Phase quantization of chaos in the semiclassical regime

Since the early stage of the study of Hamilton chaos, semiclassical quantization based on the low-order Wentzel-Kramers-Brillouin theory, the primitive semiclassical approximation to the Feynman path integrals (or the so-called Van Vleck propagator), and their variants have been suffering from difficulties such as divergence in the correlation function, nonconvergence in the trace formula, and so on. These difficulties have been hampering the progress of quantum chaos, and it is widely recognized that the essential drawback of these semiclassical theories commonly originates from the erroneous feature of the amplitude factors in their applications to classically chaotic systems. This forms a clear contrast to the success of the Einstein-Brillouin-Keller quantization condition for regular (integrable) systems. We show here that energy quantization of chaos in semiclassical regime is, in principle, possible in terms of constructive and destructive interference of phases alone, and the role of the semiclassical amplitude factor is indeed negligibly small, as long as it is not highly oscillatory. To do so, we first sketch the mechanism of semiclassical quantization of energy spectrum with the Fourier analysis of phase interference in a time correlation function, from which the amplitude factor is practically factored out due to its slowly varying nature. In this argument there is no distinction between integrability and nonintegrability of classical dynamics. Then we present numerical evidence that chaos can be indeed quantized by means of amplitude-free quasicorrelation functions and Heller's frozen Gaussian method. This is called phase quantization. Finally, we revisit the work of Yamashita and Takatsuka [Prog. Theor. Phys. Suppl. 161, 56 (2007)] who have shown explicitly that the semiclassical spectrum is quite insensitive to smooth modification (rescaling) of the amplitude factor. At the same time, we note that the phase quantization naturally breaks down when the oscillatory nature of the amplitude factor is comparable to that of the phases. Such a case generally appears when the Planck constant of a large magnitude pushes the dynamics out of the semiclassical regime.     

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.     

Energy spectrum of the Manning-Rosen potential including centrifugal term solved by exact and proper quantization rules

The energy spectrum of the Manning-Rosen potential including centrifugal term in higher dimensions is presented by exact quantization rule approach. The result is compared with that by proper quantization rule method. It is found that the latter is better than that of the exact quantization rule. We find that the interdimensional degeneracy exists for the states in different dimensions. For the special case D = 3, the results agree well with those obtained by other methods.     

Classical photodissociation dynamics with Bohr quantization

The standard classical expression of the state-resolved photodissociation cross section is not consistent with an efficient Bohr quantization of product internal motions. A new and strictly equivalent expression not suffering from this drawback is proposed. This expression opens the way to more realistic classical simulations of direct polyatomic photodissociations in the quantum regime where only a few states are available to the products.     

A semiclassical model for the vibrational excitation of spherical-top molecules in collisions with ions

A simple theory is presented to explain the previously observed single-mode vibrational excitation of the spherical-top molecules CH₄, CF₄ and SF₆ in collisions with H⁺ and Li⁺. The theory is based on a three-dimensional forced-oscillator model which has been modified to take account of many independent harmonic oscillators. For small-angle collisions the linear driving forces are the dipole-, polarizability- and quadrupole-derivatives taken from IR, Raman spectroscopy and simple estimates, respectively. To explain the results at larger angles near and beyond the rainbow it has been necessary to introduce short-range repulsive forces between the ions and the outer atoms of the molecule. For small angles both the predicted first moment of the energy transfer and the time-of-flight spectra agree quantitatively with the experimental results. At large angles, for which only the first moment of the energy is available, good qualitative agreement is obtained after a slight adjustment of the potential parameters. The energy transfer as a function of time is calculated and shows a different oscillatory behavior for the proton and Li⁺-ion systems. Also the effect of intra-mode coupling is investigated and shown to have only a small effect on the overall energy transfer. The paper closes with a discussion of the implications of these experiments and the possible role of rotational excitation. The field strengths in these ion scattering experiments are shown to be greater than in the strongest focused Q-switched laser pulses.     

Order from chaos with molecular beams

The use of molecular beam methods in the elucidation of collision processes in gases is traced from its beginnings early in this century. Some results which have been obtained are described. Limitations as well as advantages of the method are pointed out. Prospects for the future are predicted. Based on an Academy lecture delivered at Bangalore on 23 February 1979.     

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.     

Thin-layer chromatography with high-boiling mobile phases.1. Investigation of the flow of the high-boiling mobile phases

This paper considers some aspects of a new TLC technique using a molten mobile phase which is solid under ambient conditions. The flow of high-boiling mobile phase at elevated temperature in thin-layer chromatography has been investigated and it is shown that the equation Z_f² = kt is not applicable to migration of the high boiling mobile phase front. The flow stability of the high-boiling mobile phase is noted. It is suggested on the basis of studies of concentration profiles of the solidified mobile phases by scanning photometers that the shape of the mobile phase concentration profile be taken into account in calculation of R_f Values.     

Analysis of the unit cell dimensions of phases with the ThMn12 structure II: Rare earth-manganese phases

It is shown that the M-N(2) and M-N(3) contacts control the cell dimensions of binary rare earth-manganese phases with the ThMn₁₂ (MN₁₂) structure. Such behaviour is quite different from that found for other binary and ternary phases with this structure, and it recalls similar differences found for rare earth-manganese phases with the MgCu₂ structure compared with the rare earth phases of other transition metals. The cell dimensions of YMn₁₂ are also discussed.     

A semiclassical molecular dynamics of the photochromic ring-opening reaction of spiropyran

The photochromic ring-opening reaction of spiropyran(SP) has been investigated by a realistic semiclassical dynamics simulation,accompanied by SA3-CASSCF(12 10)/MS-CASPT2 potential energy curves(PECs) of S0–S2.The main simulation results show the dominate pathway corresponds to the ringopening process of trans-SP to form the most stable merocyanine(MC) product.These findings provide more important complementarity for interpreting experimental observations.     

Energy spectrum for a modified Rosen-Morse potential solved by proper quantization rule and its thermodynamic properties

We apply our recently proposed proper quantization rule, ò_xA^x_Bk(x) dx -ò_x0A^x_0Bk₀(x) dx=np{\int_{x_A}^{x_B}k(x) dx -\int_{x_{0A}}^{x_{0B}}k_0(x) dx=n\pi} , where _boxclose_boxclose_boxclose_boxclose_boxclose_boxclose_boxclose_boxclose/{k(x)=\sqrt{2 M [E-V(x) ]}/\hbar} to obtain the energy spectrum of the modified Rosen-Morse potential. The beauty and symmetry of this proper rule come from its meaning—whenever the number of the nodes of f(x){\phi(x)} or the number of the nodes of the wave function ψ(x) increases by one, the momentum integral ò_xA^x_B k(x)dx{\int_{x_A}^{x_B} k(x)dx} will increase by π. Based on this new approach, we present a vibrational high temperature partition function in order to study thermodynamic functions such as the vibrational mean energy U, specific heat C, free energy F and entropy S. It is surprising to note that the specific heat C (k = 1) first increases with β and arrives to the maximum value and then decreases with it. However, it is shown that the entropy S (k = 1) first increases with the deepness of potential well λ and then decreases with it.     

Emulsions with structured continuous phases

Recent research on the role of the continuous phase in emulsions is reviewed. Special attention is given to the structured continuous phases such as lyotropic and thermotropic liquid crystals and gel networks. The implications in the formation, stability and properties of emulsions are discussed. Some recent applications, particularly in templating for the preparation of new materials, are also introduced.     

Measuring nonadiabaticity of molecular quantum dynamics with quantum fidelity and with its efficient semiclassical approximation

We propose to measure nonadiabaticity of molecular quantum dynamics rigorously with the quantum fidelity between the Born-Oppenheimer and fully nonadiabatic dynamics. It is shown that this measure of nonadiabaticity applies in situations where other criteria, such as the energy gap criterion or the extent of population transfer, fail. We further propose to estimate this quantum fidelity efficiently with a generalization of the dephasing representation to multiple surfaces. Two variants of the multiple-surface dephasing representation (MSDR) are introduced, in which the nuclei are propagated either with the fewest-switches surface hopping or with the locally mean field dynamics (LMFD). The LMFD can be interpreted as the Ehrenfest dynamics of an ensemble of nuclear trajectories, and has been used previously in the nonadiabatic semiclassical initial value representation. In addition to propagating an ensemble of classical trajectories, the MSDR requires evaluating nonadiabatic couplings and solving the Schro?dinger (or more generally, the quantum Liouville-von Neumann) equation for a single discrete degree of freedom. The MSDR can be also used in the diabatic basis to measure the importance of the diabatic couplings. The method is tested on three model problems introduced by Tully and on a two-surface model of dissociation of NaI.