Energy quantization of chaos with the semiclassical phases alone

The mechanism of energy quantization is studied for classical dynamics on a highly anharmonic potential, ranging from integrable, mixed, and chaotic motions. The quantum eigenstates (standing waves) are created by the phase factors (the action integrals and the Maslov index) irrespective of the integrability, when the amplitude factors are relatively slowly varying. Indeed we show numerically that the time Fourier transform of an approximate semiclassical correlation function in which the amplitude factors are totally removed reproduces the spectral positions (energy eigenvalues) accurately in chaotic regime. Quantization with the phase information alone brings about dramatic simplification to molecular science, since the amplitude factors in the lowest order semiclassical approximation diverge exponentially in a chaotic domain.     

Symmetry-adapted correlation function for semiclassical quantization

We study a very simple method to incorporate quantum-mechanical symmetries, including the permutational symmetry on an equal footing with spatial symmetries, into the semiclassical calculation of correlation functions. This method is applied to the calculation of energy spectra to verify its validity by reproducing quantum energy levels for systems of bosons (symmetrized) and fermions (antisymmetrized). The mechanism of how the phase-space structure of classical dynamics is linked with the relevant quantum symmetry is discussed.     

Interference and quantization in semiclassical response functions

Application of the Herman-Kluk semiclassical propagator to the calculation of spectroscopic response functions for anharmonic oscillators has demonstrated the quantitative accuracy of these approximate dynamics. In this approach, spectroscopic response functions are expressed as multiple phase-space integrals over pairs of classical trajectories and their associated stability matrices. Here we analyze the Herman-Kluk semiclassical approximation to a linear response function and determine the origin of the capacity of this method to reproduce quantum effects in a response function from classical dynamical information. Our analysis identifies those classical trajectories that contribute most significantly to the response function on different time scales. This finding motivates a procedure for computing the linear response function in which the interference between pairs of classical trajectories is treated approximately, resulting in an integral over a single average trajectory, as in a purely classical calculation.     

Green's function semiclassical quantization of non-closed quasiperiodic classical trajectories

A semiclassical quantization procedure is introduced within the Green's function formalism for non-closed quasiperiodic trajectories which have previously defied quantization with this formalism. The results are interpreted in terms of the uncertainty principle and lead to conjectures concerning the semiclassical quantization of ergodic trajectories.     

On the validity range of the Born-Oppenheimer approximation: a semiclassical study for all-particle quantization of three-body Coulomb systems

The validity range of the Born-Oppenheimer (BO) approximation is studied with respect to the variation of the mass (m) of negatively charged particle by substituting an electron (e) with muon (mu) and antiproton (p) in hydrogen molecule cation. With the use of semiclassical quantization applied to these (ppe), (ppmu), and (ppp) under a constrained geometry, we estimate the energy difference of the non-BO vibronic ground state from the BO counterpart. It is found that the error in the BO approximation scales to the power of 3/2 to the mass of negative particles, that is, m(1.5). The origin of this clear-cut relation is analyzed based on the original perturbation theory due to Born and Oppenheimer, with which we show that the fifth order term proportional to m(5/4) is zero and thereby the first correction to the BO approximation should arise from the sixth order term that is proportional to m(6/4). Therefore, the validity range of the Born-Oppenheimer approximation is wider than that often mistakenly claimed to be proportional to m(1/4).     

A semiclassical theory for nonseparable rovibrational motions in curved space and its application to energy quantization of nonrigid molecules

The nonseparability of vibrational and rotational motions of a nonrigid molecule placed in the rotationally isotropic space induces several important effects on the dynamics of intramolecular energy flow and chemical reaction. However, most of these studies have been performed within the framework of classical mechanics. We present a semiclassical theory for the motions of such nonrigid molecules and apply to the energy quantization of three body atomic cluster. It is shown numerically that the semiclassical spectum given without the correct account of the rotational symmetry suffers from unnecessary broadening of the resultant spectral lines and moreover from spurious peaks.     

Thiol-stabilized CdSe and CdTe nanocrystals in the size quantization regime: Synthesis,optical and structural properties

We report on the recently developed method for the synthesis, optical, and structural properties of CdSe and CdTe nanocrystals. They were formed in aqueous solutions at moderate temperatures by a wet chemical route in the presence of thiol molecules as effective stabilizing agents. The size-selective precipitation technique was applied for the post-preparative nanoparticle fractionation into a series of CdSe and CdTe nanocrystals with extremely narrow size distributions exhibiting mean cluster sizes in the range of 2 to 4 nm. The nature of stabilizing agent (mercaptoalcohols and mercaptoacids) had an important influence on the particle size and determines largely the photoluminescence properties. The nanocrystals were characterized by means of UV-vis absorption and photoluminescence spectroscopy, X-ray diffraction, and high resolution transmission electron microscopy (HRTEM).     

Phase separation of symmetric polymer mixtures in a common good solvent in the semidilute concentration regime

Monte Carlo simulations of lattice models of binary (AB) symmetric polymer mixtures (chain lengthsN _A=N _B=N) in a common good solvent are carried out and the phase diagrams and critical properties of the unmixing transitions are estimated and interpreted in terms of recent theories. Polymers are modeled by self-avoiding walks of lengthN=16, 32 and 64 on the simple cubic lattice. Data for vacancy concentrations of _V=0.6, 0.8 and 0.85 are analyzed. It is shown that forN=16, _V=0.85 no phase separation occurs, down to the lowest temperature, while forN=32, _V=0.85 still phase separation occurs but no longer is complete. Our results are compatible with a scaling theory based on a renormalization of the Flory-Huggins -parameter due to blob effects.Dedicated to Prof. Fischer on the occasion of his 65th birthday.Both of the authors are deeply indebted to Prof. E. W. Fischer, not only for help, support, and enlightening discussions, but even for bringing them together and suggesting to them to collaborate! By this catalytic action he actually played a decisive role in creating a longstanding and successful collaboration, the latest results of which are presented below.     

Quantum effects in semiclassical treatment of molecular reactions

By the use of the known Coulomb wavefunctions, “quantal” Coulomb streamlines are defined for the eikonal of the system. Comparisons of such streamlines with the classical Coulomb trajectories are made. Use of such “quantum” Coulomb streamlines is proposed for reaction systems containing the Coulomb interaction U. The derived equations of motion are of particular use for cases where the residual potential V₁ (V₁ = V ? U) has either no classical forbidden region or has relatively small effects on trajectories.     

Ambiguities in the semiclassical assignment of the asymmetric rotor rotational quantum numbers

The semiclassical quantization of the rigid asymmetric rotor is revisited in the context of classical inelastic collisions. It is shown that the standard bin histogram method, widely used in quasiclassical trajectory calculations involving linear target molecules, cannot be generalized to the case of asymmetric top molecules owing to ambiguities in the assignment of the final classical action to a particular rotational quantum state. These ambiguities result from pairs of states which are indistinguishable within the bin histogram approach at all the common levels of semiclassical theory. A single value of the classical action can thus correspond to two different quantum states, preventing the distinction between these states in the calculation of rotational cross sections. Our results are illustrated for the rotational states J=1-4 of the water molecule at its equilibrium geometry.     

Herman-Kluk semiclassical dynamics of molecular rotations in laser fields

The action-angle mapping algorithm [R. Saha and M. Ovchinnikov, J. Chem. Phys. 124, 204112 (2006)] is utilized to provide a Herman-Kluk semiclassical initial value representation (SC-IVR) treatment of quantum dynamics of systems with non-Cartesian degrees of freedom. The non-Cartesian system under investigation is a linear rotor molecule in static electric and pulsed laser field. The results demonstrate that the SC-IVR procedure described in this work provides an accurate representation of quantum rotational dynamics of the system.     

Optimized Monte Carlo sampling in forward-backward semiclassical dynamics

Forward-backward semiclassical dynamics (FBSD) provides a rigorous and powerful methodology for calculating time correlation functions in condensed phase systems characterized by substantial quantum mechanical effects associated with zero-point motion, quantum dispersion, or identical particle exchange symmetries. The efficiency of these simulations arises from the use of classical trajectories to capture all dynamical information. However, full quantization of the density operator makes these calculations rather expensive compared to fully classical molecular dynamics simulations. This article discusses the convergence properties of various correlation functions and introduces an optimal Monte Carlo sampling scheme that leads to a significant reduction of statistical error. A simple and efficient procedure for normalizing the FBSD results is also discussed. Illustrative examples on model systems are presented.     

A semiclassical model of polarisation forces in atomic scattering

Low-energy elastic, differential and integral, collision cross sections for electron scattering from rare gas atoms are calculated using a simplified model treatment of the short-range polarisation forces already applied successfully to helium atoms (De Fazio et al., 1994). Static and exchange contributions to the total interaction are treated exactly, while a global semiclassical model provides a simple procedure for the short-range damping of all the long-range adiabatic terms which asymptotically lead to the polarisation forces. The results are compared with several available experimental findings for Ne and Ar atoms. The higher order terms in the perturbation expansion are found to have little effect on the cross sections and then only at low energies and in the small-angle region.Von Humboldt Stiftung Forschungspreisträger (1992)     

First principles semiclassical calculations of vibrational eigenfunctions

Vibrational eigenfunctions are calculated on-the-fly using semiclassical methods in conjunction with ab initio density functional theory classical trajectories. Various semiclassical approximations based on the time-dependent representation of the eigenfunctions are tested on an analytical potential describing the chemisorption of CO on Cu(100). Then, first principles semiclassical vibrational eigenfunctions are calculated for the CO(2) molecule and its accuracy evaluated. The multiple coherent states initial value representations semiclassical method recently developed by us has shown with only six ab initio trajectories to evaluate eigenvalues and eigenfunctions at the accuracy level of thousands trajectory semiclassical initial value representation simulations.     

The massey parameter expansion for the semiclassical scattering matrix

The Massey parameter expansion method for the semiclassical scattering matrix is proposed, using the Magnus formalism. The first Massey parameter, a unitary member of this expansion, is found for the interaction hamiltonian, depending on time, as exp(-τ) and τ^?n. Other possible applications of the proposed method are discussed.     

All-forward semiclassical simulations of nonlinear response functions

We propose a quantum trajectory algorithm for computing nonlinear response functions of condensed phase molecular systems based on a time-ordered expansion of the density matrix. The nth-order response function is expressed as a sum of 2(n) impulsive response pathways representing trajectories involving zero, one, and up to n interactions with short external pulses. These are evaluated using a forward propagation algorithm based upon a Liouville space extension of the Bohmian propagation method.     

A semiclassical study of wave packet dynamics in anharmonic potentials

Classical and semiclassical methods are developed to calculate and invert the wave packet motion measured in pump-probe experiments. With classical propagation of the Wigner distribution of the initial wave packet created by the pump pulse, we predict the approximate probe signal with slightly displaced recurrence peaks, and derive a set of first-order canonical perturbation expressions to relate the temporal features of the signal to the characteristics of the potential surface. A reduced dynamics scheme based on the Gaussian assumption leads to the correct center of mass motion but does not describe the evolution of the shape of the wave packet accurately. To incorporate the quantum interference into classical trajectories, we propose a final-value representation semiclassical method, specifically designed for the purpose of computing pump-probe signals, and demonstrate its efficiency and accuracy with a Morse oscillator and two kinetically coupled Morse oscillators. For the case of one-color pump probe, a simple phase-space quantization scheme is devised to reproduce the temporal profile at the left-turning point without actual wave packet propagation, revealing a quantum mechanical perspective of the nearly classical pump-probe signal.