S-Matrix Theory

Rost J M 1998 Semiclassical s-matrix theory for atomic fragmentation Phys. Rep. 297 272-344... 

Tanner D J and Weeks D E 1993 Wave packet correlation function formulation of scattering theory—the quantum analog of classical S-matrix theory J. Chem. Phys. 98 3884... 

Function Formulation of Scattering Theory The Quantum Analog of Classical S-Matrix Theory. 

Chew spoke of the S-Matrix Theory with Regge Poles, a concept80 that soon became a stable acquisition in the general picture of subnuclear particles,81 and Tavkhelidze discussed the "Simplest Dynamic Models of Composite Particles. ... 

Gray, S.K. and Child, M.S. (1984). Photodissociation within classical S matrix theory. Hyperbolic umbilic uniform approximation and application to CH3I + tuv —> CH3 + I, Mol. Phys. 51, 189-210. 

In Fig. 4.8 the effect of the initial-state wave functions is explored, for the case where the crucial electron-electron interaction is the two-body Coulomb interaction (4.14a) and for the case where this interaction is the two-body contact interaction (4.14d), which is not restricted to the position of the ion. In both cases, the form factor includes the function (4.23), which favors momenta such that pi + p2 is large. This is clearly visible for the contact interaction (4.14d) and less so for the Coulomb interaction (4.14a) whose form factor also includes the factor (4.19), which favors pi = 0 (or p2 = 0)- We conclude that (i) the effect of the specific bound state of the second electron is marginal and (ii) that a pure two-body interaction, be it of Coulomb type as in (4.14a) or contact type as in (4.14d), yields a rather poor description of the data. A three-body effective interaction, which only acts if the second electron is positioned at the ion, provides superior results, notably the three-body contact interaction (4.14b), cf. the left-hand panel (d). This points to the significance of the interaction of the electrons with the ion, which so far has not been incorporated into the S-matrix theory beyond the very approximate description via effective three-body interactions such as (4.14b) or (4.14c). 

Classical S-matrix theory applied to photodissociation of 538 CH3I. Semiclassical calculations found to be in good agreement with exact quantum calculations and to provide aid in understanding of qualitative features of photodissociation... 

This paper reviews this classical S-matrix theory, i.e. the semiclassical theory of inelastic and reactive scattering which combines exact classical mechanics (i.e. numerically computed trajectories) with the quantum principle of superposition. It is always possible, and in some applications may even be desirable, to apply the basic semiclassical model with approximate dynamics Cross7 has discussed the simplifications that result in classical S-matrix theory if one treats the dynamics within the sudden approximation, for example, and shown how this relates to some of his earlier work8 on inelastic scattering. For the most part, however, this review will emphasize the use of exact classical dynamics and avoid discussion of various dynamical models and approximations, the reason being to focus on the nature and validity of the basic semiclassical idea itself, i.e., classical dynamics plus quantum superposition. Actually, all quantum effects—being a direct result of the superposition of probability amplitudes—are contained (at least qualitatively) within the semiclassical model, and the primary question to be answered regards the quantitative accuracy of the description. 

The remarks in the previous paragraph apply, of course, only to the case of electronically adiabatic molecular collisions for which all degrees of freedom refer to the motion of nuclei (i.e. translation, rotation and vibration) if transitions between different electronic states are also involved, then there is no way to avoid dealing with an explicit mixture of a quantum description of some degrees of freedom (electronic) and a classical description of the others.9 The description of such non-adiabatic electronic transitions within the framework of classical S-matrix theory has been discussed at length in the earlier review9 and is not included here. 

With regard to reactive processes, there have been several applications of classical S-matrix theory to the... 

To see how Feshbach resonances appear in classical S-matrix theory, consider the collinear H + Cl2 collision as studied by Rankin and Miller.47 Fig. 8 shows the quantum number function n2(q,) for one region of ql the function is smooth, these trajectories being direct . The remaining interval of ql leads to complex trajectories, those which spend a number of additional vibrational periods in the interaction region for this region of qY values the final vibrational quantum number changes dramatically with small changes in q,. The S-matrix for the particular transition indicated in Fig. 8 thus has the form... 

As another application of classical S-matrix theory it is interesting to see how the scattering of atoms from a solid surface is described. (The extension to scattering of molecules should also be clear.) This has been worked out by Doll49 and closely parallels Wolken s50 quantum mechanical formulation of the problem. 

The final application of classical S-matrix theory to be discussed is the description of photodissociation of a complex (e.g. triatomic) molecule. The completely classical description, essentially the half-collision model of Holdy, Klutz and Wilson,54 is discussed first, and then the semiclassical version of the theory is presented. A completely quantum mechanical description of the process has been developed in detail recently by Shapiro,55 The quantity of interest is the transition dipole,... 

Classical S-matrix theory has been applied to classically forbidden processes in A + BC collision systems, both collinear and three-dimensional models, reactive as well as non-reactive processes having been studied. This section discusses some of these results. 

Doll64 has applied classical S-matrix theory to the collinear A + BC collision where atoms A and B interact via a hard sphere collision this is the model studied quantum mechanically by Shuler and Zwanzig.65 Doll treats classically allowed and forbidden processes and finds good agreement between semiclassical and quantum mechanical transition probabilities. This is a remarkable achievement for the semiclassical theory, for the hard sphere interaction is far from the smooth potential that one normally assumes to be necessary for the dynamics to be classical-like. 

Dimensionally, Pni defined by (162) is a collinear-like vibrational transition probability [cf. (144)] which depends parametrically on the initial conditions of the other degrees of freedom. The analogy to a collinear collision is purely formal, however, for there are no dynamical approximations which have been introduced the only approximations involved, beyond that of classical S-matrix theory itself, are the neglect of interference terms between different trajectories that contribute to the same S-matrix element and the assumption... 

In many cases, too, the semiclassical model provides a quantitative description of the quantum effects in molecular systems, although there will surely be situations for which it fails quantitatively or is at best awkward to apply. From the numerical examples which have been carried out thus far— and more are needed before a definitive conclusion can be reached—it appears that the most practically useful contribution of classical S-matrix theory is the ability to describe classically forbidden processes i.e. although completely classical (e.g. Monte Carlo) methods seem to be adequate for treating classically allowed processes, they are not meaningful for classically forbidden ones. (Purely classical treatments will not of course describe quantum interference effects which are present in classically allowed processes, but under most practical conditions these are quenched.) The semiclassical approach thus widens the class of phenomena to which classical trajectory methods can be applied. 

M.L. Goldberger, K.M. Watson, Lifetime and decay of unstable particles in S-matrix theory, Phys. Rev. 136 (1964) B1472. 

J. R. Stine and R. A. Marcus, Chem. Phys. Lett., 29, 575 (1974). Semiclassical S Matrix Theory for a Compound State Resonance in the Reactive Collinear H H2 Collision. 

Nuclear Magnetic Relaxation A Reprint Volume, 1961 S-Matrix Theory of Strong Interactions A Lecture Note and Reprint Volume, 1961... 

Another direction of research that was fostered by the KPS work was the development of semiclassical theories of chemical reactions. This development arose because the QCT method is an ad hoc procedure for mimicking quantum effects in chemical reaction dynamics wherein quantization is imposed initially and finally but not in-between. In semiclassical methods, one imposes the > 0 limit of quantum mechanics in a consistent way throughout the reactive collision process. The search for a consistent semiclassical theory eventually produced classical S-matrix theory [14], which is a topic of continuing interest in gas-phase dynamics [15], and it also led to the development of Gaussian wave-packet methods for simulating chemical reactions [16]. 

Theory. Usually we do not solve the fundamental equations directly. We use a theory, for example, Har-tree-Fock theory [3], Moller-Plesset perturbation theory [4], coupled-cluster theory [5], Kohn s [6, 7], Newton s [8], or Schlessinger s [9] variational principle for scattering amplitudes, the quasiclassical trajectory method [10], the trajectory surface hopping method [11], classical S-matrix theory [12], the close-coupling approximation... 

In practical applications of the above classical dynamics of electrons to the classical S-matrix theory, the choice of the initial phase degrees of freedom, qK is crucial for a path thus initiated to be able to reach the desired final condition on uk. In order to realize the special condition nK ti) = 6k,a and (tz) = k,i3, the electronic Hamiltonian of Eq. (4.50) has to be further converted to Langer-modified form. 

Tannor DJ, Weeks DE (1993) Wave packet correlation function foimulation of scattering theory the quantum analog of classical s-matrix theory. J Chem Phys 98(5) 3884... 

Classical S-matrix theory [67, 73] is a rigorous semiclassical theory, rigorous in that it incorporates the full classical mechanics for all degrees of... 

T. F. George and W. H. Miller, Classical S-matrix theory of reactive tunneling Linear H -h H2 collisions, J. Chem. Phys. 57 2458 (1972). 

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