Dimensionality Hamiltonian dynamics

Using this transformation, it has been shown in Refs. [54,72] that the effective-mode Hamiltonian Heg by itself reproduces the short-time dynamics of the overall system exactly. This is reflected by an expansion of the propagator, for which it can be shown that the first few terms of the expansion - relating to the first three moments of the overall Hamiltonian - are exactly reproduced by the reduced-dimensional Hamiltonian Heg. 

If the phonon distribution of the model Eq. (8) spans a dense spectrum - as is generally the case for the extended systems under consideration, which are effectively infinite-dimensional - the dynamics induced by the Hamiltonian will eventually exhibit a dissipative character. However, the effective-mode construction demonstrates that the shortest time scales are fully determined by few effective modes, and by the coherent dynamics induced by these modes. The overall picture thus corresponds to a Brownian oscillator type dynamics, and is markedly non-Markovian [81,82],... 

In order to understand the problem of finding TS with three or more DOFs, it is useful to address the question of dimensionalities, in configuration and phase space. In classical, Hamiltonian dynamics, transition states are grounded on the idea that certain surfaces (more precisely, certain manifolds) act as barriers in phase space. It is possible to devise barriers in phase space, since in phase space, in contrast to configuration space, two trajectories never cross [uniqueness of solutions of ODEs, see Eq. (4)]. In order to construct a barrier in phase space, the first step is to construct a manifold if that is made of a set of trajectories [8]. 

We are beginning to understand the real dynamics of global diffusion in the phase space of many-dimensional Hamiltonian systems. From here we are going to travel around the vast world created by the chaotic dynamics of nonlinear systems. 

Anomalous diffusion was first investigated in a one-dimensional chaotic map to describe enhanced diffusion in Josephson junctions [21], and it is observed in many systems both numerically [16,18,22-24] and experimentally [25], Anomalous diffusion is also observed in Hamiltonian dynamical systems. It is explained as due to power-type distribution functions [22,26,27] of trapping and untrapping times of the orbit in the self-similar hierarchy of cylindrical cantori [28]. 

The dynamical system thus reduces to a one-dimensional Hamiltonian one with a... 

In the recent past, analytical research in Celestial Mechanics has centred on KAM theory and its applications to the dynamics of low dimensional Hamiltonian systems. Results were used to interpret observed solutions to three body problems. Order was expected and chaos or disorder the exception. Researchers turned to the curious exception, designing analytical models to study the chaotic behaviour at resonances and the effects of resonant overlaps. Numerical simulations were completed with ever longer integration times, in attempts to explore the manifestations of chaos. These methods improved our understanding but left much unexplained phenomena. 

To reiterate, we prefer to describe the one-dimensional model first because of its mathematical simplicity in comparison to the three-dimensional model. From a strictly historical point of view, the situation is slightly more involved. The vibron model was officially introduced in 1981 by lachello [26]. In his work one can find the fundamental idea of the dynamical symmetry, based on U(4), for realizing an algebraic version of the three-dimensional Hamiltonian operator of a single diatomic molecule. After this work, many other realizations followed (see specific... 

In this paper, we discuss semi-implicit/implicit integration methods for highly oscillatory Hamiltonian systems. Such systems arise, for example, in molecular dynamics [1] and in the finite dimensional truncation of Hamiltonian partial differential equations. Classical discretization methods, such as the Verlet method [19], require step-sizes k smaller than the period e of the fast oscillations. Then these methods find pointwise accurate approximate solutions. But the time-step restriction implies an enormous computational burden. Furthermore, in many cases the high-frequency responses are of little or no interest. Consequently, various researchers have considered the use of scini-implicit/implicit methods, e.g. [6, 11, 9, 16, 18, 12, 13, 8, 17, 3]. 

Most microscopic theories of adsorption and desorption are based on the lattice gas model. One assumes that the surface of a sohd can be divided into two-dimensional cells, labelled i, for which one introduces microscopic variables Hi = 1 or 0, depending on whether cell i is occupied by an adsorbed gas particle or not. (The connection with magnetic systems is made by a transformation to spin variables cr, = 2n, — 1.) In its simplest form a lattice gas model is restricted to the submonolayer regime and to gas-solid systems in which the surface structure and the adsorption sites do not change as a function of coverage. To introduce the dynamics of the system one writes down a model Hamiltonian which, for the simplest system of a one-component adsorbate with one adsorption site per unit cell, is... 

We recall, from elementary classical mechanics, that symmetry properties of the Lagrangian (or Hamiltonian) generally imply the existence of conserved quantities. If the Lagrangian is invariant under time displacement, for example, then the energy is conserved similarly, translation invariance implies momentum conservation. More generally, Noether s Theorem states that for each continuous N-dimensional group of transformations that commutes with the dynamics, there exist N conserved quantities. 

The standard effective spectroscopic Fermi resonant Hamiltonian allows more complicated types of behavior. The full three-dimensional aspects of the monodromy remain to be worked out, but it was shown, with the help of the Xiao—KeUman [28, 29] catastrophe map, that four main dynamical regimes apply, and that successive polyads of a given molecule may pass from one regime to another. 

The point q = p = 0 (or P = Q = 0) is a fixed point of the dynamics in the reactive mode. In the full-dimensional dynamics, it corresponds to all trajectories in which only the motion in the bath modes is excited. These trajectories are characterized by the property that they remain confined to the neighborhood of the saddle point for all time. They correspond to a bound state in the continuum, and thus to the transition state in the sense of Ref. 20. Because it is described by the two independent conditions q = 0 and p = 0, the set of all initial conditions that give rise to trajectories in the transition state forms a manifold of dimension 2/V — 2 in the full 2/V-dimensional phase space. It is called the central manifold of the saddle point. The central manifold is subdivided into level sets of the Hamiltonian in Eq. (5), each of which has dimension 2N — 1. These energy shells are normally hyperbolic invariant manifolds (NHIM) of the dynamical system [88]. Following Ref. 34, we use the term NHIM to refer to these objects. In the special case of the two-dimensional system, every NHIM has dimension one. It reduces to a periodic orbit and reproduces the well-known PODS [20-22]. 

This is exactly the autonomous linearized Hamiltonian (7), the dynamics of which was discussed in detail in Section II. One therefore finds the TS dividing surface and the full set of invariant manifolds described earlier one-dimensional stable and unstable manifolds corresponding to the dynamics of the variables A<2i and APt, respectively, and a central manifold of dimension 2N — 2 that itself decomposes into two-dimensional invariant subspaces spanned by APj and AQj. However, all these manifolds are now moving manifolds that are attached to the TS trajectory. Their actual location in phase space at any given time is obtained from their description in terms of relative coordinates by the time-dependent shift of origin, Eq. (42). 

A more general description of the effects of vibronic coupling can be made using the model Hamiltonian developed by Koppel, Domcke and Cederbaum [65], The basic idea is the same as that used in Section III.C, that is to assume a quasidiabatic representation, and to develop a Hamiltonian in this picture. It is a useful model, providing a simple yet accurate analytical expression for the coupled PES manifold, and identifying the modes essential for the non-adiabatic effects. As a result it can be used for comparing how well different dynamics methods perform for non-adiabatic systems. It has, for example, been used to perform benchmark full-dimensional (24-mode) quantum dynamics calculations... 

The most general Hamiltonian with dynamic symmetry (II) again has a form similar to Eq. (2.43), with both linear and quadratic terms. This is a peculiar feature of one-dimensional problems. In order to simplify the discussion of three-dimensional problems, we prefer to consider a Hamiltonian with only quadratic terms... 

A general potential V(r) corresponds to a generic algebraic Hamiltonian (2.29). In the most general case the solution cannot be obtained in explicit form but requires the diagonalization of a matrix. The matrix is (N + 1) dimensional. An alternative approach, useful in the case in which the potential does not deviate too much from a case with dynamical symmetry, is to expand it in terms of the limiting potential. For the Morse potential, this implies an expansion of the type (1.7)... 

The eigenvalue problem for the Hamiltonian H [Eq. (2.92)] can be solved in closed form whenever H does not contain all the elements but only a subset of them, the invariant or Casimir operators. For three-dimensional problems there are two such situations corresponding to the two chains discussed in the preceding sections. We begin with chain (I). Restricting oneself only to terms up to quadratic in the elements of the algebra, one can write the most general Hamiltonian with dynamic symmetry (I) as... 

In this section, we investigate the dynamics of entanglement in one-dimensional spin systems with a time-dependent magnetic field. The Hamiltonian for such a system is given by [98]... 

In this contribution we extend our previous studies on the HB spectroscopy and dynamics of salicylaldimine (SA, see Fig. 1) focussing on the influence of isotopic H/D substitution on the IVR dynamics. In the following Section 2 the CRS Hamiltonian is briefly introduced. In Section 3 numerical results of a seven-dimensional (7D) wave packet propagation are discussed. 

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