2/nI sin

Q ) l l 2 sin(i8

+ n/4)

(6)

Here k 2 = 2,uE/h2 and a denotes the classical turning point. Thus for angles at which Eq. (6) is valid,

f(O) = (ik)-'(27~sind)-''~

z

1

M= -

[I,&(6) - ~ , ( e ) ] e x p ( - i ~ ~ ) (7) ~3

where I,.$(@

= JoX

A112exp{i[2q(A)+ 2 M h k 20 i-x/4]}dA

(8)

in which either the upper or the lower signs are to be taken together. Equation (8) displays the two main semiclassical characteristics. The integration over I corresponds, according to Eq. (4), to an integral over all

SEMICLASSICAL HEAVY-PARTICLE COLLISIONS

235

relevant classical trajectories, distinguished according to impact parameter, and the exponent in the integrand is determined by the quantity 2q(i) k id3, which may be identified with the classical action integral (Child, 1974a); the ambiguity of sign arises from the conventional restrictions

O
0<0<7L

I . Stationary Phase (Primitive, Semiclassical) Approximation

Under the normal molecular scattering conditions q(i.) is large and strongly dependent on I , and the A values of central significance for the theory are those at which the stationary phase condition is satisfied, which may be seen to specify precisely those trajectories for which the classical deflection x(i) gives rise to the same scattering angle 8 = Ixl(mod 2n), given the semiclassical identity derived from Eq. (9,

This intimate connection between the phase shift and classical deflection is illustrated in Fig. 3 for a typical molecular system. Scattering experiments are normally performed on systems for which the interatomic well depth E is comparable with thermal energies. Otherwise, spectroscopic investigation is usually more convenient ;even the vibrationalrotational spectrum of Ar2 has recently been determined (Tanaka and Yoshino, 1970; Colbourn and Douglas, 1976).This means that in the normal experiment the rainbow angle xr in Fig. 3 satisfies xr < -71, so that the stationary phase condition (9) can be satisfied for real values of 2 only for M = 0, when there are typically three branches to the solution of Eq. (9), as illustrated in the lowest part of Fig. 3. A simple stationary phase evaluation of the integral for I&(@ at these points yields

f(4=

c

v=o,b,c

f,(@exp[irv(f3)1

(11 )

where fy(8) is the square root of the corresponding classical differential cross section f,(Q) = ( d ~ / d f i ) ;=~(bldb/dxl/sin ~ f3)'j2

(12)

and ~ " ( 0is)the phase of the integrand in Eq. (8)at L = A,. This triple-branched interference accounts for much of the structure observed in the differential cross section do/dQ = shown in Fig. 4, but corrections are required as 0 + 0 and 8 71 due to the breakdown of Eq. (6), and at angles close to the rainbow angle f3 N lxrl due to coalescence of the branches &(8) and --f

M . S. Child

236

I

I I I I I I I

I I

I

I I I I

I

I

I

I

I I I

I I

I

I

I

FIG.3. The semiclassical connection between the phase shift q, and classical deflection 0, for a Lennard-Jones potential. I, and I, denote the rainbow and glory angular momenta, respectively. Note the existence of the three I values, l,, I,, and lc, having a common scattering [Taken from Child (1974a) with permission.] angle 0 =

&(0). This invalidates the stationary phase approximation and leads to divergence of the a and b branch contributions to f ( 0 ) because Idb/dxI -+ co as x x r . +

2. Uniform Approximation

This behavior at the classical rainbow singularity is characteristic of the simplest (fold) catastrophe in Thorn’s (1969) classification. The proper

SEMICLASSICAL HEAVY-PARTICLE COLLISIONS

237

8 (dcg) FIG.4. Semiclassical interference structure in the elastic differential cross section. The

rainbow angle, which is shown by the outermost point of inflection, lies at approximately 41’. [Taken from Berry and Mount (1972)with permission.] (Copyright by The Institute of Physics.)

uniform mapping is onto an Airy function (Berry, 1966). This is achieved by employing the equation 5.3

-

c(e)x+ y , ( q

= 2rl(4

+ A e + n/4

(13)

to define a variable transformation ,i(.~) over the a and b branch regions of E., with the parameters [ ( O ) and y,(d) determined by the requirement that the stationary phase points on the two sides of Eq. (13) should be in one-toone correspondence. This implies that c(6) and y,(6) may be expressed in terms of the same two stationary phase exponents ya(@ and y b ( 6 ) as those which appear in the primitive semiclassical approximation (11): Yr(O)

= +[Yb(O) =

{$[?b(s)

+

(14)

- ra(e)]}”3

(15)

Transformation to x as the integration variable on the right-hand side of Eq. (8) then yields (Berry, 1966)

f ( @ = L(Q)~

+ f,(d)~

X Ci~r(d)] P

XCiycCe)] P

where

+ f b ( e ) ] Ai[-l(0)]

= n”2(c”4(0)[fa(@)

+ i<-1’4(o)[h(o) - fb(@]

Ai’[-c(0)])

(16)

238

M . S . Child

and Ai( -x) and Ai’( - x) denote the Airy function and its first derivative (Abramowitz and Stegun, 1965). It is readily verified by use of standard asymptotic approximations for large [(O) that Eq. (16) reduces exactly to Eq. (1l), so that Eq. (16) may be taken to cover the entire intermediate angle region. The accuracy of this uniform Airy approximation is displayed in Fig. 4. The most important features are:

(1) The location of the rainbow angle, given by the outermost point of inflection, which depends on the ratio of the collision energy to the potential well depth. (2) The period of the slow oscillations (supernumerary rainbows), which arises from interference between the a and b branches of the deflection function. These therefore depend on the path difference between a and b type classical trajectories at the same scattering angle and on the de Broglie wavelength. The result is that the spacing between the supernumerary rainbows depends inversely on the product of the range of the potential and the square root of the reduced mass. (3) The period of the rapid oscillations, which arises from interference between the rainbow structure and the repulsive c branch component of the scattering amplitude. Extensions of the theory to cover the small-angle region where Eq. (6) becomes invalid are discussed by Bernstein (1966), Berry (1969),and Child (1974a). The above analysis applies to the scattering of heavy particles at energies above the orbiting limit, below which the occurrence of shape resonances can give rise to additional complications as qualitatively discussed by Buck (1976);the general theory in relation to the spectroscopic implications of these orbiting resonances has been reviewed by Child (1974b).

3. Diflraction Oscillations The scattering of light atoms raises special problems due to the small number of significant terms in Eq. (1).The dominant feature of the differential cross section is a series of roughly equally spaced “diffraction oscillations” bearing no relations to the classical rainbow angle as illustrated in Fig. 5 [see also Chen et al. (1973) for experimental differential cross sections for the scattering of helium on other inert gases]. These oscillations, which are observed even for a purely repulsive potential, have been attributed by Zahr and Miller (1975) to interference between the normal repulsive branch of the classical deflection function and other classically forbidden branches (arising from complex points of stationary phase), but no full semiclassical theory has been developed.

SEMICLASSICAL HEAVY-PARTICLE COLLISIONS

239

FIG. 5. Diffraction oscillations in the helium and neon differential scattering cross section. The persistence of regular slow oscillations at high angles cannot be accommodated within the semiclassical approximation. [Adapted from Chen et crl. (1973).]

4. Symmetry Oscillations

A final additional source of quantum oscillations arises from the exchange symmetry of identical particles, for which the scattering amplitude must be symmetrized or antisymmetrized for bosons and fermions, respectively, by the equation

f*(4= .f(4 f f ( .

-

4

(17)

The differential cross section is therefore necessarily symmetric in the forward and backward directions, and interference may arise between all branches off(0) with those off(71 - 0). The spacing of the resulting symmetry oscillations is again inversely dependent on the reduced mass, and so they are most readily observed in the scattering of light atoms. Several examples are discussed by Buck (1976). B. TOTALCROSSSECTION The theory of the total cross section (Bernstein, 1966: Berry, 1969) is based on the optical theorem, 471 o ( E ) = - Imf(0) k

=

471 k

1 (21 + I)sin2ql

1=0

The lower part of Fig. 3 shows that there are two contributions to the classical scattering at 6 = 0: one from large-impact parameters and the other from angular momenta close to Ig at which the b and c branches of the

240

M . S . Child

-

deflection function coalesce. Of these the former, o,(E) is dominant. Bernstein (1966) shows for an asymptotically inverse power potential V(r) - C(')/f, that the strong dependence of the high-angular-momentum semiclassical phase shifts, Vl

-

+ 51 )1 - s

(19)

implies that the o,(E) cross section may be estimated as .nb*', where b* is the impact parameter [b* = (I* + 3)/k] at which qI falls below 0.5, in other words, at which the classical action falls below 0.5h. In this approximation, (20) , p ( s ) is a pure number where o is the collision velocity, o = ( ~ E / P ) " ~and [ p ( s ) = 8.083 for s = 6, for example]. Corrections are, however, required at very high energies because the limiting phase shift no longer belongs to the long-range scattering. o,(E)

= ~(S)[C"'/AV]~""')

1. Glory Oscillations

The second, glory contribution to o(E)may be evaluated by applying the stationary phase approximation to Eq. (18),because by the semiclassical correspondence relation (10) between the derivative of the phase shift and the scattering angle, the phase shift must pass through a maximum q,(E) at the glory angular momentum I,. This phase is therefore reflected in the relative phases of the two contributions to the total cross section (Bernstein, 1966): o ( E ) = ~(s)[C(')/AU]'/('- - (4n/k)f,(0)cos[2qg(E)- 3 ~ / 4 ]

(21)

where IfJ0)l has the same classical interpretation as lf,(O)I defined by Eq. (12). Experimental consequences of this general behavior are illustrated in Fig. 6. The velocity dependence of the first term in this expression provides the best experimental information on coefficient of the long-range attractive part of the potential. The total number of glory oscillations arising from the second term may be shown to count the number of rotationless bound states supported by the potential (Bernstein, 1966), and the spacing of these oscillations contains information on the product of the well depth and potential range parameter (Bernstein and La Budde, 1973, Greene and Mason, 1972,1973; Buck, 1976). 2. Symmetry Oscillations

Additional quantum oscillations in the total cross section may be caused by orbiting resonances, as observed by Schutte et al. (1972; see also Child,

SEMICLASSICAL HEAVY-PARTICLE COLLISIONS

24 1

10’

30 qv04

tt

3 2

I

0

I

0

005

-

I

I

I

I

0.10

015

(KIB)’”

0.20

0.25

FIG. 6 . Glory oscillations and orbiting resonance structure in the total cross section for the Lennard-Jones potential, V ( r )= - 4 ~ [ ( u / r ) ” - ( ~ / r ) ~at] , collision energy E , with K = E / E , A = ( 2 p E ) ’ ’o/h, B = 2puZ/hZ.[Taken from Buck (1976) with permission.] (Copyright by John Wiley & Sons, Ltd.)

1974b), and by exchange effects in the scattering of identical particles. The latter are relevant to the semiclassical inversion methods discussed below in providing experimental information on the energy dependence of the s-wave phase shift qo(E). The theory given by Helbing (1968) recognizes that the necessary symmetrization or antisymmetrization of the scattering amplitude [see Eq. (17)] introduces the possibility of interference with the backward glories at 8 = 71. Furthermore, any repulsive core potential always leads to backward scattering 8 = 71 at 1 = 0, and by the semiclassical correspondence relation (10) the phase shift curve must have a derivative of 71/2 at this point: q,(E) = qo(E)

+

$711 -

ic12 + . . .

(22)

Finally, 1 must change by 2 between successive partial waves because only even or odd 1 values contribute according to whether one applies Bose or Fermi-Dirac statistics. Hence, since the phase appears in (1) as exp(2iq,), successive terms combine exactly in phase apart from corrections due to the term d2in Eq. (22). It follows that the backward glory contribution to symmetrized or antisymmetrized total cross section also depends on this phase (Helbing, 1968),

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M . S. Child

FIG. 7. Symmetry oscillations in the total helium-helium wattering cross sections. [Taken from Buck (1976) with permission.] (Copyright by John Wiley & Sons, Ltd.)

where tan4

=

[I

+2/(7c~)~’~]-~

(24)

and the upper and lower signs in Eq. (23) apply to Bose and Fermi statistics, respectively. This means that the symmetry oscillations for two isotopes obeying opposite statistics will be exactly out of phase, and that the positions of the maxima and minima may be used to determine the energy dependence of the 1 = 0, s-wave phase shift q,(E). This procedure has been applied to the scattering of two helium atoms, for which the experimental results (Feltgen et al., 1973) are shown in Fig. 7. There are no normal glory oscillations in this diagram because the potential well depth for He, is too small.

C. SEMICLASSICAL INVERSION PROCEDURES I . Firsov Inversion

The most complete semiclassical inversion procedure is that which makes use of the variation in the phase shift q ( A ) or classical deflection x([) derived from the differential cross section. In either case the radial integration variable in Eq. (5) is replaced by a new variable x, defined by the equation

x ( . )

= r2[l - V ( r ) / E ]

(25)

SEMICLASSICAL HEAVY-PARTICLE COLLISIONS

243

and the classical impact parameter b = A/k is introduced in place of A. Hence, for example, Eq. (10) may be written

6:

b(dy/dx)(x - b2)-l/’ dx

(26)

y(x) = h(x/r2) = In[l - V ( r) /E ]

(27)

X(b) = where

Equation (26) is then inverted by an abelian transformation (Child, 1974a) to determine the function y(x) in the form

so that according to Eq. (27),

The inverse function x ( r ) determines the potential by means of Eq. (25). This classical procedure due to Firsov (1953) requires that the energy should be above the orbiting limit; otherwise dy/dx diverges over the integration range and the abelian transformation is invalid. A variant applied by Sabatier (1965), Vollmer and Kruger (1968), and Vollmer (1969) employs twice the derivative of the phase shift in place of ~ ( bin) Eqs. (28) and (29), as justified by Eq. (10). The extraction of the deflection function ~ ( l or ) the phase shift y(l) from the differential cross section presents some difficulty. The most satisfactory procedure due to Buck (1971) consists in optimizing the parameters in three separate piecewise approximations to the deflection function over the small-l, rainbow, and large-l regions, respectively. Application of this method to the high-quality data now available for mercury alkali systems (see Fig. 8) proves that the inverted potential is quite stable over a moderate range of collision energies, as shown in Fig. 9. Other methods related to parameterization of the phase shift have been suggested by Vollmer (1969), Remler (1971), and Klingbeil (1972). 2. Inversion of the s- Wave Phase Shift

Partial information about the potential may also be extracted from knowledge of the energy variation of the s-wave phase shift r-+ r

[l

-

V(r)/E]’/’ dr - kr

+4 4

M . S . Child

244 25

\

20

i

15

I5

10

5

a

lo

tn,

IS

10

5

2:

2(

l!

/d

!

l

5

10

15

20

25

30

e

35

40

FIG.8. Measured c.m. differential cross sections for the scattering of sodium and mercury. [Taken from Buck and Pauly (1970) with permission.]

SEMICLASSICAL HEAVY-PARTICLE COLLISIONS

245

t 7 5

‘4

6

7

8

9

10

FIG.9. The Na-Hg potential obtained by inversion of the data in Fig. 8. Data are superimposed for the five different energies by use of the following symbols: 0,E = 2.87 x ergs; 0, E = 3.01 x ergs; A, E = 3.45 x E= ergs; 0, ergs; +, E = 3.13 x 4.02 x ergs. The solid line indicates the Lennard-Jones potential with the same potential minimum. [Taken from Buck and Pauly (1970) with permission.]

300

L

-c;

I

I

I

-

-

3I

2.0

I

2.5

I

3.0

rrk FIG. 10. The He4-He4 potential. Circles denote points obtained by inversion of the data in Fig. 7. The other curves are obtained by molecular beam scattering data: -, Farrar and Lee (1972); ---, Bennewitz et al. (1972); x x x , Gegenbach et a/. (1973); and by gaseous properties: -.-, Beck (1968). [Taken from Buck (1974) with permission.]

246

M . S. Child

which may be seen as a natural extension of the semiclassical quantum number derived from the Bohr quantization condition k n ( E )= -

jb[l - V(r)/E]1’2- 21 ~

7 1 0

An abelian transformation of Eq. (30) similar to that applied to Eq. (21) yields (Miller, 1969, 1971; Feltgen et al., 1973; Buck, 1974) the following expression for the energy dependence of the classical turning point a( U ) on the positive-energy part of the repulsive branch of the potential:

(32)

A comparison between the repulsive branch of the He, potential derived in this way from the symmetry oscillations shown in Fig. 5 (Feltgen et al., 1973) and that derived from the differential cross section (Farrar and Lee, 1972) is illustrated in Fig. 10.

111. Inelastic and Reactive Scattering The semiclassical treatment of inelastic and reactive scattering based on knowledge of exact classical trajectories is due to Pechukas (1969a,b), Miller (1970a,b), and Marcus (1970, 1971). The overall structure of the theory is similar to the Ford and Wheeler (1959a,b) analysis of elastic scattering. The first step is to represent the physical observable, in this case the S matrix, as an integral over a continuous family of classical trajectories. Marcus (1971) does this by exploiting the well-known connection between the Schrodinger and Hamilton-Jacobi equations (Born, 1960; Maslov, 1972) to obtain a generalized WKB form for the multidimensional wavefunction. This line has been reviewed in detail elsewhere (Child, 1974a, 1976a). The development that follows, due to Pechukas (1969a,b) and Miller (1970a,b),starts from the semiclassical limit of the Feynmann propagator (Feynmann and Hibbs, 1965). Other ramifications of this approach have been reviewed by Miller (1974, 1976b). The second step in the argument is to apply stationary phase and uniform approximation techniques to the reduction of the above integral to a form dependent only on the particular n, + n2 trajectories passing from the desired initial to the desired final state of the system. The techniques employed also show that the classical thresholds for a given process play the part of the caustics in the theory of elastic scattering, and more surprisingly that a

SEMICLASSICAL HEAVY-PARTICLE COLLISIONS

247

semiclassical description of classically forbidden events beyond these caustics may be obtained by analytical continuation of the classical equations of motion into complex time and coordinate domains. Examples given in Section II1,C show that this semiclassical description is remarkably accurate even for the most strongly forbidden classical events.

A.

~NTEGRALREPRESENTATIONS

The Feynmann propagator, denoted (qzlexp[-iff(t2

-

tl)/hllq,)

by Miller (1970a), which governs the quantum-mechanical evolution of a system from coordinatesq, at time t, to q, at time t,, is defined (Feynmann and Hibbs, 1965) by the equation (qzlexp[-iff(t,

-

~ l ) / h l ) q=J Jexp{iA[dt)l/hl d Path

(33)

where the integral is taken over all pathsq(t) leading from (qltl) to (q2t2),and A[q(t)] is the corresponding action A[q(t)l

=

1:'L[q(t), Mt),tl d t

(34)

with L[q(t), q(t), t] the classical Lagrangian (Goldstein, 1950), evaluated along the path q(t). This prescription is exact. The semiclassical reduction of Eq. (33) relies on the argument that the exponent A[q(t)]/h will typically be large under semiclassical conditions and sensitive to the path. Hence the dominant contribution to the integral will come from those paths around which the action is stationary. By Hamilton's principle (Goldstein, 1950) these are, of course, the classical paths from (qltl) to (q2t2).It is assumed for the sake of simplicity, that there is only one such path, in which case Eq. (33) reduces to
- tl

)lhllq 1 )

where S(q,t, ;q l t l ) is Hamilton's principal function obtained most frequently by integrating in Eq. (34) along the family of classical trajectories, although S(q2t2;q l t l ) may in principle also be derived by solving the HamiltonJacobi equation H [ V s,q]

+ (?S/c?t)= 0

(36)

where H(p, q) denotes the classical hamiltonian. The normalizing preexponent, containing the Van Vleck (1 928) determinant of second derivatives

248

M . S. Child

l?'S/dq, dq21, is chosen to be consistent with the unitarity of the propagator in the semiclassical limit (Fock, 1959; Miller, 1970a, 1974; Child, 1976a). The semiclassical Smatrix is derived by taking the limits t , + - 00, t2 + co, and contracting the above form for the propagator between the initial and final states +,hi)

= (qilni),

(37)

to obtain Sn,n, = J-mx

J% :

x ex~[iS(q2,ql)/hl+n,(q,) dq1 dq2

- t1)/h1lq1)(q11n2>dq1dq2

(38)

Here S(q2,4,) is used to denote the internal part of S(q2, t 2 ;q l , t , ) at times t , -+ - co, t2 + m, and it is assumed that the necessary integration over translational variables has been performed. For the sake of simplicity the argument will be developed in more detail only for a system with one internal degree of freedom, described by the Cartesian variables (p, 4). One further obvious step to complete the semiclassical description is to employ normalized WKB wave functions for the internal states (Landau and Lifshitz, 1965)

in which w is the local vibrational frequency. This reduces the S matrix element to a combination of four simpler terms:

SEMICLASSICAL HEAVY-PARTICLE COLLISIONS

249

A complete analytical description of the forced harmonic oscillator has recently been given (Pechukas and Child, 1976) in this Cartesian representation, but almost all numerical applications of the theory have employed angle action variables ( N , 8) (Goldstein, 1950)to describe the internal motion. These have the advantages that the action is closely related to the quantum number n; by the Bohr-Sommerfeld quantization condition N

= Qpdq

=

(. +

;)ll

(44)

for any oscillator example. Furthermore, since the action is a constant of the motion for the isolated system, the corresponding operator fi commutes with quantum hamiltonian E?. Hence the eigenfunctions of fl, namely,

(8ln) = $,(e)

= (271)- '/'ein'

(45)

are also eigenfunctions of fi. Transformation of the Cartesian wavefunction to this angle action form is traditionally achieved by use of the semiclassical unitary transform (Fock, 1959; Miller, 1970a; Marcus, 1973)

based on the generator F,(qd) of the corresponding classical transformation, which is designed to satisfy the equations (Goldstein, 1950). awaq

= p(4,

e),

=,/ad

=

-m, e)

(47)

where p ( q , d ) and N(q,8) are the momentum and action consistent with coordinate q and angle 8. Closer analysis (Child, 1976a) of the origin of Eq. (46) shows that an additional term -if3/2 should be included in the exponent. Equation (46) implies the following integral representation for the Cartesian wave function: =

<+> SoZ" =

<4p><+>

which is seen to bear an obvious relation to the previous WKB form when Eqs. (47) are integrated to yield Fl(4, 0) =

[P ( 4 , d ) 4

-

W q ,w 4

(49)

the first term being integrated along a line of constant N ( q , U). The precise connection is obtained by combining the result of stationary phase integration of Eq. (48) with its complex conjugate, because Eq. (45) erroneously

M . S . Child

250

implies the existence of two independent wave functions, to describe a bound one-dimensional motion. An angle rather than a Cartesian coordinate integral representation for the S matrix may now be obtained by substituting the above form for i,hn(q) in Eq. (38), performing the integrations over q1 and q2 by the stationary phase approximation. Considerable care is required in the analysis. The argument hinges on the fact that the principal function S ( q 2 ,q l ) itself plays the part of a classical generator of the dynamical transformation from initial ( p l ,ql) to final ( p 2 ,q 2 ) Cartesian variables, in the sense that W&l1= -Pl(q2,ql),

(50)

dS/&l2 = PZ(q2,ql)

where pl(q2,ql), for example, denotes the initial momentum consistent with a classical trajectory between coordinates q1 to q 2 , and similarly for the final momentum p 2 ( q 2 ,ql). The transformation will not be followed in detail, but it may be illuminating to describe the first step, which is to identify the stationary phase values of q1 and q2 as solutions of the equations

(as/%,)+ (aFl/dql) = 0,

(dSldq2)

-

(dFl/%,)

=0

(51)

or, according to Eqs. (47) and (50), -Pl(qZ,ql)

+ P(q1,81) = O,

PZ(q2,ql)

-

P(q2,82) =

(52)

These identities require that q1 and q2 should be chosen such that the initial and final momenta pl(q2,ql) and p 2 ( q 2 ,q l ) along the trajectory should be consistent with q1 and the set angle el, and q2 and the set angle 8,, respectively. The required final result is obtained by performing a quadratic expansion of the exponent about this point, but the immediate result will not be given because it has been found convenient in practical computations to replace the angle 8 by a modified variable -

0 = 8 - mR/P

(53) where ( P ,R ) denote the conjugate momentum and coordinate for the translation motion (Miller, 1970; Wong and Marcus, 1971).This new variable 0, which is a constant of the motion in the asymptotic regions, may be regarded as the conjugate variable to the action N in a system in which the time t rather than the translational variable R is employed as the second independent coordinate (Child, 1976a). The final expression obtained for the S matrix in this 8 representation may be written

25 1

SEMICLASSICAL HEAVY-PARTICLE COLLISIONS

where the function $d2, 0,) is closely related to the previous Cartesian action S(q, ,q l ) :

%L%) = S [ Y , ( ~ , ~ ~ l ) ~ 4 ,+ ~~ ~ 2, ~[ ~~1l~( 1~ , > ~ l ) > ~ l l - F1 [ q 2 ( 8 2 ,

el)>821

(55)

Defined in this way, g(8,,Gl) may also be shown by use of Eqs. (47) and (50) to act as a generator for the transformation from initial to final action angle variables, in the sense that

df,fT8,

= - Nl(8,,8,),

dg/ia8, = N2(8Z,Qi)

(56)

The above double-angle representation for S, 182 is clearly implied by the arguments of Fock (1959), Miller (1970a), and Marcus (1973), and has been explicitly displayed for the forced harmonic oscillator by Pechukas and Child (1976). The connection between the semiclassical phases in Cartesian and angle action systems has also been discussed by Fraser et al. (1975). A form similar to Eq. (54) but corrected by a factor K(n,)K(n,), where N K(n) = (2n)-1’4(n!/NNe-N)1’2,

=n

+7 1

(57)

has been derived from the standard generating function for Hermite polynomials (Abramowitz and Stegun, 1965) by Ovchinnikova (1974, 1975). Direct numerical evaluation of the S matrix by use of Eq. (54) would involve double quadrature over an integrand determined by knowledge of the classical trajectories between all combinations 0, --+ 0,. A simpler but less symmetrical form may be obtained (Miller, 1970b) by performing the integral with respect to by stationary phase. It is also permissible to replace the second integration variable 8, by Q1, because the end 8,of any classical trajectory may be taken to be functionally dependent on G, for any given initial action N , . The resulting “initial-value representation”

el

s,,,, = (2nl-l s,’” ( a ~ , / a ~exp[i4(8,)1 ),~~ d

~ ,

(58)

@(el)= S”[8,(N1,O,),0,] - N , B , + N,B,(N,

8,)

(59)

where coincides with the integral form obtained by Marcus (1970, 1971) by generalized WKB solution of the Schrodinger equation, except that Marcus uses the symbol A for the phase, and an angle variable iij defined in the range (O,1) in place of 8. A number of numerical transition probabilities have been obtained by direct quadrature in this initial-value representation (Miller, 1970b; Wong and Marcus, 1971; Kreek and Marcus, 1974), but considerable effort has been devoted to developing uniform analytical approximations dependent

M . S . Child

252

only on knowledge of the special trajectories leading from the correct initial to the correct final action (or quantum number). The following section is devoted to these uniform approximations. A comparison between these uniform results and the above quadratures is given in Section II1,D. B.

STATIONARY PHASE AND UNIFORM

APPROXIMATIONS

The above integral representations for the S matrix bear a close resemblance to that employed for the scattering amplitude in the discussion of elastic scattering. First, the integrand depends on a quantity determined by the classical equations of motion, in this case the action a,) in Eq. (54) or the closely related phase (D(8,)in Eq. (58) and in the elastic scattering case and 8, at present, and the phase shift y~,. Second, the integration variables L in the previous case) span the full range allowed by the physical situation. Finally, within each integration range there are discrete values of the relevant variable ??"), say, or L"), which defines trajectories leading from the desired initial state to the desired final state of the system; these are shown below to correspond to the stationary phase points of the integrand. The purpose of the present section is show how a variety of uniform approximations for the S matrix dependent on knowledge of only these special trajectories may be constructed. Figure 11 may be used to clarify the discussion. This shows the final quantum number n2 as a continuous function of the initial phase angle at a constant value of n,. As can be seen there are always two stationary phase starting angles and elb)for values of n,, designated n!, say, lying in the range nmin< n! < n,,,, with c?n,/c'G, positive at @') and negative at It is useful to adopt this convention for the choice of label a or b. Values of n , outside this range are termed classically inaccessible, but may still be treated by extension of the classical theory as discussed in the following section.

s(Gl,

(el

sl

e'f)

s'p).

1. Stationary Phase Approximation

The simplest approximation dependent only on the stationary phase trajectories is obtained by simple stationary phase evaluation of Eq. (58) (Miller, 1970a), S,,,,

where

+ iz/4) + P,lt2exp(iQb/ti- iz/4)

= P,'I2 exp(i@.,/h

(60)

25 3

SEMICLASSICAL HEAVY-PARTICLE COLLISIONS

nma’

n2

,min

FIG.11. The final “quantum number” nz as a continuous function of the initial oscillator phase angle 8. nmaxand nmindenote the upper and lower classically accessible values. Note the existence of two trajectories, with initial angles Oc0) and 8(b)for classically accessible values of n 2 .

with the derivative evaluated at @’, and the sign of the term f n / 4 being that of (dn,/dO,),,. Pa and P b therefore have a simple classical interpretation as the density of classical trajectories at final quantum number n 2 . This primitive semiclassical result is the direct analog of Eq. (1l),derived for the simpler case of purely elastic scattering. It suffers from a similar defect in that it diverges at the caustics of the classical motion, which in this case are the classical thresholds nmin and nmax.Figure 11 shows that the two trajectories come together and that dn,/dO, = 0 at these points, so that Pa and P , go to infinity. The phase difference (Q, - @.,)/h governing the semiclassical oscillations in the transition probability

P,,

=

Pa + P b + 2(PaPb)”2sin[(@, - Qa)/h]

(62)

may be given a simple graphical interpretation, as shown in Fig. 12. This illustrates how the closed curve C, representing the initial asymptotic state of the system is translated and distorted to C; by the collision, subject by Liouville’s theorem (Goldstein, 1950) to conservation of the area enclosed. The dashed curve C, denotes a classically accessible final state n 2 . The points ( A , B ) and (A’,B’)the initial and final points, respectively, on the

254

M . S . Child

FIG. 12. The dynamical transformation in phase space induced by a typical collision. The curves C, and C2 describe asymptotic states with quantum numbers n , and n 2 . respectively. The collision induces the transition C , + C‘,, such that the phase difference between the two stationary phase trajectories AA’ and BB’ is given by the shaded area in the diagram.

n, + n2 trajectories. The relevant phase difference - @a may be shown (Pechukas and Child, 1976; Child 1978; Child and Hunt, 1977) to be equal to the area of the smaller segment into which the translate C; is divided by Cz. (Here it is assumed that n1 is the smaller of the two quantum numbers.) This means, since the area common to C, and C; is by the Bohr quantization condition equal to (2n, l)h, that the maximum phase difference in Eq. (62) is ( n , -$)Tc.In other words, there can be at most n, 1 maxima in the variation of P,, either as a function of final quantum number n2 at given energy or as a function of collision energy E at given n 2 .

+

+

+

2. Uniform Airy Approximation

The confluence of two stationary phase points leads to the simplest (fold) type of catastrophe in Thom’s (1969) classification, which is handled as in the case of the rainbow singularity in elastic scattering by means of a variable transformation O,(x), which maps the exponent in Eq (58) onto a function cubic in x : (63) @ ( ~ , ) /= h 9x3 - t(n,)x A(n,)

+

This lead to the following uniform Airy approximation (Connor and Marcus, 1971): S,,“, - ,1PeiA[(p;/2

+ PLi2)(li4Ai( - 5)

-

i(PAl2 - Pi/2)5-114 Ai’( - t)] (64)

SEMICLASSICAL HEAVY-PARTICLE COLLISIONS

255

with the parameters ( ( n 2 ) and A(n2)given in terms of the stationary phase exponents by

This reduces for ( >> 1 to the previous primitive semiclassical form given by Eq. (60).

3. Uniform Bessel Approximation Account has been taken in the above derivation of the presence of a maximum or a minimum in Fig. 11, but the periodicity of the function n2(8,,n,) has been ignored. This may become serious in cases of nearly elastic behavior, when the gap between nminand n,,, becomes so small that the stationary phase regions around the special trajectories overlap with both. This situation falls outside Thom’s (1969) catastrophe classification but it may be handled (Stine and Marcus, 1973), by a mapping 8,(y) such that @(e,)/h= A(n2)- 5(n2)cos 27ry

with m

=

In, -

-

2nmy

rill and A(n2)and C(nJ determined by the equations A = i ( @ b + @.,)/A (5’ - mz)1/2 + marccos(m/() = @.,)/h +(@b -

(67)

(68)

The final expression for the S matrix element is

S,,,, = ( ~ / 2 ) ” ~ e ’ ” [ ( P+ t ’ Pb”2)((2 ~ - m2)”“5,(() - i(P,”2

- Pb””((i“2

-

m”- 1/45, m(O1

(69)

where 5,(() and 5;(() denote the mth-order Bessel function and its first derivative. This also reduces to the primitive semiclassical form for 5 >> 1. Figure 13 gives a comparison between the exact transition probabilities and the above primitive semiclassical, uniform Airy Bessel approximations, for the special case of a forced harmonic oscillator for which the theory may be handled analytically throughout (Pechukas and Child, 1976). The parameter a measures the interaction strength. This illustrates the relatively crude nature of the primitive semiclassical approximation for transitions from n = 0 state. The uniform Airy approximation shows a marked improvement except, as expected, for the 0 - 0 transition at weak interaction strengths. The uniform Bessel approximation is seen to remedy this defect, but to give a progressively worse description as the interaction strength increases.

256

M . S . Child \ \

\ \ ',,Primitive

\.

1

Airy

I I

I I 1

I

\

\ \

\

\ \

\

I

I

I

2

FIG. 13. Comparison between the primitive semiclassical uniform Airy, uniform Bessel, and exact transition probabilities for the forced harmonic oscillator (a) 0 --t 1 transition; (b) 0 + 0 transition. The strength parameter c( is the fourier component of the forcing term at the oscillator frequency. [Taken from Pechukas and Child (1976) with permission.]

SEMICLASSICAL HEAVY-PARTICLE COLLISIONS

257

It is evident that the above approximations cover all eventualities, but that no single expression derived from Eq. (64) is universally applicable. More recently Child and Hunt (1977), following arguments similar to those of Ovchinnikova (1973, have derived a uniform Laguerre approximation from the double-integral representation (54), designed to be equally applicable to all oscillator excitation problems. This is more complicated to describe but as easy to apply as the uniform Bessel approximation. Comparison between all these forms is made in Section 111, D. Reference may also be made to a variety of other uniform approximations designed for use with systems having more than one degree of freedom, and for situations giving rise to more than two stationary phase trajectories (Connor, 1973a-d, 1974a,b; Marcus, 1972; Kreek et al., 1974, 1975). Two general discussions of the relevances of Thom’s (1969) catastrophe theory to the structure of uniform approximations have also been given (Connor, 1976a; Berry, 1976). C. CLASSICALLY FORBIDDEN EVENTS One of the most remarkable achievements of the theory (Miller and George, 1972a,b; George and Miller, 1972a,b; Stine and Marcus, 1972) has been to obtain a semiclassical description of events such as tunneling through a potential barrier or collisional excitation to vibrational states that are dynamically inaccessible by classical mechanics. The emphasis here is on dynamical inaccessibility, but not violation of any conservation law. There is no conservation law restricting motion to one side or other of a potential barrier; it is simply that the normal laws governing interconversion of kinetic and potential energy prevent the particle from passing through. Equally, there may be sufficient total energy to populate a given vibrational state, but the available interaction between oscillator and collision partner may be too weak to cause the relevant transition. The resolution of this paradox lies in analytic continuation of the classical equations of motion into the complex time plane and to complex values of any nonphysically measurable variables. This means that only real values of the internal action are acceptable because these correspond to the quantum numbers, but that the angle variables, which cannot be simultaneously measured in quantum mechanics, may be complex. This analytic continuation is already familiar in the WKB theory of one-dimensional tunneling based on the transmission factor exp( - J ( p ( d q ) determined by an imaginary momentum ilpl in the barrier region. It is also suggested by the presence of the maximum and minimum in Fig. 9 that analytic continuation of the solution of the equation %(9A

=

4

(70)

M . S . Child

258

which has two real roots in the classical region, will yield two complex solutions for the initial angle variable when ng is classically inaccessible. These ideas may be underlined by more detailed analysis of two soluble models. The first is the problem of passage through a quadratic barrier (Miller and George, 1972a), V ( q )=

(71)

-L 2 G 1 2

at a negative energy -AE, subject to the boundary condition p < 0 for t < 0. The solution of the classical equations is readily shown to be 4

-(2AE/lc)’12 coshw*t,

=

p

-(2AE/p)”’sinhw*t

=

(72)

where (73)

w* = (ti/p)1’2

The coordinate q therefore remains negative at all times, while the momentum changes sign at t = 0 as the particle bounces back from the barrier in accordance with classical experience. Suppose, however, that the time experiences an imaginary increment iz/w* during the motion, so that finally t = t‘

+ in/w*

(74)

with t’ real. Then according to Eq. (72) 4 p

= =

- ( ~ A E / K ) ”cosh(w*t’ ~ + in) = (2Af?/K)”2 cash w*t’ -(2AE/p)’” sinh(w*t’ + in)= (2AE/p)’12 sinhw*t’

(75)

The signs of both p and 4 have changed and the particle has passed through the barrier. It is readily verified on computing the action that the semiclassical phase associated with the motion simultaneously acquires an imaginary component ~ m [ ~ q ql)/til= ,, Im

J-

“J

+ i n / g*

oo

p q dt

= 71 A E / ~ W *

(76)

giving rise to the correct first-order WKB transmission factor exp( - n AE/hw*) for the problem. There is, of course, a second complex conjugate trajectory that also passes through the barrier, but leads to an exponential increase in the amplitude of the wave function. This is rejected on physical grounds (Miller and George, 1972a). A more detailed analysis of this quadratic barrier passage problem has been given by Child (1976b). The second example is the forced harmonic oscillator, with hamiltonian H ( p , q ) = +p2

+ +q2

-

f(t)q

(77)

in a system of units for which the mass, force constant, and vibrational

259

SEMICLASSICAL, HEAVY -PARTICLE COLLISIONS

frequency are equal to unity. It is assumed that the forcing term vanishes at t = f co,that it is an even function of time, and that the system starts in the state ( N , , 0,) so that, at time t + - co, q = (2N1)l/,cos(t

+ O1),

p

=

-(2N1)112sin(t

The classical equations may be shown to yield, at t + + K, q

= (2N,)'I2cos(t

+ 0,) + a sin t,

p

=

+ 8,)

-(2N1)1/2 sin(t + 0,)

+ acos t

(78) (79)

where c( is the Fourier component of the forcing function. The final action is therefore

N,

= +(p2

+ q 2 ) = N,

-

a ( 2 ~ , ) 'sindl /~

+ +a2

(80)

and the maximum and minimum classically accessible values, obtained at

O1

=

-n/2 and 8,

= 4 2 , respectively, are given

by (81)

N 2 = (Nil2

Real values of N , outside this range may however be obtained by choosing the initial angle to lie along one or other of the lines 8, = f n/2 + i07 in the complex angle plane, so that

N,

=N

, 3- ~ 1 ( 2 N , ) "cash ~ 0;'

+ *a2

(82)

The semiclassical phase associated with these complex trajectories again acquires a progressively increasing imaginary part as N , moves away from the classical region (Pcchukas and Child, 1976). There are again two complex conjugate trajectories for each classically forbidden transition, one consistent with an exponentially small and the other with an exponentially large transition probability. The mathematical argument for rejection of the latter is somewhat clearer than in the tunneling case. It is that the integration path for stationary phase (or steepest descents) evaluation of the integral in Eq. (58) can pass through only one of the complex stationary phase points, and the chosen point is always that leading to an exponentially small value for the integral (see Child, 1976a).This has already been taken into account in obtaining the uniform approximations given in Section III,B, since these approximations are specifically designed t o bridge the classical threshold regions. This analytical discussion demonstrates the existence of physically meaningful complex solutions of the classical equations of motion. The numerical determination of such complex trajectories in real applications initially posed some stability problems (Stine and Marcus, 1972; Miller and George, 1972a,b), but the results obtained are in close agreement with exact quantum-mechanical values (see Tables I and 11). Calculations of this type are particularly relevant to studies of vibrational energy transfer and the

260

M . S . Child

chemically reactive exchange of light atoms, which are dominated in the thermal energy range by events that are forbidden by classical mechanics. D. NUMERICAL APPLICATIONS AND

CONCLUSIONS

Tables I and I1 list sample results for the excitation of harmonic and Morse oscillators subject to exponential interactions according to the models of Secrest and Johnson (1966) and Clark and Dickinson (1973). Results are given for the primitive semiclassical (PSC), uniform Airy, uniform Bessel, and uniform Laguerre approximations ; the heading Quadrature includes results for numerical quadrature in Eqs. (58), where these are available. Entries marked by an asterisk are classically inaccessible. A more extensive tabulation of this type, which also covers other less sophisticated approximations, has been given by Duff and Truhlar (1975). It is evident that the primitive semiclassical approximation is always relatively crude, at least for small n, values, but that the uniform Airy expression shows a marked improvement except for the diagonal n , + n , transitions at low collision energies. These are adequately covered by the TABLE I HARMONIC OSCILLATOR TRANSITION PROBABILITIES SECREST AND JOHNSON (1966) '

n,

o* 0 0 0

o* 1 1 1 1 I* 2 2 2 2 2*

n,

0 1 2 3 4 1 2 3 4 5 2 3 4 5 6

Primitiveb -

0.422 0.416 0.359 ~

0.290 0.009 0.168 0.285 -

0.208 0.020 0.165 0.262 ~

IN THE

MODELOF

Airyb

Bessel'

Laguerre'

Exactd

0.058 0.211 0.381 0.266 0.075 0.287 0.01 1 0.174 0.240 0.062 0.206 0.017 0.170 0.194 0.045

0.334 0.205 0.380 0.264 0.0851 0.284 0.012 0.175 0.239 0.0756 0.203 0.016 0.167 0.193 0.0367

0.0523 0.219 0.366 0.267 0.0887 0.281 0.010 0.170 0.240 0.0766 0.204 0.017 0.169 0.194 0.0370

(0.0599) 0.218 0.366 0.267 0.0891 (0.286) 0.009 0.170 0.240 0.0769 (0.207 0.018 0.169 0.194 0.0371

The energy unit is half the vibrational quantum; m = 2/3, OL = 3/10, E = 20. Entries marked with asterisk are classically inaccessible. Values in parentheses were obtained by difference. Child and Hunt (1977). Miller (1970b). Secrest and Johnson (1966).

26 1

SEMICLASSICAL HEAVY-PARTICLE COLLISIONS

TABLE I1

HARMONIC OSCILLATOR TRANSITION PROBABILITIES" n,

nz

Airy'

Bessel'

Laguerre'

O*

1 2 2 3 3

1.08(-1) 1.20(-3) 4.41(-2) 1.51(-5) 1.48(-3)

1.03(-1) 1.15(-3) 4.16(-2) 1.43(-5) 1.33(-3)

1.08(-1) 1.22(-3) 4.16(-2) 1.45(-5) 1.33(-2)

O*

1*

1*

2*

Quadratured ~

5.3(-2) 2.5(-4) 1.7(-3)

~

4.3(-2) 1.8(-6) 4.6(-4)

Exact' 1.07( - 1) 1.22(-3) 4.18(-2) 1.46(-5) 1.33(-3)

" The second column under Quadrature gives Pn>",;m = 2/3, d~ = 3/10, E = 8. Values in parentheses were obtained by difference. ' Stine and Marcus (1972). ' Child and Hunt (1977). Wong and Marcus (1971). Secrest and Johnson (1966).

uniform Bessel formula, but this decreases in accuracy as the transition probability falls below unity. The uniform Laguerre approximation is seen to be consistently more accurate than either the Airy or Bessel approximation. Finally, quadrature results give moderate accuracy for the classically accessible transitions but become progressively less accurate outside this region. Two results for the n , -+n2 and n2 -+ n , transitions are given in each case because the integral in Eq. (58) is not symmetric in n, and n 2 .The reason for the greater accuracy in the classically accessible case is probably that the integration contour is necessarily taken along the real GI axis and hence passes through the points of stationary phase if the transition is classically accessible but not if it is outside the classically accessible range. Overall, the Airy uniform approximation is generally recommended on grounds of simplicity, but the Laguerre uniform approximation is to be preferred for the highest accuracy. The above results refer to excitation of one internal degree freedom. The general theory is equally valid in more complicated situations, but application of the powerful uniform approximations is complicated by the necessity to find the special n, + n2 trajectories, the direct search for which becomes prohibitive for as few as three degrees of freedom. For this reason only a few fragmentary results have been reported for the vibrationally and rotationally inelastic scattering of a diatomic molecule (Doll and Miller, 1972). One way around this difficulty is to employ a partial averaging procedure whereby the rotational motion is treated by purely classical Monte Carlo techniques, and only the vibrational part of the problem is treated by the full semiclassical method (Doll and Miller, 1972; Miller and Raczkowski, 1973; Raczkowski and Miller, 1974). Another solution is to revert to numerical quadrature for the multiple-integral initial-value representation analogous to Eq. (58) (Kreek and Marcus, 1974). Despite these

262

M . S . Child

difficulties Fitz and Marcus (1973, 1975) have been able to develop a full semiclassical treatment of collisional line broading. The semiclassical theory has also been compared with exact quantummechanical results for the collinear (all atoms constrained to lie on a line) hydrogen atom exchange reaction (Duff and Truhlar, 1973; Bowman and Kuppermann, 1973). Two special problems have been identified. The first concerns quantum-mechanical tunneling in nonseparable systems, because the use of complex classical trajectories (George and Miller, 1972a,b) yields a reaction threshold above that obtained by quantum mechanics. This problem has been reinvestigated by Hornstein and Miller (1974) but the situation is still not satisfactory. Nevertheless, considerable progress has been made toward developing a reliable semiclassical version of transitionstate theory for the chemical reaction rate constant (Miller, 1975; Chapman et a/., 1975; Miller, 1976a, 1977).The second difficulty concerns the treatment of Feshbach resonances observed in this reaction but not adequately described by the semiclassical calculation of Bowman and Kuppermann (1973).Fuller analysis by Stine and Marcus (1974) shows that a quantitative description may be obtained by following a series of multiple collisions within a collision complex.

IV. Nonadiabatic Transitions The theory of nonadiabatic transitions applies to situations where the Born-Oppenheimer separation of nuclear and electronic degrees of freedom breaks down. The basic theory was formulated by Landau (1932), Zener (1932), and Stuckelberg ( I 932), but serious doubts on its general application were cast by the criticisms of Bates (1960) and Coulson and Zalewski (1962) concerning the inflexibility of the Landau-Zener model. Recent developments have been to obtain appropriate validity criteria and to increase the flexibility of the model by emphasizing its topological structure. This has led to emphasis on the significance of certain complex transition points at which the adiabatic potential curves intersect. The key papers on the two-state model are by Bykovskii et al. (1964),Demkov (1964),Dubrovskii (1964),Delos and Thorson (1972, 1974), and Crothers (1971),and the review by Nikitin (1968). Applications of the two-state theory to analysis of the inelastic differential cross section and generalizations to more complicated situations are outlined in Sections IV,B and IV,C. Particular attention is given to the description of two-state “surface hopping” processes in systems with several nuclear degrees of freedom. The theory due to Tully and Preston (1971)and extended by Miller and George (1972a,b) is based on the assumption that since any

SEMICLASSICAL HEAVY -PARTICLE COLLISIONS

263

classical trajectory must cut a one-dimensional section through the intersecting surfaces, any problem may be reduced to a combination of singlecurve crossings. The reader is referred to reviews by Tully (1976) for more detail of the theory and by Baede (1976)for a wider account of its application.

A. ONE-DIMENSIONAL TWO-STATE MODEL The time-independent equations for a typical two-state problem may be written

where

k,Z(R)= 2 p [ E - qi(R)]/h2 U i j ( R )= 2PLl/j(R)/h2

and V ( R )is the matrix of the electronic hamiltonian in the basis of asymptotic electronic states. It is assumed in what follows that Vll(R) < V22(R)at infinite separation. Equations (83) define the exact quantum-mechanical problem. An equivalent time-dependent semiclassical form may be based on the assumed knowledge of a classical trajectory R(T)with velocity variation v(z) for the relative motion, in terms of which Eqs. (83) may be reduced to

where the elements q j ( z )denote Kj(R)evaluated along R(z).The arguments used by Bates and Crothers (1970)and Delos et al. (1972)in justifying Eq. (86) make use of the approximation

which will be used below to relate a number of equivalent results. The above equations are in the diabatic picture [see Smith (1969) and Lichten (1963) for a precise definition]. The equivalent adiabatic representation is obtained by transforming to a parametrically time-dependent

264

M . S. Child

The necessary unitary transform may be written (Levine et al., 1969)

w=

(

cos 8(z), sin 6(z),

-sin Q(z) cos QT)

(89)

where the angle 8(z), which is also used below to define a new independent variable t, is given by t = COt28(T) = -[Vi,(T) - Vzz(T)]/21/12(2) (90) The equations of motion in this adiabatic representation become (Delos and Thorson, 1972)

The coupling therefore depends on the time derivative of the mixing angle O(T), and hence on the rate of change of the electronic wave function. The key quantity d%/dz may be written

thereby drawing attention to the times zC,z,*[which are necessarily complex according to Eq. (88)] at which the adiabatic terms V,(T) intersect, because do/& clearly diverges at these points. Their location continues to dominate the structure of the theory even in the mathematically more convenient diabatic representation (86), where their role is less immediately apparent. The most convenient development for present purposes is based on the variable t defined by Eq. (90) as the independent variable, in terms of which Delos and Thorson (1972) show by introduction of the functions T ( t )= Vl z[z(t)l h dt/dz yl( t ) = [T(t)]-

exp[

that Eq. ( 5 5 ) may be reduced to T2(t)(l+ t 2 )

-

iT(t) +

-

i J'to T(t')t'dt']c1[z(t)]

(93)

265

SEMICLASSICAL HEAVY-PARTICLE COLLISIONS

The transition points now lie at t = & i. Further simplification results from the Landau-Zener curve-crossing model defined by the equations

GZ(4

-

Vll(d V,,(z)

=(F, =

F,)[R(d

-

V,, = const

Rx1 = (Fl - F,)u(z

- 7,)

(95)

where u denotes the nuclear velocity and z, is the time at the crossing point. In this case T ( t )= 2V~,/hv(F, - F,) = T ,

= const

(96)

with the result that Eq. (94) reduces to an equation of Weber form (Abramocitz and Stegun, 1965):

dZY1 + [ T i ( l dt2

~

+ t2)

-

i T o ] y ,= 0

(97)

As emphasized by Delos and Thorson (1972), the same will be true of any model in which the function T(t) is constant over the effective transition region. The problem is therefore mathematically equivalent to transmission through a quadratic barrier - T i t 2 at the complex energy T i - iT,,and the solution may be expressed in terms of parabolic cylinder functions of complex order. Manipulation of the standard asymptotic forms of these functions yields the familiar Landau-Zener transition probability (Landau, 1932; Zener, 1932; Delos and Thorson, 1972), which is given below as a special case of a more general result. The generalization is due to Dubrovskii (1964) and was followed in a different form by Child (1971). It is based on the argument that deviations from the strict Landau-Zener model or from constancy of the function T ( t ) will not affect the fundamental complex barrier transmission structure of the problem, providing the product T2(t)(l+ t 2 )has only two zeros (transition points) close to the real axis. Hence it is permissible to map the general Eq. (94) onto the quadratic model (97) by use of a variable transformation due to Miller and Good (1953). Analysis of the model case is quite lengthy because account must be taken of transitions occurring during both inward and outward motion ( - 00 < z < 0 with u < 0, respectively). The final results for the S matrix elements take the following forms if the classical turning point, corresponding to z = 0, lies a region such that It1 >> IT: - iTol: S,, S,, S,,

+ (1 - e-2"6)exp(- 2 f

2ix)] exp(2iq1) = SZl= 2ieCff6(1- e-2n6)1'2sin(r+ x)exp(iG, + iq,) = [ e - 2 K 6+ (1 - e-'"')exp(2ir + 2ix] exp(2iq,) = [e-21Ld

-

(98)

266

M . S. Child

where the parameters 6, r,q l , q2,x,and qk may be expressed in the following equivalent forms, related by Eqs. (87), (90), and (93): 1

6 =Im 71

Ji T(t)(l + t

1 271

1 h

=-1m-p

1 2n

= - Im

r = - 2 Re =

-h

[V+(Z) -

V-(z)]dz

c

I,o, i

T(t)(1 + t 2 ) ' I 2 d t -

V+(t)]dz

J ~k +' ( R )dR a+

~~c a-

k - ( R )dR

x = arg T(i6) + 6 - 6 In 6 + n/4 q1 = y q2 = y +

(99)

[k-(R) - k+(R)]dR

JR:

ReIJi"[V-(z) 1 Re[

y dt

p i

1

+ r = y - + R e [l rycc k+(R)dR a+ a+

-

= y+ -

- JR U -c

Re[JRC a + k + ( R ) d R-

s"' a-

1

k_(R)dR

1

(102)

k-(R)dR

where y * are the WKB phase shifts in the two adiabatic channels. It is readily verified that Eq. (99) reduces in the Landau-Zener approximation to (5Lz

=

To12 = V:,/hv(F, - F J

(103)

The derivation of these equations has followed Landau (1932), Dubrovskii (1964) and Delos and Thorson (1972), although the original result including the phase term r but not x was derived by Stuckelberg (1932) by a phase integral approach more recently discussed by Kotova (1969), Thorson et al. (1971),Crothers (1971),and Dubrovskii and Fischer Hjalmars (1974).Similar results have been obtained in the Landau-Zener model by transformations of Eqs. (83)to the momentum representation (Ovchinnikova, 1964; Bykovskii et al., 1964; Nikitin, 1968; Child, 1969; Bandrank and Child, 1970). The only differences are that the broken phase shifts ijl,q2 are replaced by the true diabatic WKB phase shifts y 1 and the Stuckelberg interference term r is given in the present notation in the mixed diabatic-adiabatic form (Bandrank and Child, 1970)

SEMICLASSICAL HEAVY-PARTICLE COLLISIONS

267

Reservations about this mixed prescription have been expressed by Crothers (1975), but numerical differences with the form given by Eq. (100) are likely to be small under conditions where Eq. (98) is valid. Recent efforts have been devoted to assessing the validity of the results in the light of important criticisms by Bates (1960) of the flexibility of the crude Landau-Zener model. It is clear from the above discussion that the number and positions of the complex transition points (z,,z,*) at which the nonadiabatic coupling (do/&) diverges is of paramount importance. These could be detected as points close to the real axis, where there is a rapid change in the composition of the electronic wave function. The validity of the Stuckelberg-Landau-Zener equations (99)-( 104) depends on (1) the existence of one complex conjugate pair (zc,z,*) on each (inward or outward) part of the trajectory, and (2) an adequate separation between these pairs, in the sense that the usual large argument asymptotic expansions for the parabolic cylinder functions may be applied in the intervening classical turningpoint region. The implication in terms of l- and 6 is that (Bykovskii et al., 1964)

r >> 1,

rp >> 5

(105)

These conditions break down at energies close to a curve crossing because -+ 0, but a perturbation formula valid for 6 << 1 has been derived to cover this region by Nikitin (1968) and extended by Miller (1968) and Child (1975). Eu and Tsien (1972) have clearly demonstrated the inadequacy of the Stuckelberg-Landau-Zener model at general interaction strengths in this threshold region. The Stuckelberg-Landau-Zener formulas are normally applied to curvecrossing situations, but another important class of processes termed Demkov (1964) or perturbed symmetric resonance transitions (Crothers, 1971, 1973) also gives rise to two pairs of complex transition points. These are characterized by the rapid onset of overwhelming coupling between two nearly coincident asymptotic states. The problem, which could also occur in some curve-crossing situations, is that the mixing angle O(T) passes from 0 + n/4 rather than 0 + n/2 as assumed above, because JV,,(z)( remains large compared with IV,,(z) V,,(s)l in the internal region. Hence the limit t = cot 26 + - cc is physically inaccessible, with the result that the large-argument parabolic cylinder function expansions cannot be used in the inner region. Crothers (1972) has overcome this difficulty by deriving a new expansion valid for large order and large argument by use of which the following S matrix elements have been derived (Crothers, 1971, 1976):

r

s,, = ~ ~ - 2 n+ ed- 2 i r S,,

= S,, =

s2,= [e-2rr6

+

] exp(2iy",)/[e~~"' 11 isechnhsinrexp(iy"+ + G - ) + e + 2 i r ] exp( - 2ii7,)/[e-2r6 + 11

(106)

M . S. Child

268

with the parameters 6, r, Fjl, and Fjz again given by Eqs. (99)-(102). These results clearly reduce to the Stuckelberg-Landau-Zener form under nearly << 1). adiabatic conditions (eCZrrd B. INELASTICATOM-ATOMSCATTERING

Nearly all applications and generalizations of the two-state model have been inspired by the structure of Eq. (98), the main features of which persist in Eq. (106). The important point is that each S matrix element may be interpreted in terms of interference between two types of trajectory. For example, the first term ascribes a probability amplitude e-2ndto a trajectory with a phase 2ij1 'v 2y1, approximately equal to that determined by the diabatic potential I/, while the second term has amplitude (1 - eCznd)and phase 2(@,- r - x) = 2(+ - x) attributable to motion under the lower adiabatic potential V- (see Fig. 14). Both of these are elastic processes. Similarly, in the inelastic case two terms arise from the sine function, both - e-z*d)liz, but the phase of one term of which have amplitude 2eCrr6(1 q1 q 2 r x = q1 q+ x is interpreted as arising from motion first under Vll followed by a switch to V , , while the second phase Fjl + q2 r - x 2 y- + yz - x would correspond to motion first under V- and then under I/,, . This interpretation falls within the general pattern of WKB phase accumulation dependent on the potential in question, coupled with a transition amplitude ePndor [l - e-zna]liz for each diabatic or adiabatic

,,

+ + +

+ +

FIG. 14. The diabatic [VIl(R) and V Z 2 ( R ) ]and adiabatic [ V + ( R ) and V _ ( R ) ]potential

curves.

SEMICLASSICAL HEAVY-PARTICLE COLLISIONS

269

passage through the transition region. There is also a phase correction fx accompanying adiabatic trajectories governed by V,, respectively. The most direct application is to the interpretation of the atom-atom inelastic differential cross section as discussed by Olson and Smith (1971), Kotova and Ovchinnikova (1971), Delvigne and Los (1972), 1973), Bobbio et al. (1973), and Delos (1974).The procedure for reduction of the scattering amplitude a .

Aj(0) = (2iki)-

1 (21 + l)P,(cos0 ) [ S i j ( I )

-

I=O

Sij]

(107)

is a direct generalization of that described in Section II,A for the case of elastic scattering, except that the S matrix elements, given for each 1 by Eq. (98) [or (106) if the transition is of Demkov (1964) type], contain two branches. Thus it is convenient by analogy with Eq. (10) to define two deflection functions for each type of process and to take account of interference between them. In the case of elastic scattering from channel 1, one would define

fill(/)

= 2dq,/dl,

0--(1)

= 2d(y", -

r - z)/dl

(108)

and in the case of inelastic scattering,

ep)+= d(ql + qz + r + x)/dr,

&z(i)

=

d(il1

+ q2 r - x)/dl -

(109)

where the subscripts are intended to indicate the type of trajectory involved. Typical forms for these functions adapted from Delos (1974) and the corresponding computed differential cross sections are illustrated in Fig. 15. The computations employed the Landau-Zener approximation for 1 values at which Eqs. (105) were satisfied and exact numerical solutions of Eqs. (83) for values close to 1, at which the classical turning points coincide with the crossing point. The upper and lower pairs of diagrams refer to elastic and inelastic scattering, respectively, the left-hand figures being calculated for weak coupling (small Vlz) and the right-hand figures for strong coupling (large V,J. The ordinate in the lower part of each diagram is pij(0) = O(sin O)(doij/dQ)

(110)

The transition from nearly, diabatic to nearly adiabatic elastic scattering is seen to cause a marked change in the cross section. In the former case the almost monotonic O1 branch o f f ; ,(d) is dominant because e-2rrsE 1, and the only structure arises from interference with the weak 0 - - branch. In the adiabatic limit, however, the dominant deflection function 8- - ( I ) shows both a maximum and a minimum, leading to a complicated double-rainbow pattern between emin and Om,,. The weak Stuckelberg oscillations for 0 > Om,,

270

M . S . Child

(C)

(d 1

FIG. 15. Deflection functions and differential cross sections for a model two-state problem. The ordinate ~ , ~ ( fisl )the modified differential cross section 0 sin O(doij/dfl).(a, c) Nearly diabatic (weak) and (b, d) nearly adiabatic (strong) coupling. [Adapted from Delos (1974), with permission.]

arise from interference with the branch. The inelastic differential cross sections show a much simpler interference pattern because the two branches of S12(l)both have the same amplitude coefficient 2e-""")[l - e-2n""'I l'' > where the 1 dependence of 6(1) arises from the radial velocity component

27 1

SEMICLASSICAL HEAVY-PARTICLE COLLISIONS

v(/) of the scattering trajectory with angular momentum 1, according to the Landau-Zener formula roughly

6(/) =

W W W I

-

F2)

(111)

This means that the 8,+ and 8-2 branches come together with equal amplitude, and that all the observed oscillations may be attributed to interference between the upper and lower branches of the resulting smoothly connected figure. The difference between the envelope in the weak- and strong-coupling cases reflects the way in which e-zac')[l - e-2*6(1)]1'2 varies with 1. In the weak-coupling case 6(1) is small except for a small 1 range close to the cutoff point I,, at which u ( / ) is zero. Hence the envelope of S12(l)shows a slow increase with 1 followed by a cutoff that may be too sharp to be lead to any depletion of p12(6)at 6(lx).In the strong-coupling case, on the other hand, V12 is large; hence the exponent 6(1) is also large and more strongly dependent on 1. The decline of the envelope of S, , ( 1 ) with increasing I therefore extends over all I values, leading to the strongest relative depletion at 6 z Q(1,). Similar calculations have been performed by Olson and Smith (1971) in interpreting the He' + Ne (2p) -+ He+ + Ne ( 2 ~ 5 3 scross ) section at 709 eV; by Delvigne and Los (1973) in relation to the charge exchange K I + K + + I - ; and by Faist and Bernstein (1976) in relation to the inelastic scattering of halogen atoms.

+

C . SURFACE-HOPPING PROCESSES Extension of the theory to nonadiabatic transitions occurring in systems with more than one nuclear degree of freedom has also been inspired by the structure of the Stuckelberg-Landau-Zener equations, (see Tully, 1976, for a detailed review) but the types of possible potential surface intersections are inevitably more complicated than those encountered in the simple diatomic case. The possibility of a real intersection (at which the nonadiabatic coupling diverges) between two N-dimensional adiabatic potential surfaces, on a surface of dimensionality N - 1 must always be taken into account [Teller, 1937; Herzberg and Longuet-Higgins, 1963; see also the controversy between Naqvi (1972),Naqvi and Byers Brown (1972),and Longuet-Higgins (1975)]. Such conical intersections are frequently symmetry determined, as in the equilateral triangular configurations of H 3 (Herzberg and LonguetHiggins, 1963; Nikitin, 1968), but this is not necessarily the case; computations by Grice (1976) demonstrate intersections of this type between the lowest two surfaces for the triatomic mixed alkali atom systems. A more common situation, however, is that of an avoided line intersection, arising in two-dimensional nuclear coordinate space from an interaction between

272

M . S . Child

two diabatic states with coincident energies along the line of intersection between the two surfaces. Classic examples of this type are found in the systems H: (Tully and Preston, 1971), M + X, (Baede, 1976), and X H, (Kormonicki, et al., 1976). There is a wealth of literature on bound dynamical motion at a conical intersection leading to the dynamical Jahn-Teller effect (Herzberg, 1967). Nikitin (1968) has outlined an approximate semiclassical treatment for use in the scattering context. This involves the assumption of a linear constantvelocity classical trajectory by means of which the multidimensional nuclear dynamical equations may be reduced to the time-dependent Landau-Zener model. In other words, knowledge of the classical trajectory has been used to cut a one-dimensional section through the multidimensional potential surface manifold. This idea has been accepted and extended to more general situations in two different ways by Tully and Preston (1971) and Miller and George (1972a,b). Both approaches adopt a classical treatment for the motion on one or other of the two adiabatic potential surfaces but differences arise in computing the transition probability from one surface to the other. Tully and Preston (1971) argue in favor of a simple Landau-Zener probability e-nd , whenever the trajectory crosses the diabatic intersection line or seam, with 6 given by means of Eq. (103) in terms of the interaction strength Vl,, and the components of velocity and potential gradient in the direction normal to this seam. This scheme also requires minor adjustments to the normal velocity component after the transition, in order to conserve energy and angular momentum. Among many calculations of this type, those by Baede et al. (1973)and Auerbach et al. (1973)may be cited as providing considerable insight into the mechanism of the charge exchange reactions

+

M + + XYM+ + X -

+Y

The procedure proposed by Miller and George (1972a) and most extensively described by Lin et al. (1974) is more sophisticated, in that the semiclassical phase is accumulated by calculating the classical action along the trajectory and that the trajectory is diverted into the complex plane in order to circumnavigate the appropriate complex transition points. The imaginary part of the action calculated in this way is a direct generalization of 6 given by Eq. (99) rather than the Landau-Zener form given by (103). Allowance is therefore made for deviation from the Landau-Zener model in the transition region. A further advantage is that the transition from one adiabatic surface to another is made at a point at which the two surfaces intersect;

273

SEMICLASSICAL HEAVY -PARTICLE COLLISIONS

hence energy and angular momentum are automatically conserved. There are, however, considerable computational difficulties associated with the use of complex trajectories, and with location of the relevant transition points in the complex time plane. Komornicki et al. (1976)have therefore developed a scheme intermediate between the simple Tully and Preston (1971) method and the full complex trajectory approach, whereby the trajectory is integrated in real time on either surface, but the transition parameter 6 is calculated according to Eqs. (100) by analytic continuation of the two surfaces in a direction roughly normal to the line of avoided intersections. This again necessitates a small velocity correction on passing from one surface to the other in order to maintain energy conservation. A comparison between results obtained by this decoupled approximation, the full complex trajectory method, and numerical solution of quantum-mechanical equations is given in Fig. 16. This refers to quenching of spin-orbit excited fluorine atoms by collision with H 2 . It is evident that both the semiclassical treatments are in good order of magnitude agreement with the quantum-mechanical results but that the treatment of the phase terms is not quite correct.

+

FIG. 16. Transition probabilities as a function of energy for the F(2P,,2) H2(u,) + F(’P,,,) + H2(u2) system. The probability applies to a transition between the u = 0 levels of the two electronic states. The solid line denotes the exact quantum-mechanical results. The other curves are derived by various semiclassical approximations. [Taken from Komornicki ef a/. (1976) with permission.]

274

M . S . Child

Two other broadly semiclassical approaches to the theory of nonadiabatic transitions in multidimensional systems have been suggested. The first, due to Bauer et al. (1969), employs an internal state expansion to reduce the problem to a network of intersecting one-dimensional curves at each of which the branching ratio is calculated by the Landau-Zener formula. This is open to the objection that the width of any given transition zone may span several adjacent crossing points, so that the conditions for application of the Landau-Zener model become invalid. At the opposite extreme, Kendall and Grice (1972) have argued that a single transition zone may encompass all significant crossing points, in which case the theory may be modeled on a single average electronic matrix element V, J R ) governing the transition probability between the two electronic states, with the internal state distribution within each electronic manifold determined by the Franck-Condon principle in terms of the overlap between the internal wave functions. Quantitative validity criteria on this picture have been given by Child (1973). There have, however, been few accurate numerical studies of realistic heavyparticle systems against which to test either of the above hypotheses.

V. Summary There is now a well-recognized distinction between “classically allowed” and “classically forbidden” events in the theory of molecular scattering. The most important semiclassical correction to classical mechanics is to ascribe a phase to each trajectory dependent on the classical action, which is real for the allowed events and complex for those which are forbidden. This leads to interference effects in the allowed regime and to exponentially small transition probabilities in situations forbidden by classical mechanics. The treatment of the classical threshold region raise special problems, which must be handled by means of the now well-developed class of uniform approximations, designed to pass smoothly from the oscillatory (classically allowed) to the exponentially damped (classically forbidden) region. The major exceptions to this general semiclassical behavior occur when either the effective mass or the available interaction is sufficiently small that the classical action involved is comparable with h. More serious quantum corrections must then be applied. The magnitude of the total scattering cross section summed over all events is normally limited in the molecular context by the uncertainty principle, for example, but this quantity is seldom of chemical interest except in the case of elastic scattering, when it happens that the Born and WKB approximations give the same expression for the high-angular-momentum phase shift (Child, 1974a). The more general breakdown of the semiclassical description of the elastic scattering of helium and neon atoms is, however, evident in Fig. 5.

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Another more practical question is whether the predicted semiclassical interference effects, which contain valuable information about the forces involved, will be observable under realistic experimental conditions. The results cited in Sections I1 and IV confirm that such oscillations will generally be observable in scattering experiments involving atoms, by currently available experimental techniques. The situation with respect to inelastic and reactive collisions is less clear. Nonresonant vibrational energy transfer between most small molecules is classically forbidden at thermal energies, so that oscillatory interference effects are precluded, but the calculations reported in Section III,D indicate that the classically forbidden version of the theory would be applicable. Experiments involving rotational energy transfer are complicated by the wide thermal distribution over initial states in most experiments, but the molecular beam techniques of Toennies (1974b) and Reuss (1976) appear to offer the highest resolution. Turning to the reactive situation, the complications due to the breadth of the initial rotational state distribution will probably preclude the observation of interference structure in either the product angular differential cross section or the rotational state distribution, but there is some indication of an effect of this type in the experimental product vibrational state distribution shown in Fig. 2. Further experiments involving vibrationally excited reactants could be very revealing. Finally there is no question that a quantum-mechanical or semiclassical theory is required to account for tunneling at the reactive scattering threshold for light atom systems. This could be vital for a correct estimate of the thermal energy chemical reaction rate constant. REFERENCES Abramowitz. M., and Stegun, 1. A. (1965). "Handbook of Mathematical Functions." Dover, London. Auerbach, D. J . , Hubers, M. M., Baede, A . P. M.. and Los, J . (1973). C/7em. Phys. 2, 107-1 18. Baede, A. P. M. (1976). A d i . Chem. Phys. 30,463-536. Baede, A. P. M., Auerbach, D. J., and Los, J. (1973). Physica (Utrecht) 64,134-148. Bailey, R. T., and Cruickshank, F. R. (1974). "Molecular Spectroscopy (R. F. Barrow, D. A. Long, and D. J. Miller, eds.), Vol. 2, Spec. Period. Rep., Vol. 2, pp. 262-356. Chem. Soc., London. Balint-Kurti. G. G. (1976). A d . Chcm. Phys. 30, 137-184. Balint-Kurti, G. G., and Levine, R. D. (1970). Chem. Phys. Lett. 7, 107-111. Bandrauk, A. D., and Child, M. S. (1970). M o l . Phys. 19, 95-111. Bates, D. R. (1960). Proc. R. Soc. London, Ser. A 251,22-31. Bates, D. R., and Crothers, D. S. F. (1970). Proc. R. Soc. London, Ser. A 315, 465-478. Bauer, E., Fisher, E. R., and Gilmore, F. R. (1969). J. Chem. Phys. 51,4173-4181. Beck, D. (1968). Mol. Phys. 14, 311-315. Beck, D. (1970). PTOC.. Inr. Sch. P/r~..s."Enrico F~7i7i"44. I . Bennewitr. H . G., Busse, H., Dohmann, H. D., Oates, D. E., and Schrader, W. (1972). Z . PI7y.s. 253, 435. Bernstein, R. B. (1960). J . Chem. Phys. 33, 795-804.

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