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The reactive island (RI) theory of chemical reaction rates is modified so that a precise knowledge of the transition state is not required. This revised form of the theory is applied to a realistic two (J=0) and three degree of freedom model (J≠0) of the isomerization HCN⇄CNH. The detailed reaction dynamics is examined and understood in terms of the cylindrical manifolds embedded in the molecular phase space. Of special interest we find a T-shaped conformer in addition to the HCN and CNH moieties at J=0. Rates of unimolecular decay are examined by numerical simulation and theory. Good agreement is found between theory and simulation except for a case where a substantial amount of phase space is consumed by regular motion. In all cases studied, the theory is a significant improvement over the Rice–Ramsperger–Kassel–Marcus (RRKM) theory, which is up to three orders of magnitude too large. [ABSTRACT FROM AUTHOR]

In a series of recent publications we discussed the concept of cylindrical manifolds and their role in mediating the reaction dynamics of chemical reactions. The cylindrical manifolds were used to develop a chemical reaction rate theory we called reactive island (RI) theory. RI theory was cast in terms of the map dynamics between n Poincaré mapping planes—which were referred to as the ‘‘n-map.’’ Therefore ‘‘time’’ did not explicitly appear within n-map RI theory. In this paper we extend n-map RI theory to the time domain. The formal theory, cast as a master equation, is used to obtain the temporal RI model. Temporal RI theory is applied to a two degree-of-freedom system exhibiting dynamical chaos. The results of temporal RI theory are compared with classical Rice–Ramsberger–Kassel–Marcus (RRKM) theory and it is found that even under, presumably, ‘‘ideal’’ dynamical conditions, RRKM theory can be in serious error with numerical simulation. It is also seen that temporal RI theory accurately accounts for the rates at energies far above the barrier, where RRKM theory is not expected to be applicable. We also discuss some rigorous comparisons between the decay rates obtained from n-map and temporal RI theories. [ABSTRACT FROM AUTHOR]

In Paper I we discussed the existence of cylindrical manifolds embedded in phase space which mediate the dynamics of chemical reactions. A kinetic theory of population decays and decay rate constants was developed which we called ‘‘reactive island’’ (RI) theory. In this paper we discuss the details of the numerical implementation of the theory and then apply it to several molecular models (with two coupled degrees of freedom) representing isomerization between two and three states. Numerical simulations of population decays and asymptotic decay rate constants are compared to the RI theoretical predictions as well as the predictions from the Purely Random Theory (PRT) and Transition State Theory (TST) of reactions. Of the ten systems studied we find that RI theory is generally in good to excellent agreement with the numerical simulations. Only one system exhibits significant deviation between the RI and numerical results. This deviation is seen to be a result of a strong intraconformer dynamical bottleneck. Finally, we compare the theoretical prediction and the numerical simulation for the average n-map mapping time Trxn and find that the agreement, within numerical error, is exact irrespective of the character of the dynamics (i.e., chaotic or regular). [ABSTRACT FROM AUTHOR]

A microcanonical kinetic theory of reactions based upon the structure within phase space is developed. It is shown that the dynamics of reaction across an energetic barrier is mediated by invariant manifolds embedded in phase space that have the geometry of simple cylinders. The ideas are developed by considering molecular systems modeled by two vibrational degrees of freedom, a reaction coordinate and a ‘‘bath’’ coordinate. The kinetic theory is constructed by focusing on the dynamics between n mapping planes (‘‘n-map’’) and the ‘‘reactive island’’ (RI) structure within them. We discuss how the structure of the conformer population decay in isomerization reactions can be obtained from the RI kinetic model. Formal solutions of the kinetic equations are discussed with specific attention given towards the calculation of the isomerization reaction rate. The formal theory is developed in Paper I of this series. Numerical considerations and applications to the reaction dynamics of model molecular systems with two degrees of freedom will be given in Paper II and extension of the theory and applications to multidimensional systems will be given in Paper III. [ABSTRACT FROM AUTHOR]

A projection operator formalism is used to quantify the normal/resonant character of eigenstates of a vibrational Hamiltonian with two degrees of freedom. We show how the application of the formalism quantifies the resonant character of these eigenstates, in the form of a probability, and how this probability is related to the width of the resonance in classical phase space. Normal and resonant quantum numbers shared by a given eigenstate are found to be canonically related. [ABSTRACT FROM AUTHOR]

In this paper we focus on the detailed nonlinear classical dynamics of conformational isomerization. In particular we concentrate on systems which admit phase space structures we call ‘‘reactive islands.’’ Our calculations are on a two degree of freedom model of the molecule 3-phospholene with an experimentally fit potential energy surface by Harthcock and Laane. The reactive islands (RIS) are embedded within and are part of chaotic regions of phase space. We find that the RIS are constructed from a linear stability analysis of the period 1 orbit at the transition state or approximated by a similar analysis on reactive periodic orbits. The two approaches converge as the order of the reactive periodic orbit increases. It is found that the fully constructed RIS have well defined regions of reactivity and thus mediate the process of conformational isomerization.The overlap areas of the RIS give important kinetic information such as probabilities for trapped to reactive motion, reactive to trapped motion, and primary back reaction. The implications of this work lie in precisely identifying the microscopic dynamical pathway for isomerization. The results of this work and its relation to the related work of Gray and Rice as well as its relation to contemporary work in modern dynamics is also discussed. [ABSTRACT FROM AUTHOR]

A high degree of structure and therefore order in chaos is found to exist in the detailed dynamical pathways to conformational isomerization. It is shown that this structure can be used to determine the probabilities associated with the dynamical pathways to reaction, trapping, and back reaction. An earlier publication described the mediation of the dynamics of 3-phospholene by phase space structures we called ‘‘reactive islands’’ (RIS)21. In this paper we extend the physical and mathematical properties of RIS and develop the corresponding kinetic theory. RIS theory is applied to a model of a hindered rotor and 3-phospholene. It is shown that the RIS kinetic model accurately predicts trajectory simulations of conformer population decay. Comparisons with standard RRKM theory are included. A discussion on the extension of RIS theory to quantum reactive dynamics and its relevance to laboratory experiments is also included. [ABSTRACT FROM AUTHOR]

In this paper a semiclassical quantization method is developed whose formulation and application is simple, general and accurate. We call this method semiclassical quantization by circuit counting (SQCC). SQCC is based upon evaluating the action integral along a trajectory until it undergoes a near recurrence. Several near recurrences are used to obtain a set of independent equations which can then be inverted to yield the actions for the trajectory. Once the actions are obtained a linear extrapolation is used to calculate semiclassical energies. We apply the method to two model systems: (1) three uncoupled Morse oscillators and (2) a realistic three degree of freedom model of SO2. SQCC is found to work very well for these systems. We discuss some of the advantages and disadvantages of SQCC and its application to systems with more than three coupled molecular degrees of freedom. [ABSTRACT FROM AUTHOR]

In this paper we extend the arbitrary trajectory approach to semiclassical quantization developed in an earlier paper. A numerical method is developed and tested that uses a second-order correction to extrapolate several energy eigenvalues from a single trajectory. The method is based upon finding the second-order action derivatives of the Hamiltonian. We test the scheme on a Hamiltonian previously analyzed by Swimm and Delos. We find that about 100 energy eigenvalues can be fairly accurately obtained with three reference trajectories (24 total trajectories). Strengths, weaknesses, and future applications of the method are discussed. [ABSTRACT FROM AUTHOR]

Classical resonances arising from the interaction of three nonlinearly coupled oscillators are studied from both a theoretical and numerical perspective. In particular, our study focuses on ternary classical resonances defined by n1ω1 +n2ω2 -n3ω3 =0. We discuss some of the experimental and quantum mechanical consequences of binary and ternary classical resonances (e.g., Fermi resonances and vibration–rotation coupling). Numerically we show that it is possible to construct a three-dimensional map such that ternary classical resonances can be systematically found. Theoretically, we show that canonical transformations exist between resonant and nonresonant motion. These transformations predict various structural features of the three-dimensional numerical maps which are subsequently observed in a model numerical calculation. Finally we argue that the methods and ideas presented in this paper are generic and can be used for more general systems. [ABSTRACT FROM AUTHOR]

We examine the classical limit (h→0) of the semiclassical spectral quantization method (SQ) proposed in an earlier publication. Using the classical limit analysis, equations are derived which should improve the accuracy of SQ for finite h. Our analysis is focused on problems involving coupled and uncoupled oscillator systems. Special attention is given to the situation when the unperturbed oscillators are anharmonic. Application to cases where classical resonances are present are also discussed. [ABSTRACT FROM AUTHOR]

Presents a correction to the article 'Cylindrical Manifolds and Reactive Island Kinetic Theory in the Time Domain.'