We investigate the consequences of quasiperiodic and ergodic classical dynamics on predissociation rates from bound vibrational states into a continuum. We find a strong correlation between the quasiperiodic/ergodic motion and rate constants for certain locations of the predissociationsurface. Other locations for the predissociative surface are insensitive to the difference between classical motion which is confined to small regions of phase space and motion which samples a large portion of its available phase space.
We examine the concept of nodal breakup of wave functions as a criterion for quantum mechanical ergodicity. We find that complex nodal structure of wave functions is not sufficient to determine quantum mechanical ergodicity. The influence of classical resonances [which manifest themselves as classical resonance zones (CRZ)] may also be responsible for the seeming complexity of nodal structure. We quantify this by reexamining one of the two systems studied by Stratt, Handy, and Miller [J. Chem. Phys. 71, 3311 (1974)] from both a quantum mechanical and classical point of view. We conclude that quasiperiodic classical motion can account for highly distorted quantum eigenstates. One should always keep this in mind when addressing questions regarding quantum mechanical ergodicity.
Semiclassical quantization of the quasiperiodic vibrational motion of molecules is usually based on Einstein–Brillouin–Keller (EBK) conditions for the quantization of the classical actions. Explicit use of the EBK conditions for molecular systems of K degrees of freedom requires K quantization conditions. Therefore, explicit use of the EBK conditions becomes increasingly difficult if not impossible for polyatomic systems of three or more degrees of freedom. In this paper we propose a semiclassical quantization method which makes explicit use of phase coherence of the de Broglie wave associated with the trajectory rather than the EBK conditions. We show that taking advantage of phase coherence reduces the K quantization conditions to a single quantum condition—regardless of the number of degrees of freedom. For reasons that will become obvious we call this method ‘‘spectral quantization.’’ Polyatomic vibrational wave functions and energy eigenvalues are generated from quasiperiodic classical trajectories. The spectral method is applied to an ABA linear triatomic molecule with two degrees of freedom and to an anharmonic model of the molecule cyanoacetylene. The usefulness of the technique is demonstrated in this latter calculation since the cyanoacetylene model will have four coupled vibrational degrees of freedom.
A time dependent wave packet method is presented for the rapid calculation of the properties of systems in thermal equilibrium and is applied, as an illustration, to electronic spectra. The thawed Gaussian approximation to quantum wave packet dynamics combined with evaluation of the density matrix operator by imaginary time propagation is shown to give exact electronic spectra for harmonic potentials and excellent results for both a Morse potential and for the band contours of the three transitions of the visible electronic absorptionspectrum of the iodine molecule. The method, in principle, can be extended to many atoms (e.g., condensed phases) and to other properties (e.g., infrared and Raman spectra and thermodynamic variables).
Radiationless transitions in polyatomic molecules prove to be quite amendable to a semiclassical treatment both below and above crossings between the potential surfaces involved in the transition. Below such crossings, tunneling integrals are easily performed which give good estimates of the dependence of the nonradiative rate on the energy gap and excess energy in the electronic state. Above the surface crossing, the transitions become classically allowed and a Tully–Preston surface hopping model suffices. We find that a nonlinear dependence of ln(knr) vs E plots is the rule rather than the exception. The ln(knr) vs E plots tend to flatten out with increasing energy. This effect can occur below surface crossings, but is most dramatic when a surface crossing is reached. The recent beam results of Smalley and co‐workers on pyrazine and pyrimidine are seen to be a possible case of this simple behavior.
We present in this paper a coordinate independent semiclassical quantization method. We demonstrate that in order to extract accurate eigenvalues and eigenfunctions the trajectory does not necessarily have to reside on the quantizing torus, rather, one can use information obtained on arbitrary tori. Because the method is coordinate independent, no difficulty is encountered in quantizing within classical resonance zones. Furthermore, nearby eigenstates and eigenvalues (nearby in action space) may be extracted from the same trajectory—this is especially convenient when the density of states becomes large.
A method is presented for using a single preresonance Raman spectrum and an absorptionspectrum to obtain changes in equilibrium geometry upon electronic excitation. The relative displacements along each of the vibrational normal coordinates are obtained from the Raman intensities, while the overall scaling of the displacements is determined by the absorption band shape. The absorption spectra, as well as Raman excitation profiles, are calculated using either a sum over vibronic states or a formally equivalent time‐dependent method [S.‐Y. Lee and E. J. Heller, J. Chem. Phys. 71, 4777 (1979)]. The time‐dependent method is computationally much faster than the vibronic sum for large multidimensional systems. Our analysis, which assumes isolated molecules and separable, harmonic surfaces, yields a good fit to the vapor phase absorptionspectrum of trans‐hexatriene with a Lorentzian linewidth of 175 cm−1. However, the diffuse absorptionspectrum of isoprene cannot be adequately reproduced using Lorentzian line shapes, even when all 33 normal modes are included. Finite temperature and excited state frequency changes are also found to have little effect on the calculated band shapes. These results suggest that inhomogeneous broadening may be a major factor, but calculations using Gaussian broadening fail to accurately reproduce the experimental spectrum.
While liquid solution reactions are much more important in chemistry, gas phase reactions are much better understood. Given the central importance of solution reactions to inorganic, organic, industrial and biochemistry, it is rather surprising that, as yet, there is not a single such reaction whose molecular dynamics are understood in detail. Theoretical and experimental evidence already makes clear that much of the important molecular dynamic action in solution reactions occurs on the picosecond and subpicosecond time scales. The dihalogen photodissociation and recombination reactions, X2 + hv→X + X→X2, involving the simplest possible molecular reactants and products, diatomics, and in rare gas solution involving only two elements, seem excellent candidates for study.
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.