We extend the Mixed Quantum-Classical Initial Value Representation (MQC-IVR), a semiclassical method for computing real-time correlation functions, to electronically nonadiabatic systems using the Meyer-MillerStock-Thoss (MMST) Hamiltonian to treat electronic and nuclear degrees of freedom (dofs) within a consistent dynamic framework. We introduce an efficient symplectic integration scheme, the MInt algorithm, for numerical time-evolution of the nuclear and electronic phase space variables as well as the Monodromy matrix, under the non-separable MMST Hamiltonian. We then calculate the probability of transmission through a curve-crossing in model two-level systems and show that in the quantum limit MQC-IVR is in good agreement with the exact quantum results, whereas in the classical limit the method yields results in keeping with mean-field approaches like the Linearized Semiclassical IVR. Finally, exploiting the ability of MQC-IVR to quantize different dofs to different extents, we present a detailed study of the extents to which quantizing the nuclear and electronic dofs improves numerical convergence properties without significant loss of accuracy.

The general approach to classical unimolecular reaction rates due to Thiele is revisited in light of recent advances in the phase space formulation of transition state theory for multidimensional systems. Key concepts, such as the phase space dividing surface separating reactants from products, the average gap time, and the volume of phase space associated with reactive trajectories, are both rigorously defined and readily computed within the phase space approach. We analyze in detail the gap time distribution and associated reactant lifetime distribution for the isomerization reaction HCN CNH, previously studied using the methods of phase space transition state theory. Both algebraic ͑power law͒ and exponential decay regimes have been identified. Statistical estimates of the isomerization rate are compared with the numerically determined decay rate. Correcting the RRKM estimate to account for the measure of the reactant phase space region occupied by trapped trajectories results in a drastic overestimate of the isomerization rate. Compensating but as yet not fully understood trapping mechanisms in the reactant region serve to slow the escape rate sufficiently that the uncorrected RRKM estimate turns out to be reasonably accurate, at least at the particular energy studied. Examination of the decay properties of subensembles of trajectories that exit the HCN well through either of two available symmetry related product channels shows that the complete trajectory ensemble effectively attains the full symmetry of the system phase space on a short time scale t Շ 0.5 ps, after which the product branching ratio is 1:1, the "statistical" value. At intermediate times, this statistical product ratio is accompanied by nonexponential ͑nonstatistical͒ decay. We point out close parallels between the dynamical behavior inferred from the gap time distribution for HCN and nonstatistical behavior recently identified in reactions of some organic molecules.

Abstract. We analyse the classical dynamics o f near-collinear electron configurations in helium. The dynamics turns out to be fully chaotic. An application o f periodic orbit quantization techniques yields the energy of doubly excited states with high accuracy. The analysis shows that near-collinear intra-shell resonances are associated with an asymmetric stretch like molion of the electron pair rather than the symmetric stretch motion along the Wannier ridge.The failure of the Copenhagen School (see e.g. van Vleck 1922) to obtain a reasonable estimate of the ground-state energy of the helium atom was a cornerstone in the evolution of quantum mechanics. Nowadays we know the essential shortcomings of the old quantum theory (Leopold and Percival 1980, section 3.4 in Gutzwiller 1990) : (i) the role of conjugate points along classical trajectories and their importance for the approach to wave mechanics were not properly accounted for;(ii) the precise role of periodic trajectories when the classical dynamics are nonintegrable or even chaotic was unknown.Even though it is widely known that the old quantum theory failed in the early days of quantum mechanics, it is less well known that the principal obstacles to the determination of the ground-state energy of the helium atom were overcome about a decade ago (Leopold and Percival 1980). Nevertheless, a proper semiclassical treatment of the helium atom is still an outstanding problem of the basic theory (as is a proper quantum description). The helium atom therefore remains the essential touchstone of semiclassical mechanics, even though considerable progress has been achieved very recently in applications for other chaotic atomic systems (Friedrich and Wintgen 1989, Cvitanovii: and Eckhardt 1989, Tanner et a/ 1991).The purpose of this letter is threefold. We first analyse the classical dynamics for near-collinear arrangements of the two electrons in the helium atom. We find strong evidence that the resulting motion is fully chaotic in the corresponding symmetry plane whereas the linearized motion o f the plane is stable. We then apply modern semiclassical techniques to quantize the chaotic dynamics and obtain the energies of certain doubly excited states. Finally, our results (together with numerically highly accurate quantum mechanical calculations) unambiguously show that the widely accepted viewpoint of electron pair propagation along the Wannier ridge for doubly excited

The method of algebraic quantization, a semiclassical analog of Van Vleck perturbation theory, is applied to multidimensional resonant, nonresonant, and nearly resonant systems. perturb, a special purpose program written in C, is utilized to implement classical perturbation theory efficiently to high order. States corresponding to both regular and chaotic classical regimes are quantized, and accurate eigenvalues obtained in both cases. Various quantization rules are compared, and a novel symmetry preserving rule is given which leads to good agreement with quantum mechanics. The method is able to reproduce purely quantum mechanical splittings to very good accuracy. Algebraic quantization combined with Padé resummation is used to determine energy eigenvalues for a resonant system with five degrees of freedom.

In this review we discuss the recently described roaming mechanism for chemical reactions from the point of view of nonlinear dynamical systems in phase space. The recognition of the roaming phenomenon shows the need for further developments in our fundamental understanding of basic reaction dynamics, as is made clear by considering some questions that cut across most studies of roaming: Is the dynamics statistical? Can transition state theory be applied to estimate roaming reaction rates? What role do saddle points on the potential energy surface play in explaining the behavior of roaming trajectories? How do we construct a dividing surface that is appropriate for describing the transformation from reactants to products for roaming trajectories? How should we define the roaming region? We show that the phase space perspective on reaction dynamics provides the setting in which these questions can be properly framed and answered. We illustrate these ideas by considering photodissociation of formaldehyde. The phase-space formulation allows an unambiguous description of all possible reactive events, which also allows us to uncover the phase space mechanism that explains which trajectories roam, as opposed to evolving toward a different reactive event.

We explore both classical and quantum dynamics of a model potential exhibiting a caldera: that is, a shallow potential well with two pairs of symmetry related index one saddles associated with entrance/exit channels. Classical trajectory simulations at several different energies confirm the existence of the "dynamical matching" phenomenon originally proposed by Carpenter, where the momentum direction associated with an incoming trajectory initiated at a high energy saddle point determines to a considerable extent the outcome of the reaction (passage through the diametrically opposing exit channel). By studying a "stretched" version of the caldera model, we have uncovered a generalized dynamical matching: bundles of trajectories can reflect off a hard potential wall so as to end up exiting predominantly through the transition state opposite the reflection point. We also investigate the effects of dissipation on the classical dynamics. In addition to classical trajectory studies, we examine the dynamics of quantum wave packets on the caldera potential (stretched and unstretched). These computations reveal a quantum mechanical analogue of the "dynamical matching" phenomenon, where the initial expectation value of the momentum direction for the wave packet determines the exit channel through which most of the probability density passes to product.

We study reaction dynamics on a model potential energy surface exhibiting post-transition state bifurcation in the vicinity of a valley ridge inflection (VRI) point. We compute fractional yields of products reached after the VRI region is traversed, both with and without dissipation. It is found that apparently minor variations in the potential lead to significant changes in the reaction dynamics. Moreover, when dissipative effects are incorporated, the product ratio depends in a complicated and highly non-monotonic fashion on the dissipation parameter. Dynamics in the vicinity of the VRI point itself play essentially no role in determining the product ratio, except in the highly dissipative regime.

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