ALBERT

Notes: It is known that a new kind of large amplitude motion gives rise to as a very characteristic mode of weak chaos. This is essentially an unpredictable and intermittent motion taking place in a thin quasiseparatrix which wanders among several very clear vibrational modes. In this paper, we study the spectroscopic characterization of the quantum version of this large amplitude motion in terms of the dynamics of a wave packet, which is prepared in a narrow energy-range so that it is localized along a thin quasiseparatrix. In particular, we discuss possible significance of the weak chaos in single molecule spectroscopy, for which the spectra are supposed to be averaged neither in ensemble of molecules nor in time. That this wavepacket state is unusual originates from the extremely long-time behavior and the strong sensitivity to the initial condition at which the wave packet is prepared. The weak chaos combined with the statistical nature of quantum mechanics brings about a notion of unreproducibility in the spectrum. Conversely, it is anticipated that one of the distinguished features inherent to a single molecule spectroscopy manifests itself when weak chaos is observed. © 1995 American Institute of Physics.

Notes: A new method to calculate eigenfunctions and eigenvalues in a given energy range is proposed, which can therefore be applied to highly excited states of electronic and/or vibrational states of a molecule. The spectral components of a wave packet that lie outside the energy range are projected out through the time evolution; that is, the packet is screened onto the energy range. If the range includes only a single root, the corresponding eigenfunction is screened first, and the eigenvalue follows as its expectation value. For a case where there is more than a single root, several methods can be figured out. One typical and effective procedure is to construct local basis functions in terms of the aforementioned energy screened wave packets to represent the Hamiltonian in them and to diagonalize it. The concept to construct a local basis was originally developed by Neuhauser [J. Chem. Phys. 93, 2611 (1990)]. The present method performs it in a more efficient and theoretically satisfactory way. © 1995 American Institute of Physics.

Notes: The validity of the physical premise of the Rice–Ramsperger–Kassel–Marcus (RRKM) theory is investigated in terms of the classical dynamics of isomerization reaction in Ar7-like molecules (clusters). The passage times of classical trajectories through the potential basins of isomers in the structural transitions are examined. In the high energy region corresponding to the so-called liquidlike phase, remarkable uniformity of the average passage times has been found. That is, the average passage time is characterized only by a basin through which a trajectory is currently passing and, hence, does not depend on the next visiting basins. This behavior is out of accord with the ordinary chemical law in that the "reaction rates'' do not seem to depend on the height of the individual potential barriers. We ascribe this seemingly strange uniformity to the strong mixing (chaos) lying behind the rate process. That is, as soon as a classical path enters a basin, it gets involved into a chaotic zone in which many paths having different channels are entangled among each other, and effectively (in the statistical sense) loses its memory about which basin it came from and where it should visit next time. This model is verified by confirming that the populations of the lifetime of transition from one basin to others are expressed in exponential functions, which should have very similar exponents to each other in each passing-through basin. The inverse of the exponent is essentially proportional to the average passage time, and consequently brings about the uniformity. These populations set a foundation for the multichannel generalization of the RRKM theory. Two cases of the non-RRKM behaviors have been studied. One is a nonstatistical behavior in the low energy region such as the so-called coexistence phase. The other is the short-time behavior. It is well established [M. Berblinger and C. Schlier, J. Chem. Phys. 101, 4750 (1994)] that in a relatively simple and small system such as H+3, the so-called direct paths, which lead to dissociation before the phase-space mixing is completed, increase the probability of short-time passage. In contrast, we have found in our Ar7-like molecules that trajectories of short passage time are fewer than expected by the statistical theory. It is conceived that somewhat a long time in the initial stage of the isomerization is spent by a trajectory to find its ways out to the next basins. © 1996 American Institute of Physics.

Notes: In a previous study of isomerization dynamics of clusters as a chaotic conservative system, we proposed a temperature, called the microcanonical temperature [C. Seko and K. Takatsuka, J. Chem. Phys. 104, 8613 (1996)], which is expected to characterize a phase space distribution on a constant energy plane. In contrast to the standard view of equal a priori distribution in phase space, we note a fact that this distribution usually becomes sharply localized with a single peak, if projected onto the potential energy coordinate. The microcanonical temperature is defined as a kinetic energy at which this projected distribution takes the maximum value. Then the most probable statistical events should be dominated by those components in vicinity of the peak, provided that the projected distribution is singly and sharply peaked and the associated dynamics is ergodic. The microcanonical temperature can be similarly redefined in the individual potential basins. Here in the present article a numerical fact is highlighted that the inverse of the lifetime of an isomer bears an Arrhenius-type relation with thus defined local microcanonical temperature assigned to the corresponding potential basin. We present an analysis of how the Arrhenius relation can arise. © 2000 American Institute of Physics.

Notes: It is well-known that a single cluster like Ar7 undergoes "melting'' from solidlike to liquidlike states as the energy is increased, the transition of which is not as sudden as the ordinary phase transition though and has a somewhat broad energy range in which solid and liquid coexist. We study a very anomalous dynamics of the coexistence region in the structural isomerization. It is explicitly shown that the time-series of the structural changes both in the purely solidlike and liquidlike phases are stationary, while the coexistence region is found to generate a strongly nonstationary dynamics. The calculated distribution of the residing times for the cluster to stay in one of the possible structures exhibits a nonexponential form having a large hole around the zero lifetime in the coexistence region. Motivated by these strange behaviors, we have calculated the phase-space volumes that are assigned to the individual potential basins, and verified directly that while the pure liquid region is of ergodic nature, the dynamics in the coexistence region is indeed strongly nonergodic. The steep rises of the Lindemann index and the maximum Liapunov exponent in the coexistence region, which were reported before by other authors, are found to be ascribed to the statistical nature rather than the dynamical properties as opposed to the picture suggested by the physical meaning of the indices. It also turns out that the energy range for the coexistence region should be taken wider than considered before and thus extends beyond the "melting point'' that is defined usually on the basis of the Lindemann index. Therefore it is appropriate to divide the coexistence region into two subphases. A "temperature'' in a microcanonical ensemble is defined so as to characterize the distribution of phase-space volume on a given energy plane. Based on this distribution, we describe a statistical reason why the onset energy of the melting is much higher than those of the transition states. © 1996 American Institute of Physics.

Notes: A quasisemiclassical trajectory method (QSCT) is proposed, which provides a practical procedure to study tunneling chemical reaction dynamics in multidimensional systems. QSCT incorporates the semiclassical tunneling paths that are generated by our previously proposed method [K. Takatsuka and H. Ushiyama, Phys. Rev. A 51, 4353 (1995)] into the so-called quasiclassical trajectory method, whereby the chemical reactions in a wide energy range are calculated in a systematic way. The accuracy of this approach is tested with the system of collinear H+H–H reaction on the so-called LSTH (Liu–Siegbahn–Truhlar–Horowitz) potential surface. The thermal rate constants at 1–3000 K are calculated. The resultant reaction rates are compared with the quantum mechanical values [Bondi et al., J. Chem. Phys. 76, 4986 (1982)], showing that they are in a systematically good agreement in this wide temperature range. We have also examined the dependence of the reaction probability on the initial sampling of the quasiclassical method. The contribution of the paths of dynamical tunneling to chemical reaction above the reaction threshold is estimated for the first time. © 1998 American Institute of Physics.

Notes: In isomerization dynamics of Ar7-like molecules of high energy, which is in the so-called liquidlike phase, a peculiar characteristic has been observed [K. Takatsuka and C. Seko, J. Chem. Phys. 105, 10356 (1996)], that is, the occurrence of a given geometric isomerization in a short lifetime is less frequent than expected by an exponential distribution based on the mixing in dynamics. This behavior is exactly the reverse to those observed in the simpler system such as dissociation reaction of H3+ [M. Berblinger and C. Schlier, J. Chem. Phys. 101, 4750 (1994)], in which many of the so-called direct paths are ejected before the mixing takes effect and thereby the short lifetime isomerization (or dissociation) occurs more frequently than the exponential distribution. The former fact implies that the classical trajectories take somewhat longer time (induction time) to find their ways out to the other isomers in phase space, and therefore it can be a prototype of the so-called slow dynamics that is frequently observed in large and complicated molecular systems. The present paper discusses a possible mechanism to describe the present induction phenomenon. We first show a numerical fact that an ensemble of trajectories turns into a stage very quickly that can be regarded as a diffusion process getting out of a potential basin, if projected onto a one-dimensional configuration space. Thus, a natural idea arises that the induction time should be a consequence for the group of trajectories to be transported to the reaction regions, or transition regions, with a limited speed. In contrast, the standard statistical theories assume that the population in a transition region that is lost to the product side is to be supplied instantaneously from the reactant region. We present a simple diffusion model to examine the above idea. It has been found that the frequencies of isomerization can be reproduced in a good quantitative level by the estimate in terms of the first passage time based on the calculated diffusion coefficients and related quantities. The remarkable uniformity of the average passage-times (lifetimes) that was previously found by us is also described well in this simple model. © 1998 American Institute of Physics.

Notes: We report a new kind of "dynamical tunneling" that can be observed in chaotic molecular vibration. The present phenomenon has been found in eigenfunctions quantized in a thin quasiseparatrix (chaotic zone) in phase space. On the classical Poincaré section corresponding to this situation, two or more unstable (hyperbolic) fixed points coexist and are connected through the so-called heteroclinic crossings, whereby the entire quasiseparatrix is generated. When the quasiseparatrix is thin enough, each of the hyperbolic fixed points is surrounded by the relatively "wide lake" of chaos due to the infinite and violent crossings between the stable and unstable manifolds, and these lakes are in turn connected by "narrow canals." Our finding is, in spite of the fact that the narrow canals are classically allowed for the trajectories to pass through fast, wave packets can be quantized predominantly as "quasistanding-waves" in each lake area and hence can be mostly localized to remain there for much longer time than the corresponding classical trajectories do. In other words, the wave packets are localized in the vicinity of the classically unstable fixed points due to the quantum effect. However, a pair of these "localized" wave packets are eventually delocalized into the other lakes, and thereby form a pair of eigenfunctions (purely standing waves) with a small level splitting. Thus the present phenomenon can be characterized as a tunneling between the states of quantum localization in an oscillator problem. © 1998 American Institute of Physics.

Notes: The effects of multidimensionality in the quantum mechanical tunneling of chemical reactions are investigated. The aim of the present report is twofold. In the first place, we construct a new semiclassical theory to describe the tunneling by incorporating nonclassical solutions of the time-dependent Hamilton–Jacobi equation into the Feynman kernel. A systematic class of complex-valued (nonclassical) solutions for the time-independent Hamilton–Jacobi equation has been found that are generated along non-Newtonian paths in real-valued configuration space [K. Takatsuka and H. Ushiyama, Phys. Rev. A 51, 4353 (1995)]. In the present paper, the straightforward extension is applied to the time-dependent Hamilton–Jacobi equation, the solutions of which describe the tunneling in chemical reactions. It is shown that no damping factor due to the tunneling arises from the preexponential factor in the thus obtained nonclassical kernel, since it is still real valued, aside from the complex phase due to the Maslov index, and moreover its functional form is essentially the same as in the nontunneling case. Thus only the imaginary part of the action integral is responsible for the damping. A quasiclassical treatment of the semiclassical mechanics is developed to characterize the real-valued tunneling paths. In the second-half of this paper, some typical tunneling reactions in collinear three atomic systems on the LEPS (London–Eyring–Polanyi–Sato) potential surface are investigated in terms of our semiclassical theory. The effect of the initial energy distribution among the vibrational and translational modes is investigated asking which is preferable for tunneling and what is the resultant distribution of the energy in the product molecules. The following two factors to control the tunneling reactions are mainly examined as our first case study: (a) the mass effects featuring heavy–light–heavy and light–heavy–light patterns and (b) the anisotropy of the potential surface, namely, the early or late barrier. Tunneling paths of the types of Marcus–Coltrin and Miller–George are both generated spontaneously. A path of Marcus–Coltrin type takes a major role when the translational energy dominates in tunneling, while that of Miller–George type is dominant in a case where the vibrational excitation is important. As a distinguished feature of the multidimensionality in tunneling, we have identified what we call a tunneling tube, in which a bunch of the tunneling paths are involved emanating from the so-called caustic line. It turns out that the width of the tunneling tube determines in part the final energy distribution among the product vibrational modes. © 1997 American Institute of Physics.