Lee and Heller’s time‐dependent theory of resonance Raman scattering is reviewed. This theory is formally identical to the traditional Kramer–Heisenberg–Dirac (KHD) theory but, in its wave packet interpretation, the time‐dependent theory has distinct calculational and conceptual advantages over the KHD sum‐over‐states method. For polyatomics with large Franck–Condon displacements and Duschinsky rotations, where typically the KHD sum is over 1010 states with complicated Franck–Condon factors, these advantages are most pronounced. Wave packet propagation on general harmonic potential surfaces (Franck–Condon displacement, frequency shifts, and Duschinsky rotation) is exact. Formulas for the propagated wave packet are given for various levels of harmonic sophistication. The role of symmetry in the wave packet dynamics is discussed and explicit formulas are derived for the overlap of the moving wave packet ‖φi(t)〉 with the final state of interest ‖φf〉. The half Fourier transform of this overlap gives the Raman amplitude α. The transform method of Tonks and Page, relating absorption and Raman excitation profiles, is shown to arise naturally in our approach. We show excitation profiles calculated by the time‐dependent theory for multidimensional harmonic potential surfaces with and without Duschinsky rotation. For the no‐Duschinsky cases, we compare our results with the profiles of Myers and Mathies and of Champion and Albrecht, which were calculated by a sum‐over‐states; we then discuss some discrepancies between the latter’s results and ours.
A time‐dependent semiclassical method for generating energy‐dependent photodissociation partial cross sections is presented. The method is based on the Wigner equivalent formulation of quantum mechanics with the semiclassical limit arising from one dynamical approximation: the replacement of the quantum Liouville operator by its h/→0 limit. The results of the present scheme for the collinear dissociation of ICN on a single dissociativesurface are compared to those obtained from a distorted‐wave analysis and a semiclassical wave packet propagation. A model calculation employing standard trajectory techniques indicates that the present method has several distinct advantages over the traditional quasiclassical approach.
It is shown that the recent results and error trends of Lee and Scully and Brown and Heller are explicable in terms of the ’’dangerous cross term’’ analysis. Extended quantum state suffer larger errors in the Wigner method.
A new and convenient semiclassical method is proposed. It relies only upon classical trajectories and Gaussian integrals. It seems to work very well for the model molecular vibrational spectra investigated here. It should be applicable to a wide variety of processes and can be variationally improved if necessary.
A new technique is developed to generate semiclassical wave functions. The method uses only information already available from a standard semiclassical quantization of a system. Linear superpositions of Gaussian coherent states that lie along quantizing classical trajectories are used, with phases given by the action integrals plus a Maslov‐type correction. Wave functionsgenerated in this way suffer from none of the problems with caustics that primitive semiclassical wave functions encounter. The semiclassical wave functions are convenient for subsequent use in applications, e.g., molecular spectra. By generatingwave functions for several simple systems, we show that under most circumstances these wave functions are very accurate approximations to the true quantum states.
Tunneling involves an allowed quantum event which fails to take place classically. Dynamical tunneling is the subset of such events which do not involve a classically insurmountable potential barrier. In this paper, we present unambiguous evidence for dynamical tunneling in bound state quantum systems.The tunneling occurs between two distinct regions of classically trapped quasiperiodic motion. Close analogies are shown to exist between this situation and ordinary barrier penetration in a double minimum potential. In the cases we study, tunneling occurs between equivalent or nearly equivalent local mode motions, which have arisen out of a resonance between the symmetric and nonsymmetric stretch.
Classically periodic molecular vibration (such as a totally symmetric stretch) can be unstable against the addition of small components of other modes, depending on anharmonic coupling strengths, near resonance of fundamental frequencies, and the total energy. We report here on some very strong correspondances between classical stability of the motion and quantum spectral features, wave functions, and energy transfer. The usual concept of a vibrational Fermi resonance turns out to apply best to the case where the transition to classical instability occurs at an energy below the first resonant quantum levels (this is the case for the famous Fermi resonance in CO2). In the (probably more common) event that resonant classical instability should set in above several quanta of energy in the mode of interest, the quantum spectrum shows tell‐tale pre‐ and post‐resonant signatures which include attraction of quantum levels (rather than the usual Fermi repulsion) and other features not normally associated with Fermi resonances. Evidence is presented which suggests that certain types of periodic motion in anharmonic molecules act as ’’traps’’, and are resistant to energy exchange with other types of motion. Numerical evidence linking the classical and quantum behavior, together with a new semiclassical theory presented here provides a very explicit connection between quantum and classical anharmonic motion.
We define a simple, purely local mode model vibrational Hamiltonian which gives rise to an apparent normal modespectrum under conditions resembling a symmetric stretch Franck–Condon transition. The model clearly distinguishes the question of the intrinsic separability of the Hamiltonian from the nature of the initial conditions, or "pluck", implied by the transition moment.
This paper proposes new criteria by which to gauge the extent of quantum intramolecular randomization in isolated molecules. Several hallmarks of stochastic and nonstochastic behavior are identified, some of which are readily available from spectral data. We find that it is very important to tailor the criteria to the specific experimental situation, with the consequence that a given molecule can be labeled both stochastic and nonstochastic, even in the same general energy regime, depending on the experiment. This unsettling feature arises as a quantum analog of the necessity, in classical mechanics, of specifying the apriori known integrals of the motion before ergodic or stochastic behavior can be defined. In quantum mechanics, it is not possible to have flow or measure local properties (analog of trajectories and phase space measure) without some uncertainty in the integrals of the motion (most often the energy). This paper addresses the problems this creates for the definition of stochastic flow. Several systems are discussed which show significant differences in their quantum vs classical stochastic properties.
We present theory and numerical results for a new method for obtaining eigenfunctions and eigenvalues of molecular vibrational wave functions. The method combines aspects of the semiclassical nature of vibrational motion and variational, abinitio techniques. Localized complex Gaussian wave functions, whose parameters are chosen according to classical phase space criteria are employed in standard numerical basis set diagonalization routines. The Gaussians are extremely convenient as regards construction of Hamiltonian matrix elements, computation of derived properties such as Franck–Condon factors, and interpretation of results in terms of classical motion. The basis set is not tied to any zeroth order Hamiltonian and is readily adaptable to arbitrary smooth potentials of any dimension.
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.