Semiclassical Study Research Articles

Local behaviour of evaporating stars and black holes around the total evaporation event

Many models in which the object under study loses all its mass have appeared in the literature. This can be found from evaporating stars to incipient black holes, all the way to evaporating black holes. In this paper we try a semiclassical study of these evaporating models centred on the evaporating event itself. We analyse their common properties, behaviours and possibilities. Specifically, we pay special attention to the evaporating models as a means of avoiding singularities during the collapse. In the case of any pre-existing non-spacelike curvature singularity, we show that these models tend to evaporate it. Finally, we introduce a new class of evaporating black holes.

Quantum dynamical effects in liquid water: A semiclassical study on the diffusion and the infrared absorption spectrum

The important role of liquid water in many areas of science from chemistry, physics, biology, geology to climate research, etc., has motivated numerous theoretical studies of its structure and dynamics. The significance of quantum effects on the properties of water, however, has not yet been fully resolved. In this paper we focus on quantum dynamical effects in liquid water based on the linearized semiclassical initial value representation (LSC-IVR) with a quantum version of the simple point charge/flexible (q-SPC/fw) model [Paesani et al., J. Chem. Phys. 125, 184507 (2006)] for the potential energy function. The infrared (IR) absorption spectrum and the translational diffusion constants have been obtained from the corresponding thermal correlation functions, and the effects of intermolecular and intramolecular correlations have been studied. The LSC-IVR simulation results are compared with those predicted by the centroid molecular dynamics (CMD) approach. Although the LSC-IVR and CMD results agree well for the broadband for hindered motions in liquid water, the intramolecular bending and O-H stretching peaks predicted by the LSC-IVR are blueshifted from those given by CMD; reasons for this are discussed. We also suggest that the broadband in the IR spectrum corresponding to restricted translation and libration gives more information than the diffusion constant on the nature of quantum effects on translational and rotational motions and should thus receive more attention in this regard.

A semi-classical study of the Jaynes–Cummings model

We consider the Jaynes–Cummings model of a single quantum spins coupled to a harmonic oscillator in a parameter regime where the underlying classicaldynamics exhibits an unstable equilibrium point. This state of the model is relevant tothe physics of cold atom systems, in non-equilibrium situations obtained by fastsweeping through a Feshbach resonance. We show that in this integrable systemwith two degrees of freedom, for any initial condition close to the unstable point,the classical dynamics is controlled by a singularity of the focus–focus type. Inparticular, it displays the expected monodromy, which forbids the existence ofglobal action-angle coordinates. Explicit calculations of the joint spectrum ofconserved quantities reveal the monodromy at the quantum level, as a dislocation inthe lattice of eigenvalues. We perform a detailed semi-classical analysis of theassociated eigenstates. Whereas most of the levels are well described by the usualBohr–Sommerfeld quantization rules, properly adapted to polar coordinates, we showhow these rules are modified in the vicinity of the critical level. The spectraldecomposition of the classically unstable state is computed, and is found to bedominated by the critical WKB states. This provides a useful tool with which toanalyze the quantum dynamics starting from this particular state, which exhibits anaperiodic sequence of solitonic pulses with a rather well defined characteristicfrequency.

Trans–cis photoisomerization of azobenzene by n → π * excitation: A semiclassical dynamics study

A realistic dynamics simulation study is reported for the trans–cis photoisomerization of azobenzene triggered by the n → π * excitation and the results show that the formation of cis isomer follows the rotational motion around the N N bond. The simulation find that the CNN bond angle bending vibrations also play a significant role in the vibronic coupling between the HOMO and LUMO, which essentially leads a nonadiabatic transition of the molecule to the electronic ground state.

Semiclassical study of the one-dimensional hydrogen molecule

The hydrogen molecule, as a restricted four-body problem, is a mixed chaotic system and is studied in this work as an extension of the helium atom. Several types of one-dimensional periodic orbits have been studied for this two-fixed-center system starting from some known orbits for the one-fixed-center system (helium atom). The hydrogen molecule has been studied by means of a semiclassical formalism. The single quantization of some periodic orbits are shown. These orbits are used to form a global quantization of the hydrogen molecule. Electronic energies of this nonintegrable molecular system are obtained with nonsingle semiclassical quantization.

Adiabatic States Derived from a Spin-Coupled Diabatic Transformation: Semiclassical Trajectory Study of Photodissociation of HBr and the Construction of Potential Curves for LiBr+

The development of spin-coupled diabatic representations for theoretical semiclassical treatments of photodissociation dynamics is an important practical goal, and some of the assumptions required to carry this out may be validated by applications to simple systems. With this objective, we report here a study of the photodissociation dynamics of the prototypical HBr system using semiclassical trajectory methods. The valence (spin-free) potential energy curves and the permanent and transition dipole moments were computed using high-level ab initio methods and were transformed to a spin-coupled diabatic representation. The spin-orbit coupling used in the transformation was taken as that of atomic bromine at all internuclear distances. Adiabatic potential energy curves, nonadiabatic couplings and transition dipole moments were then obtained from the diabatic ones and were used in all the dynamics calculations. Nonadiabatic photodissociation probabilities were computed using three semiclassical trajectory methods, namely, coherent switching with decay of mixing (CSDM), fewest switches with time uncertainty (FSTU), and its recently developed variant with stochastic decoherence (FTSU/SD), each combined with semiclassical sampling of the initial vibrational state. The calculated branching fraction to the higher fine-structure level of the bromine atom is in good agreement with experiment and with more complete theoretical treatments. The present study, by comparing our new calculations to wave packet calculations with distance-dependent ab initio spin-orbit coupling, validates the semiclassical trajectory methods, the semiclassical initial state sample scheme, and the use of a distance-independent spin-orbit coupling for future applications to polyatomic photodissociation. Finally, using LiBr(+) as a model system, it is shown that accurate spin-coupled potential curves can also be constructed for odd-electron systems using the same strategy as for HBr.

Semiclassical wave packet study of anomalous isotope effect in ozone formation

We applied the semiclassical initial value representation method to calculate energies, lifetimes, and wave functions of scattering resonances in a two-dimensional potential for O+O2 collision. Such scattering states represent the metastable O3* species and play a central role in the process of ozone formation. Autocorrelation functions for scattering states were computed and then analyzed using the Prony method, which permits one to extract accurate energies and widths of the resonances. We found that the results of the semiclassical wave packet propagation agree well with fully quantum results. The focus was on the 16O16O18O isotopomer and the anomalous isotope effect associated with formation of this molecule, either through the 16O16O+18O or the 16O+16O18O channels. An interesting correlation between the local vibration mode character of the metastable states and their lifetimes was observed and explained. New insight is obtained into the mechanism by which the long-lived resonances in the delta zero-point energy part of spectrum produce the anomalously large isotope effect.

Complex trajectories in chaotic dynamical tunneling

We develop the semiclassical method of complex trajectories in application to chaotic dynamical tunneling. First, we suggest a systematic numerical technique for obtaining complex tunneling trajectories by the gradual deformation of the classical ones. This provides a natural classification of the tunneling solutions. Second, we present a heuristic procedure for sorting out the least suppressed trajectory. As an illustration, we apply our technique to the process of chaotic tunneling in a quantum mechanical model with two degrees of freedom. Our analysis reveals rich dynamics of the system. At the classical level, there exists an infinite set of unstable solutions forming a fractal structure. This structure is inherited by the complex tunneling paths and plays a central role in the semiclassical study. The process we consider exhibits the phenomenon of optimal tunneling: the suppression exponent of the tunneling probability has a local minimum at a certain energy which is thus (locally) the optimal energy for tunneling. We test the proposed method by comparison of the semiclassical results with the results of the exact quantum computations and find a good agreement.

Raychaudhuri equation in quantum gravitational optics

The equation of Raychaudhuri is one of the key concepts in the formulation of the singularity theorems introduced by Penrose and Hawking. In the present article, taking into account QED vacuum polarization, we study the propagation of a bundle of rays in a background gravitational field through the perturbative deformation of Raychaudhuri’s equation. In a sense, this could be seen as another semiclassical study in which geometry is treated classically but matter (which means the photon here) is allowed to exhibit quantum characteristics that are encoded in its coupling to the background curvature.

Curvature maxima in two-state systems: A semi-classical study

In an interacting two (electronic-) state model of molecules with one (effective) nuclear degree of freedom the normalized mean square deviation of the curvature (NMSDC) shows a temperature-maximum. The maximum increases with decrease of the interaction strength, but undergoes a sharp transition to no maximum, when the interaction term vanishes. The NMSDC maximum serves as a geometric hallmark for the underlying potential. The analogous result found by Mazzoni and Casetti [L.N. Mazzoni, L. Casetti, Phys. Rev. Lett. 97 (2006) 218104] for folding proteins can be reproduced by this elementary interacting two state model.

Ring opening reaction of 1,3-cyclohexadiene studied by semiclassical dynamics simulation

A semiclassical dynamics simulation study is reported for the ring opening reaction of 1,3-cyclohexadiene (CHD) triggered by a femtosecond-scale laser pulse. The results clearly demonstrate that, following the excitation by the laser pulse, the ring opening occurs at ∼110 fs and the molecule decays to the ground electronic state at ∼210 fs due to non-adiabatic transition of electrons from LUMO to HOMO orbitals. Isomerization of the product of the ring opening reaction, 1,3,5-hexatriene (HT), to various stable isomers are also well demonstrated by the simulations.

Semiclassical wave packet study of ozone forming reaction

We have applied the semiclassical wave packet method (SWP) to calculate energies and lifetimes of the metastable states (scattering resonances) in a simplified model of the ozone forming reaction. All values of the total angular momentum up to J=50 were analyzed. The results are compared with numerically exact quantum mechanical wave packet propagation and with results of the time-independent WKB method. The wave functions for the metastable states in the region over the well are reproduced very accurately by the SWP; in the classically forbidden region and outside of the centrifugal barrier, the SWP wave functions are qualitatively correct. Prony's method was used to extract energies and lifetimes from the autocorrelation functions. Energies of the metastable states obtained using the SWP method are accurate to within 0.1 and 2 cm(-1) for under-the-barrier and over-the-barrier states, respectively. The SWP lifetimes in the range of 0.5<tau(n)<100 ps are accurate to within 10%. A three-level model was used to investigate accuracies of different approximations for the reaction rate constant. It was shown that the majority of the metastable states in this system are either long lived (narrow resonances) which can be treated as stable, or short lived (broad resonances) which can be treated without the knowledge of their lifetimes. Only a few metastable states fall into the intermediate range where both energies and lifetimes are needed to model the kinetics. The recombination rate constant calculated with the SWP method at room temperature and pressure is in good agreement with available experimental data.

N atoms recombination on a silica surface: A global theoretical approach

The paper reports on a semiclassical dynamics study about N 2 formation after nitrogen atom recombination on a silica surface via the Eley–Rideal collisional mechanism at T = 1000 K. The molecular dynamics study was carried out using a fitted potential energy surface based on new density functional computations on clusters cleaved from the β-cristobalite unit cell. The B3LYP results obtained using silica clusters of increasing dimensions (up to Si 7O 14H 14) show that only atomic nitrogen is adsorbed (binding energy E b ≅ 2.7–2.8 eV). Then, the obtained potential energy surface has been used in a time-dependent semiclassical collisional method which is able to describe, at a very detailed level, the recombination of N atoms on the surface assisted by the phonon excitation mechanism of the substrate. The recombination coefficient γ of the Eley–Rideal recombination mechanism was calculated together with the reaction energetics.

Phase space propagators for quantum operators

The Heisenberg evolution of a given unitary operator corresponds classically to a fixed canonical transformation that is viewed through a moving coordinate system. The operators that form the bases of the Weyl representation and its Fourier transform, the chord representation are, respectively, unitary reflection and translation operators. Thus, the general semiclassical study of unitary operators allows us to propagate arbitrary operators, including density operators, i.e., the Wigner function. The various propagation kernels are different representations of the super-operators which act on the space of operators of a closed quantum system. We here present the mixed semiclassical propagator, that takes translation chords to reflection centres, or vice versa. In contrast to the centre–centre propagator that directly evolves Wigner functions, they are guaranteed to be caustic free, having a simple WKB-like universal form for a finite time, whatever the number of degrees of freedom. Special attention is given to the near-classical region of small chords, since this dominates the averages of observables evaluated through the Wigner function.

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).

The quantum Neumann model: Refined semiclassical results

We extend the semiclassical study of the Neumann model down to the deep quantum regime. A detailed study of connection formulae at the turning points allows to get good matching with the exact results for the whole range of parameters.

Vibrational energy transfer in N2–N2 collisions: A new semiclassical study

The vibrational energy relaxation in collisions between N2 molecules in the low- and medium-lying vibrationally excited levels was revisited using the semiclassical coupled-state method and the use of two different potential-energy surfaces having the same short-range potential recently determined from ab initio calculations but with different long-range interactions. Compared to the data reported in the classical work by Billing and Fisher [Chem. Phys. 43, 395 (1979)], the newly calculated vibration-to-translation rate constant K(1,0 / 0,0) is in much better agreement with the available experimental data over a large temperature interval, from T = 200 K up to T = 6000 K. Nevertheless, as far as the vibration-to-translation exchanges are concerned, the lower-temperature regime remains quite critical in that the new rate constants do not completely account for the rate constant curvature suggested by the experiments for temperatures lower than T = 500 K. The dependence of the state-selected vibration-to-vibration rate constants, K(v,v-delta v / 0,1), both upon the vibrational quantum number v and the gas temperature are calculated. The substantial deviations from previously found behaviors could have major consequences for the vibrational kinetic modeling of N2-containing gas mixtures.

Dynamical hydrogen atom tunneling in dichlorotropolone: A combined quantum, semiclassical, and classical study

Based on the Cartesian Reaction Surface framework we construct a four-dimensional potential for the tropolone derivative 3,7-dichlorotropolone, a molecule with an intramolecular O-H...O hydrogen bond. The reduced configuration space involves the in-plane hydrogen atom coordinates, a symmetric O-O vibrational mode, and an antisymmetric mode related to deformations of the seven-membered ring. The system is characterized in terms of quantum mechanical computations of the low-lying eigenstates as well as a classical and semiclassical analysis of spectra obtained via Fourier transforming autocorrelation functions. For the semiclassical analysis we utilize the amplitude-free correlation function method [K. Hotta and K. Takatsuka, J. Phys. A 36, 4785 (2003)]. Our results demonstrate substantial anharmonic couplings leading to highly correlated wave functions even at moderate energies. Furthermore, the importance of dynamical tunneling in tropolone is suggested since many low-lying states--including the ground state--lie above the classical saddle point but nevertheless appear as split pairs.

Quantum and semiclassical study of magnetic quantum dots

We study the energy level structure of two-dimensional charged particles in inhomogeneous magnetic fields. In particular, for magnetic anti-dots the magnetic field is zero inside the dot and constant outside. Such a device can be fabricated with present-day technology. We present detailed semiclassical studies of such magnetic anti-dot systems and provide a comparison with exact quantum calculations. In the semiclassical approach we apply the Berry-Tabor formula for the density of states and the Borh-Sommerfeld quantization rules. In both cases we found good agreement with the exact spectrum in the weak magnetic field limit. The energy spectrum for a given missing flux quantum is classified in six possible classes of orbits and summarized in a so-called phase diagram. We also investigate the current flow patterns of different quantum states and show the clear correspondence with classical trajectories.

Quantum-mechanical and semiclassical study of the collinear three-body Coulomb problem: Resonances

This paper continues a quantum-mechanical (QM) and semiclassical (SC) study of the collinear three-body Coulomb problem [O. I. Tolstikhin and C. Namba, 70, 062721 (2004)], extending it to resonances. Our QM treatment is based on the theory of Siegert pseudostates (SPS) implemented in hyperspherical coordinates by means of the slow/smooth variable discretization method. A consistent formulation of this approach for arbitrary dimension of configuration space and number of open channels is given. Not only resonance energy and total width, but also partial widths are defined in terms of the SPS; for the cases of one and two open channels, they are expressed in terms of the SPS eigenvalues only. The SC results are obtained in the leading-order approximation from the asymptotic solution of the problem for $h\ensuremath{\rightarrow}0$, where $h$ is a dimensionless parameter that depends only on the masses of particles, varies in the interval $0\ensuremath{\leqslant}h\ensuremath{\leqslant}1$, and has the meaning of an effective Planck's constant for the motion in hyperradius. It is shown that resonance widths as functions of $1∕h$ oscillate with an exponentially decaying amplitude and almost constant period. The SC theory qualitatively explains such a behavior as a manifestation of interference effects in the nonadiabatic process of decay of the resonance state and provides surprisingly good quantitative results even for systems with $h\ensuremath{\sim}1$.