We study the quantum mechanics of a Hamiltonian system which is classically chaotic: the stadium billiard. We have examined many of the eigenstates of the stadium, up to about the 10,000th. Complex periodic orbits play an active role in shaping the eigenstates.
A new method for obtaining molecular vibrational eigenstates using an efficient basis set made up of semiclassical eigenstates is presented. Basis functions are constructed from a ‘‘primitive’’ basis of Gaussian wave packets distributed uniformly on the phase space manifold defined by a single quasiperiodic classical trajectory (an invariant N‐torus). A uniform distribution is constructed by mapping a grid of points in the Hamilton–Jacobi angle variables, which parametrize the surface of the N‐torus, onto phase space by means of a careful Fourier analysis of the classical dynamics. These primitive Gaussians are contracted to form the semiclassical eigenstates via Fourier transform in a manner similar to that introduced by De Leon and Heller [J. Chem. Phys. 81, 5957 (1984)]. Since the semiclassical eigenstates represent an extremely good approximation of the quantum eigenstates, small matrix diagonalizations are sufficient to obtain eigenvalues ‘‘converged’’ to 4–5 significant figures. Such small diagonalizations need not include the ground vibrational state and thus can be used to find accurate eigenstates in select regions of the eigenvalue spectrum. Results for several multidimensional model Hamiltonians are presented.
We discuss a semiclassical treatment of the rigid asymmetric rotor that delivers eigenenergies as well as eigenstates. We give possibilities to improve the semiclassical wave functions to any accuracy required. The method is devised so that inclusion of vibrations is possible. As no information about energetically lower states is included in the procedure, the calculation of highly excited states is easier than with conventional quantum methods. Calculation of quantum splitting from semiclassical eigenstates is treated. We give numerical examples for every procedure developed, so that the performance of the theory can be judged.
The time dependence of spontaneous emission from isolated, highly vibrationally excited molecules is studied from the point of view of the classical and semiclassical mechanics of dissipatively perturbed Hamiltonian systems. A correlation function expression for the time‐dependent emission spectrum serves as a starting point for quasiclassical approximations. These in turn lead to an investigation of the classical dynamics of model molecular systems in which small, energy‐nonconserving terms have been added to Hamilton’s equations. Numerical calculations show rich dynamical behavior which can be qualitatively understood in terms of the resonance structure of the unperturbed system. For example, trajectories tend to be captured by zones of nonlinear resonance. This ‘‘mode locking’’ produces a characteristic cleanup of an emission spectrum that otherwise is rather congested at these energies. The close connection between spectra and dynamics suggests that the classical dynamics of dissipatively perturbed systems may provide a useful language for interpreting radiative and condensed‐phase vibrational relaxation, and possibly some types of intramolecular relaxation phenomena as well.
We begin by discussing the properties expected of eigenfunctions of a classically chaotic Hamiltonian system, using simple Correspondence Principle arguments. The properties involve nodal surfaces, coordinate and momentum space amplitude distribution, and phase space distribution. The eigenfunctions of the stadium billiard are examined, and it is found that the periodic orbits of shortest periods and smallest stability parameter profoundly affect the eigenfunctions: “scars” of higher wavefunction density surround the special periodic orbits. Finally a theory is presented for the scars, showing that they must exist, and relating them directly to the special periodic orbits. These same periodic orbits cause level density fluctuations.
The relaxation of a primary system coupled weakly to a bath of environmental modes is examined from the standpoint of recent developments in the semiclassical theory of molecular bound states. Emphasis is placed upon highly excited, strongly nonlinear (but quasiperiodic) primary systems and zero temperature baths. The starting point for the analysis is a master equation for the populations of the eigenstates of the primary system. The correspondence principle provides semiclassical approximations to the transition rates, allowing quantum state populations to be calculated from classical trajectories. A second semiclassical approximation leads to an equation of motion for a probability density in the classical action variables. As h→0, this density agrees with the density generated by running an ensemble of damped classical trajectories and averaging out the angle variables; retention of terms of order h provides smoothed quantum corrections. Numerical examples of both semiclassical approximations are presented.
The combination of abinitio calculation of electronic wave functions with a wave packet calculation of the nuclear motion is used, within the Born–Oppenheimer approximation to compute the vibrational and electronic absorption of a polyatomic molecule. A particular virtue of this approach is that high as well as low temperature spectra are both calculable. This method is applied to C2H, for which the complete active space self‐consistent field (CASSCF) method is used to determine full Born–Oppenheimer potential surfaces. Using the assumption that the A(2Π) ← X(2Σ+) absorption can be written as the sum of the A(2A’) ← X and A(2A‘) ← Xabsorptions, the spectra are determined to 60 cm−1 resolution at a temperature of 3000 K. As a result of the large thermal bending amplitude at 3000 K, the calculated spectra are broad and have little resolved structure. Two bands are resolvable, one is due to the A(2A‘) ← Xabsorption and is centered at 5500 cm−1, while the other is due to A(2A’) ← Xabsorption and is centered at 9500 cm−1. The dramatic blue shift of the A(2A’) ← X band results from the combination of the large X state thermal bending amplitude and high bending frequency of the A(2A’) state. We also determine the X state pure vibrational absorptionspectrum and show it to be of much lower intensity than the pure electronic spectrum.
Exact eigenfunctions for a two‐dimensional rigid rotor are obtained using Gaussian wave packet dynamics. The wave functions are obtained by propagating, without approximation, an infinite set of Gaussian wave packets that collectively have the correct periodicity, being coherent states appropriate to this rotational problem. This result leads to a numerical method for the semiclassical calculation of rovibrational, molecular eignestates. Also, a simple, almost classical, approximation to full wave packet dynamics is shown to give exact results: this leads to an aposteriori justification of the De Leon–Heller spectral quantization method.
The exact thermal rotational spectrum of a two‐dimensional rigid rotor is obtained using Gaussian wave packet dynamics. The spectrum is obtained by propagating, without approximation, infinite sets of Gaussian wave packets. These sets are constructed so that collectively they have the correct periodicity, and indeed, are coherent states appropriate to this problem. Also, simple, almost classical, approximations to full wave packet dynamics are shown to give results which are either exact or very nearly exact. Advantages of the use of Gaussian wave packet dynamics over conventional linear response theory are discussed.
Flow in phase space is intimately related to the concept of ergodicity. An ergodic system will sample all regions of phase space democratically in the time average, subject only to known a priori constants of the motion. For convenience, we shall refer to the accessible phase space as the “energy shell.” This shell may have some thickness if a distribution of energies are initially populated An energy envelope is defined by the initial distribution of energy and the envelope plays a role both in classical and quantum phase Space flow, as discussed before and as will become evident in the theory and examples to follow.
The exact thermal rotational spectrum of a two-dimensional rigid rotor is obtained using Gaussian wave packet dynamics. The spectrum is obtained by propagating, without approximation, infinite sets of Gaussian wave packets. These sets are constructed so that collectively they have the correct periodicity, and indeed, are coherent states appropriate to this problem. Also, simple, almost classical, approximations to full wave packet dynamics are shown to give results which are either exact or very nearly exact. Advantages of the use of Gaussian wave packet dynamics over conventional linear response theory are discussed. [ABSTRACT FROM AUTHOR]
We compare the dynamics of quantum wave packets with the dynamics of classical trajectory ensembles. The wave packets are Gaussian with expectation values of position and momenta which centers them in phase space. The classical trajectory ensembles are generated directly from the quantum wave packets via the Wigner transform. Quantum and classical dynamics are then compared using several quantum measures and the analogous classical ones derived from the Wigner equivalent formalism. Comparisons are made for several model potentials and it is found that there is generally excellent classical–quantum correspondence except for certain specific cases of tunneling and interference. In general, this correspondence is also very good in regions of phase space where there is classical chaos.
Two methods for calculating the good action variables and semiclassical eigenvalues for coupled oscillator systems are presented, both of which relate the actions to the coefficients appearing in the Fourier representation of the normal coordinates and momenta. The two methods differ in that one is based on the exact expression for the actions together with the EBK semiclassical quantization condition while the other is derived from the Sorbie–Handy (SH) approximation to the actions. However, they are also very similar in that the actions in both methods are related to the same set of Fourier coefficients and both require determining the perturbed frequencies in calculating actions. These frequencies are also determined from the Fourier representations, which means that the actions in both methods are determined from information entirely contained in the Fourier expansion of the coordinates and momenta. We show how these expansions can very conveniently be obtained from fast Fourier transform (FFT) methods and that numerical filtering methods can be used to remove spurious Fourier components associated with the finite trajectory integration duration. In the case of the SH based method, we find that the use of filtering enables us to relax the usual periodicity requirement on the calculated trajectory. Application to two standard Henon–Heiles models is considered and both are shown to give semiclassical eigenvalues in good agreement with previous calculations for nondegenerate and 1:1 resonant systems. In comparing the two methods, we find that although the exact method is quite general in its ability to be used for systems exhibiting complex resonant behavior, it converges more slowly with increasing trajectory integration duration and is more sensitive to the algorithm for choosing perturbed frequencies than the SH based method. The SH based method is less straightforward to use in studying resonant systems, but good results are obtained for 1:1 resonant systems using actions defined in terms of the complex coordinates Q1±iQ2. The SH based method is also shown to be remarkably accurate in determining high energy eigenvalues (about three‐quarters of the dissociation energy).
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.