A recent time dependent formulation of total photodissociation cross sections is exploited to give a qualitative explanation of line shapes and absorption envelopes for symmetric triatomic (XY2) vacuum uvspectra. Attention is given to the dependence of the cross section on potential surface parameters and on the nature of the initial vibrational wavefunctions. The symmetric triatomic case treated here is illustrative of techniques which can be applied to more complicated, unsymmetric polyatomic situations.
An unconventional time dependent formula for total photodissociation cross sections shows the importance of short time dynamics in direct photofragmentation. This is exploited to provide a systematic expansion in powers of h/ for the cross section. The lowest order term is a classical cross section which is shown to be an improvement upon the venerable reflection approximation. Terms to higher order in h/ lead to even greater improvements in accuracy as shown by simple numerical examples. Our formulas are directly applicable to polyatomic photofragmentation, and as a spinoff we derive the polyatomic generalization of the (diatomic) reflection method.
The theory of quantum amplitudes in the semiclassical, h/→0 limit is considerably extended by including overlaps between Gaussian coherent states (wave packets) in the primitive ’’semiclassical algebra.’’ Traditionally, semiclassical formulations have exclusively involved overlaps between eigenstates of Hermitian operators; these are delocalized in most representations. Paradoxically, the states which are most localized and particlelike in the classical limit (wave packets) are not eigenfunctions of Hermitian observables, and have been omitted from asymptotic, formal theories of the semiclassical limit. When incorporated into a generalized theory, the Gaussian wave packet states apppear to alleviate many of the practical difficulties encountered in implementation of the Hermitian, delocalized state formulation. We present the generalized theory here, and show how it contains and reduces to the Hermitian semiclassical theory in various limits.
A useful and intuitive phase space picture of common semiclassical approximations and procedures is proposed and developed. Underlying the intuitive pictures is the semiclassical, C→0 limit of the Wigner distribution, which is discussed in an Appendix. Simple classical analog phase space diagrams are suggested which represent quantum amplitudes, completeness relations, stationary phase points and integrations, caustic singularities, uniformizations, and the interesting short and long time behavior of the semiclassical propagator. Semiclassical wave packet amplitudes, now known to be on equal footing with the more usual amplitudes between delocalized eigenstates of Hermitian operators, are included in the phase space pictures. It becomes apparent why the primitive semiclassical wave packet amplitudes are to some extent uniformizing, and numerical results are presented to support this conclusion. The power of the phase space picture as an aid in formulating new approximations is illustrated.
The asymptotic (h/→0) equivalence of the Miller–Marcus classical S‐matrix theory and Gaussian wave packet dynamics is shown. This result is not suprising, but the analysis yields considerable insight into both methods. Both approaches are seen to rely upon a linear response of dynamical variables against small changes in initial conditions. However, the two theories ’’back off’’ the h/→0 limit in a very different manner. Wave packets emerge as a kind of apriori uniformization procedure as opposed to the aposteriori uniformizations of classical S‐matrix theory. In certain contexts the wave packets are shown to provide a parabolic cylinder function uniformization of the primitive semiclassical result. Wave packets in generalized coordinates are discussed. Analogy with classical S‐matrix theory suggests new procedures for through‐barrier tunneling of individual wave packets.
Explicitly time dependent methods for semiclassical dynamics are explored using variational principles. The Dirac–Frenkel–McLachlan variational principle for the time dependent Schrödinger equation and a variational correction procedure for wavefunctions and transition amplitudes are reviewed. These variational methods are shown to be promising tools for the solution of semiclassical problems where the correspondence principle, classical intuition, or experience suggest reasonable trial forms for the time dependent wavefunction. Specific trial functions are discussed for several applications, including the curve crossing problem. The useful semiclassical content of the time dependent Hartree approximation is discussed. Procedures for the variational propagation of density matrices are also derived.
We investigate the suitability of the Wigner method as a tool for semiclassical dynamics. In spite of appearances, the dynamical time evolution of Wigner phase space densities is found not to reduce to classical dynamics in most circumstances, even as h→0. In certain applications involving highly ’coherent’ density matrices, this precludes direct h‐expansion treatment of quantum corrections. However, by selective resummation of terms in the Wigner–Moyal series for the quantum phase space propagation it is possible to arrive at a revised or renormalized classicallike dynamics which solves the difficulties of the direct approach. In this paper, we review the Wigner method, qualitatively introduce the difficulties encountered in certain semiclassical applications, and derive quantitative means of surmounting these difficulties. Possible practical applications are discussed.
In this paper we develop a new approach to semiclassical dynamics which exploits the fact that extended wavefunctions for heavy particles (or particles in harmonic potentials) may be decomposed into time−dependent wave packets, which spread minimally and which execute classical or nearly classical trajectories. A Gaussian form for the wave packets is assumed and equations of motion are derived for the parameters characterizing the Gaussians. If the potential (which may be nonseparable in many coordinates) is expanded in a Taylor series about the instantaneous center of the (many−particle) wave packet, and up to quadratic terms are kept, we find the classical parameters of the wave packet (positions, momenta) obey Hamilton’s equation of motion. Quantum parameters (wave packet spread, phase factor, correlation terms, etc.) obey similar first order quantum equations. The center of the wave packet is shown to acquire a phase equal to the action integral along the classical path. State−specific quantum information is obtained from the wave packet trajectories by use of the superposition principle and projection techniques. Successful numerical application is made to the collinear He + H2 system widely used as a test case. Classically forbidden transitions are accounted for and obtained in the same manner as the classically allowed transitions; turning points present no difficulties and flux is very nearly conserved.
The “quantum trajectory” wavepacket approach to semiclassical collision dynamics is generalized to include effects which cause distortion of initially gaussian wavepackets. The generalization takes the form of a discrete phase space path “integral” or sum. A complete set of gaussian phase space localized basis functions is proposed, and in the semiclassical limit, each basis function time develops in a simple classical-like fashion. Thus, the formulation conveniently builds in the correct limiting semiclassical behavior. The phase space path sum is formally exact, numerically it appears convenient, and is apparently equally at home in quantum and semiclassical regimes.
In this paper, we propose and solve sequential coupling models for molecular dissociation of the Rice‐McLaughlin‐Jortner (RMJ) type in which the usual assumption of constant coupling among the states is replaced by an assumption of random coupling. The counter‐intuitive nonsequential branching behavior found previously for constant coupling is eliminated and we find completely sequential time dependence which obeys the phenomenological rate equations. We isolate the features of constant vs random coupling which give rise to the branching vs sequential behavior in terms of simple physical models and considerations of the coherence properties of the wavefunction. It is concluded that constant coupling is inappropriate for most molecules, and that the random coupling assumption has the effect of validating the use of a random phase approximation which in turn causes the molecule to decay as if each quasibound molecular level is coupled to its own continuum. Our conclusions do not change when we solve an extended model with many continua, with each molecular level coupled to each continuum.
The quantum mechanical collinear atom-diatom collision problem is treated via a discrete expansion of the wave-function in terms of uncoupled, distorte wave states, using the Wigner-Eisenbud R-matrix formalism. Previously unreported narrow Feshbach resonances, which reduce the inelastic transition probability, are easily found and examined. Off resonant transition probabilities are in agreement with the work of Diestler and Feuer using the same model.